%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Birman, Joan S.
%%
%% New points of view in knot theory
%%
%% publ: Bull. Amer. Math. Soc. (N.S.) 28(1993) no. 2
%% pp: 253-287
%% type: Research-Expository Paper markup: amstex file size: 126K
%% contact:jb@math.columbia.edu
%%
%% copyright: American Math. Society copyright; see end of article
%%
%% Include files necessary for this article: bull-ppt.tex
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Date: 11-DEC-1992
% : 0 \newpage: 0 \displaybreak: 0
% \eject: 0 \bye: 0 \break: 0 \allowbreak: 0
% \allowdisplaybreak: 0 \allowdisplaybreaks: 0
% \allowlinebreak: 0 \allowmathbreak: 0
% \smallpagebreak: 0 \medpagebreak: 0 \bigpagebreak: 0
% \smallbreak: 0 \medbreak: 0 \bigbreak: 0
%\goodbreak: 0
% : 0 : 0 \newline: 0
% \magnification: 0 \mag: 0
% \baselineskip: 0 \normalbaselineskip: 0
% \hsize: 1 \vsize: 0 \pagewidth: 0 \pageheight: 0
% \hoffset: 0 \voffset: 0 \hcorrection: 0
%\vcorrection: 0
% \parindent: 0 \parskip: 0
% \vfil: 0 \vfill: 0 \vskip: 0
% \smallskip: 1 \medskip: 0 \bigskip: 1
% \sl: 0 \def: 4 \let: 0 \redefine: 0
%\predefine: 0
% \tolerance: 0 \pretolerance: 0
% \font: 0 \end: 0 \noindent: 2
% ASCII 13 (Control-M Carriage return): 0
% ASCII 10 (Control-J Linefeed): 0
% ASCII 12 (Control-L Formfeed): 0
% ASCII 0 (Control-@): 0
%
% File `article.tex' (or whatever you want to name it).
%\input bulplain.tex % some setup for title, author,
%section heads, refs.
\input amstex
\documentstyle{amsppt}
\input bull-ppt
\keyedby{bull389e/mhm}
\define\np{\noindent}
\define\ID{\operatorname{id}}
\define\GL{\operatorname{GL}}
\define\TR{\operatorname{tr}}
\define \BK{\bold{K}}
% Figure caption.
%\def\caption#1{\bigskip
% \setbox0\hbox\bgroup{\bf #1\unskip}\hbox{.}\enspace}
%\def\endcaption{\egroup
% \ifdim\wd0 >\hsize
% \noindent\unhbox0 \par
% \else
% \centerline{\unhbox0 \unkern\setbox0\lastbox}%
% \fi}
\topmatter
\cvol{28}
\cvolyear{1993}
\cmonth{April}
\cyear{1993}
\cvolno{2}
\cpgs{253-287}
\title
New points of view in knot theory
\endtitle
\author
Joan S. Birman
\endauthor
\address
Department of Mathematics,
Columbia University,
New York, New York 10027\endaddress
\ml jb\@math.columbia.edu\endml
\subjclass Primary 57M25; Secondary 57N99\endsubjclass
\keywords Knots, links, knot polynomials, knot groups,
Vassiliev
invariants, R-matrices, quantum groups
\endkeywords
\thanks
This research was supported in part by NSF Grant
DMS-91-06584 and by the
US-Israel Binational Science
Foundation Grant 89-00302-2\endthanks
\thanks This manuscript is an expanded version of an
AMS-MAA Invited Address,
given on January 8, 1992, at the Joint Winter Meeting of
the AMS and the MAA in
Baltimore, Maryland\endthanks
%\shortauthor{}
%\shorttitle{}
\date September 30, 1992\enddate
\endtopmatter
\document
\heading Introduction\endheading
In this article we shall give an account of certain
developments in knot
theory which followed upon the discovery of the Jones
polynomial \cite {Jo3} in
1984. The focus of our account will be recent glimmerings
of understanding of
the topological meaning of the new invariants. A second
theme will be the
central role that braid theory has played in the subject.
A third will be the
unifying principles provided by representations of simple
Lie algebras and
their universal enveloping algebras. These choices in
emphasis are our own.
They represent, at best, particular aspects of the
far-reaching ramifications
that followed the discovery of the Jones polynomial.
We begin in \S 1 by discussing the topological
underpinnings of that
most famous of the classical knot invariants---the
Alexander polynomial.
It will serve as a model for the sort of thing one would
like to see for the
Jones polynomial. Alexander's 1928 paper ends with a hint of
things to come, in the form of a crossing-change formula
for his
polynomial, and in \S 2 we discuss how related formulas
made their
appearances in connection with the Jones polynomial and
eventually
led to the discovery of other, more general knot and link
polynomials.
A more systematic description
of these ``generalized Jones invariants" is given in \S 3,
via the
theory of R-matrices. That is where braids enter the
picture, because every generalized Jones invariant is
obtained from a
trace function on an ``R-matrix representation" of the
family of
braid groups $\lbrace B_n; n = 1,2,3,\dots\rbrace.$ The
mechanism for
finding R-matrix representations of $B_n$ is the
theory of quantum groups. For this reason,
the collection of knot and link invariants that they
determine have been called
{\it quantum group invariants \/}. We shall refer to them
here
as either quantum group invariants or {\it generalized
Jones invariants \/},
interchangeably. While the theory of R-matrices and their
construction via
quantum groups gives a coherent and uniform description
of the entire class of
invariants, the underlying ideas will be seen to be
essentially
combinatorial in nature. Thus, by the end of \S 3 the
reader should begin to
understand how it could happen that in 1990 topologists
had a fairly coherent
framework for constructing vast new families of knot and
link invariants,
possibly even enough to classify unoriented knot and link
types, without having
the slightest clue to the underlying topology.
In \S 4 we introduce an entirely new collection of
invariants, which arose out
of techniques pioneered by Arnold in singularity theory
(see the introduction
to \cite {Arn1, Arn2}). The new invariants will be seen to
have a solid basis
in a very interesting new topology, where one studies not
a {\it single \/}
knot, but a space of {\it all \/} knots. This point of
view was carried out
successfully for the case of knots by Vassiliev \cite
{V}. ({\it Remark\/}.
For simplicity, we restrict our attention in this part of
the review to knots.
The theory is, at this writing, less well developed for
links.) Vassiliev's
knot invariants are rational numbers. They lie in vector
spaces $V_i$ of
dimension $d_i, i=1,2,3,\dots$, with invariants in $V_i$
having ``order" $i$.
On the other hand, quantum group invariants are Laurent
polynomials over
$\roman {Z}$, in a single variable $q$. The relationship
between them is as
follows:
\proclaim{Theorem 1} Let ${\Cal J}_q(\BK)$ be a quantum
group
invariant for a knot type $\BK$. Let
$${\Cal P}_x(\BK) = \sum_{i=0}^{\inf} u_i(\BK)x^i$$
be the power series over the rational numbers $Q$ obtained
from ${\Cal J}_q(\BK)$ by setting $q=e^x$ and expanding
the powers of $e^x$
in their Taylor series. Then the coefficient $u_i(\BK)$ of
$x^i$ is $1$ if
$i=0$ and a Vassiliev invariant of order $i$ for each $i
\geq 1$.
\endproclaim
\np Thus Vassiliev's topology of the space of all knots
suggests the topological
underpinnings we seek for the quantum group knot invariants.
A crucial idea in the statement of Theorem 1 is that one
must pass to the power
series representation of the knot polynomials before one
can understand the
situation. This was first explained to the author and Lin
by Bar Natan. Theorem
1 was first proved for special cases in \cite {BL} and
then generalized in
\cite {Li1}. In \S\S5--7 we describe a set of ideas which
will be seen to lead
to a new and very simple proof of Theorem 1. First, in \S
5, we review how
Vassiliev's invariants, like the Jones polynomial, the
HOMFLY and Kauffman
polynomials \cite {FHL, Kau1}, and the $\bold {G}_2$
polynomial \cite {Ku} can
be described by axioms and initial data. (Actually, all
of the quantum group
invariants admit such a description, but the axioms can
be very complicated;
so such a description would probably not be
enlightening). In \S 6 we
introduce braids into the Vassiliev setting, via a new
type of object, the
monoid of singular braids. Remarkably, this monoid will be
seen to map
homomorphically (we conjecture isomorphically) into the
group algebra of the
braid group, implying that it is as fundamental an object
in mathematics as the
braid group itself. This allows us to extend every
R-matrix representation of
the braid group to the singular braid monoid. In \S7 we
use R-matrices and
singular braids to prove Theorem 1. In \S8 we return to
the topology,
discussing the beginning of a topological understanding of
the quantum group
invariants. We then discuss, briefly, a central problem:
do we now know enough
to classify knots via their algebraic invariants? We will
describe some of the
evidence which allows us to sharpen that question.
Our goal, throughout this review, is to present the material
in the most transparent and nontechnical manner possible
in order to help
readers who work in other areas to learn as much as
possible about
the state of the art in knot theory. Thus, when we give
``proofs", they
will be, at best, sketches of proofs. We hope there will
be enough detail
so that, say, a diligent graduate student who is motivated
to
read a little beyond this paper will be able to fill in
the gaps.
Among the many topics which we decided deliberately to
{\it exclude\/} from
this review for reasons of space, one stands out: it
concerns the
generalizations of the quantum group invariants and
Vassiliev invariants to
knots and links in arbitrary 3-manifolds, i.e., the
program set forth by Witten
in \cite {Wi}. That very general program is inherently
more difficult than the
special case of knots and links or simply of knots in the
3-sphere. It is an
active area of research, with new discoveries made every
day. We thought, at
first, to discuss, very briefly, the 3-manifold invariants
of Reshetikhin and
Turaev \cite {RT} and the detailed working out of special
cases of those
invariants by Kirby and Melvin \cite {KM}; however, we
then realized that we
could not include such a discussion and ignore Jeffrey's
formulas for the
Witten invariants of the lens spaces \cite {Je}.
Reluctantly, we
made the decision to restrict our attention to knots in
3-space, but still, we
have given at best a restricted picture. For example, we
could not do justice
to the topological constructions in \cite {La} and in
\cite {Koh1, Koh2}
without making this review much longer than we wanted it
to be, even though it
seems very likely that those constructions are closely
related to the central
theme of this review. We regret those and other omissions.
\heading 1. An introduction to knots and their Alexander
invariants
\endheading
We will regard a knot
$K$ as the embedded image of an oriented circle
$ S^1$ in oriented 3-space
$\Bbb{R}^3$ or $S^3$.
If, instead, we begin with $\mu \geq 2 $ copies of $S^1$,
the image (still called $K$), is a {\it link \/}.
Its {\it type \/} $\BK$ is the
topological type of the pair $(S^3,K)$,
under homeomorphisms which preserve
orientations on both $S^3$ and $K$. Knot types do not
change if we replace $S^3$ by
$\Bbb{R}^3,$ because every homeomorphism of $S^3$ is
isotopic to one which fixes one point,
and that point may be chosen to be the point at infinity.
We may visualize a knot via a {\it diagram \/}, i.e., a
projection of $K
\subset \Bbb{R}^3$ onto a generic $\Bbb{R}^2 \subset
\Bbb{R}^3$, where the
image is decorated to distinguish overpasses from
underpasses, for example, as
for highways on a map. Examples are given in Figure 1.
The example in Figure
1(a) is the unknot, represented by a round planar circle.
Figure 1(b) shows a
{\it layered \/} diagram of a knot, i.e., one which has
been drawn, using an
arbitrary but fixed starting point (which in Figure 1(c)
is the tip of the
arrow) without the use of an eraser, so that the first
passage across a double
point in the projection is always an overpass. We leave it
to the reader to
prove that a layered diagram with $\mu$ components always
represents $\mu$
unknotted, unlinked circles. From this simple fact it
follows that {\it any
\/} diagram of {\it any \/} link may be systematically
changed to a diagram for
the unlink on the same number of components by finitely
many crossing changes.
\midspace{23pc}
\caption{{\smc Figure 1}. {\rm Diagrams of the unknot.}}
The diagram in Figure 1(c) was chosen to illustrate the
subtleties of knot
diagrams. It too represents the unknot, but it is not
layered. This example was
constructed by Goeritz \cite {Go} in the mid-1930s. At
that time it was known
that a finite number of repetitions of Reidemeister's
three famous moves,
depicted (up to obvious symmetries and variations) in
Figure 2, suffice to
take any one diagram of a knot to any other. Notice that
Reidemeister's moves
are ``local" in the sense that they are restricted to
regions of the diagram
which contain at most three crossings. If one defines
the {\it complexity \/}
of a knot diagram to be its crossing number, a natural
question is whether
there is a set of moves that preserve or reduce complexity
and that, applied
repeatedly, suffice to reduce an arbitrary diagram of an
arbitrary knot to one
of minimum crossing number. The diagram in Figure 1(c)
effectively ended that
approach to the subject, because the eight moves which had
by then been
proposed as augmented Reidemeister moves did not suffice
to simplify this
diagram.
Some insight may be obtained into the question by
inspecting the ``handle move"
of Figure 3. Note that the crossing number in the diagram
of Figure 1(c) can be
reduced by an appropriate handle move. The region that is
labeled $X$ in Figure
3 is arbitrary. The handle move clearly decreases crossing
number, but a few
moments\vadjust{\fighere{12.5pc}\caption{{\smc Figure 2}.
{\rm Reidemeister moves.}}} of thought should convince
the reader that if one tries to factorize it
into a product of Reidemeister moves, for any sufficiently
complicated choice
of $X$, it will be necessary to use the second
Reidemeister move repeatedly in
order to create regions in which the third move may be
applied. The same sort
of reasoning makes it unlikely that any set of local moves
suffices to reduce
complexity. Indeed, if we knew such a set of moves, we
would have the beginning
of an algorithm for solving the knot problem, because
there are only finitely
many knot diagrams with fixed crossing number.
\topspace{6pc}
\caption{{\smc Figure 3}. {\rm A handle move.}}
In Figure 4 we have given additional examples, taken from
the beginning of a
table compiled at the end of the nineteenth century by the
physicist Peter
Guthrie Tait and coworkers, as part of a systematic effort
to classify knots.
The knots in the table are listed in order of their
crossing number, so that,
for example, $7_6$ is the 6th knot that was discovered
with 7 crossings. The
part of the table which we have shown includes all prime
knots with at most $7$
crossings, up to the symmetries defined by reversing the
orientation on either
$K$ or $S^3$. (We have shown the two trefoils, for reasons
which will become
clear shortly.) The tables, which eventually included all
prime knots defined
by diagrams with at most 13 crossings, were an ambitious
undertaking. (Aside:
Yes, it is true. Knots, like integers, have a
decomposition into an
appropriately defined product of prime knots, and this
decomposition is unique
up to order.) Their clearly stated goal \cite {Ta} was to
uncover the
underlying principles of knotting, but to the great
disappointment of all
concerned they did not even succeed in revealing a single
knot type invariant
which could be computed from a diagram. Their importance,
to this day, is due
to the fact that they provide a rich source of examples
and convincing evidence
of both the beauty and subtlety of the subject.
\topspace{37.5pc}
\caption{{\smc Figure} 4.}
Beneath each of the examples in Figure 4 we show two
famous invariants,
namely, the Alexander polynomial $A_q(\BK)$ and the
one-variable Jones
polynomial $J_q(\BK)$. Both are to be regarded as Laurent
polynomials in $q$,
the series of numbers representing the sequence of
coefficients, the bracketed
one being the constant term. Thus the lower sequence
(1,-1,1,-2, \lbrack
2\rbrack , -1,1) beneath the knot $6_1$ shows that its
Jones polynomial is
$q^{-4} - q^{-3} + q^{-2} - 2q^{-1} + 2 - q + q^2$. The
richness of structure
of both invariants is immediately clear from the sixteen
examples in our table.
There are no duplications (except for the Alexander
polynomials of the two
trefoils, which we put in deliberately to make a point).
The arrays of
subscripts and superscripts, as well as the roots (which
are not shown) and the
poles of the Jones invariant (about which almost nothing
is known), suggest
that the polynomials could encode interesting properties
of knots. Notice that
$A_q(\BK)$ is symmetric, i.e., $A_q(\BK) = \bold
{A}_{q^{-1}}(\BK)$, or,
equivalently, the array of coefficients is palendromic. On
the other hand,
$J_q(\BK)$ is not. Both polynomials take the value 1 at
$q=1$. ({\it
Remark\/}.
The Alexander polynomial is actually only determined up to
$\pm$multiplicative powers of $q$, and we have chosen to
normalize it to stress
the symmetry and so that its value at $1$ is $+1$ rather
than $-1$.)
The knots in our table are all invertible; i.e., there is
an isotopy of 3-space
which takes the oriented knot to itself with reversed
orientation. The first
knot in the tables that fails to have that property is
$8_{17}$. Neither the
Alexander nor the Jones polynomials changes when the
orientation on the knot is
changed. We shall have more to say about noninvertible
knots in \S 8.
While we understand the underlying topological meaning of
$A_q(\BK)$ very
well, we are only beginning to understand the topological
underpinnings of
$J_q(\BK)$. To begin to explain the first assertion, let
us go back to one of
the earliest problems in knot theory: to what extent does
the topological type
$\bold{X}$ of the complementary space $X = S^3 - K$ and/or
the isomorphism
class $\bold{G}$ of its fundamental group $G(K) =
\pi_1(X,x_0)$ suffice to
classify knots? The trefoil knot is almost everybody's
candidate for the
simplest example of a nontrivial knot, so it seems
remarkable that, not long
after the discovery of the fundamental group of a
topological space, Max Dehn
\cite {De} succeeded in proving that the trefoil knot and
its mirror image had
isomorphic groups, but that their knot types were
distinct. Dehn's proof is
very ingenious! This was the beginning of a long story,
with many contributions
(e.g., see \cite {Sei, Si, CGL}) which reduced repeatedly
the number of
distinct knot types which could have homeomorphic
complements and/or isomorphic
groups, until it was finally proved, very recently, that
(i) $\bold {X}$
determines $\BK$ (see \cite {GL}) and (ii) if $\BK$ is
prime, then $\bold{G}$
determines $\BK$ up to unoriented equivalence \cite {Wh}.
Thus there are at
most four distinct oriented prime knot types which have
the same knot group
\cite {Wh}. This fact will be important to us shortly.
The knot group $\bold{G}$ is finitely presented; however,
it is infinite,
torsion-free, and (if $\BK$ is not the unknot) nonabelian.
Its isomorphism
class is in general not easily understood via a direct
attack on the problem.
In such circumstances, the obvious thing to do is to pass
to the abelianized
group, but unfortunately $G/[G,G] \cong H_1(X;\roman
{Z})$ is infinite cyclic
for all knots, so it is of no use in distinguishing knots.
Passing to the
covering space $\widetilde X$ which belongs to $[G,G]$,
we note that there is a
natural action of the cyclic group $G/[G,G]$ on
$\widetilde X$ via covering
translations. The action makes the homology group
$H_1(\widetilde X;\roman
{Z})$ into a $\roman {Z}[q, q^{-1}]$-module, where $q$ is
the generator of
$G/[G,G]$. This module turns out to be finitely
generated. It's the famous
{\it Alexander module \/}. While the ring $\roman {Z}[q,
q^{-1}]$ is not a PID,
relevant aspects of the theory of modules over a PID apply
to $H_1(\widetilde
X;\roman {Z})$. In particular, it splits as a direct sum
of cyclic modules, the
first nontrivial one being $\roman
{Z}[q,q^{-1}]/A_q(\BK)$. Thus $A_q(\BK)$ is
the generator of the ``order ideal", and the smallest
nontrivial torsion
coefficient in the module $H_1(\widetilde X)$. In
particular, $A_q(\BK)$ is
very clearly an invariant of the knot group.
We regard the above description of $A_q(\BK)$ as an
excellent model for what
we might wish in a topological invariant of knots. We know
precisely what it
detects, and so we also know precisely what it {\it fails
\/} to detect. For
example, it turns out that $\pi _1(\widetilde X)$ is
finitely generated if and
only if $X$ has the structure of a surface bundle over
$S^1$, but there is no
way to tell definitively from $A_q(\BK)$ whether $\pi
_1(\widetilde X)$ is or
is not finitely generated. On the other hand, if a surface
bundle structure
exists, the genus of the surface is determined by
$A_q(\BK)$. The polynomial
$A_q(\BK)$ also generalizes in many ways. For example,
there are Alexander
invariants of links, also additional Alexander invariants
when the Alexander
module has more than one torsion coefficient. Moreover,
the entire theory
generalizes naturally to higher dimensional knots, the
generalized invariants
playing a central role in that subject.
Returning to the table in Figure 4, we remark that when a
knot is replaced by
its mirror image (i.e., the orientation on $S^3$ is
reversed), the Alexander and
Jones polynomials $A_q(\BK)$ and $J_q(\BK)$ go over to
$A_{q^{-1}}(\BK)$ and
$J_{q^{-1}}(\BK)$ respectively. As noted earlier,
$A_q(\BK)$ is invariant under
such a change, but from the simplest possible example, the
trefoil knot, we see
that $J_q(\BK)$ is not. Now recall that $\bold {G}$ does
{\it not \/} change
under changes in the orientation of $S^3$. This simple
argument shows that
$J_q(\BK)$ cannot be a group invariant! Since there are
at most four distinct
knot types that share the same knot group $\bold{G}$, a
first wild guess would
be that $J_q(\BK)$, which does detect changes in the
ambient space orientation
(but not in knot orientation), classifies unoriented knot
types; but this
cannot be true because \cite {Kan} constructs examples of
infinitely many
distinct prime knot types with the same Jones polynomial.
Thus it seems
interesting indeed to ask about the underlying topology
behind the Jones
polynomial. If it is not a knot group invariant, what can
it be? We will begin
to approach that problem by a circuitous route, taking as
a hint the central
and very surprising role of ``crossing-change formulas"
in the subject.
\heading 2. Crossing changes
\endheading
A reader who is interested in the history of mathematics
will find, on
browsing through several of Alexander's Collected Works,
that many of his
papers end with an intriguing or puzzling comment or
remark which, as it turned
out with the wisdom of hindsight, hinted at future
developments of the
subject. For example, in his famous paper on braids \cite
{Al1}, which we will
discuss in detail in \S 6, he proves that every knot or
link may be
represented as a closed braid. He then remarks (at the end
of the paper) that
this yields a construction for describing 3-manifolds via
their fibered knots;
however, he did it long before anyone had considered the
concept of a fibered
knot! Another example that is of direct interest to us
now occurs in \cite
{Al2}, where he reports on the discovery of the Alexander
polynomial. In
equation (12.2) of that paper we find observations on the
relationship between
the Alexander polynomials of three links: $K_{p_+},
K_{p_-},$ and $K_{p_0}$,
which are defined by diagrams that are identical outside
a neighborhood of a
particular double point $p$, where they differ in the
manner indicated in
Figure 5.
The formula which Alexander gives is:
$$ A_q(\BK_{p_+}) - A_q(\BK_{p_-}) = (\sqrt q - 1/{\sqrt
q}) A_q(\BK_{p_0}).
\tag 1a $$
This formula passed unnoticed for forty years. (We first
learned about
Alexander's version of it in 1970 from Mark Kidwell.)
Then, in 1968 it was
rediscovered, independently, by John Conway \cite {C}, who
added a new
observation: If you require, in addition to (1a), that:
$$ A_q(\bold {O}) = 1, \tag 1b $$
where ${\bold {O}}$ is the unknot, then (1a) and (1b)
determine $A_q(\BK)$ on
all knots, by a
recursive\vadjust{\fighere{5.5pc}\caption{{\smc Figure 5}.
{\rm Related link diagrams.}}}
procedure. To see this, the first thing to observe
is that if $\bold{O}_{\mu}$ is the $\mu$-component unlink,
then we may find related diagrams in which
$K_{p_+}, K_{p_-},$ and $K_{p_0}$ represent
$\bold {O}_{\mu}, \bold {O}_{\mu}$, and $\bold {O}_{\mu +
1}$ respectively,
as in Figure 6.
\topspace{4pc}
\caption{{\smc Figure 6}. {\rm Three related diagrams for
the unlink.}}
This fact, in conjunction with (1a), implies that
$A_q(\bold{O}) = 0$ if
$\mu \geq 2.$ Next, recall (Figure 1b) that any diagram
$K$ for any knot type
$\BK$ may be changed to a layered diagram which represents
the unknot or unlink
on the same number of components, by appropriate crossing
changes. Induction on
the number of crossing changes to the unlink then
completes the proof of
Conway's result. This led him to far-reaching
investigations of the
combinatorics of knot diagrams.
While the Jones polynomial was discovered via braid theory
(and we shall have
more to say about that shortly), Jones noticed, very early
in the game, that
his polynomial also satisfied a crossing-change formula,
vis:
$$ q^{-1}J_q(\BK_{p_+}) - qJ_q(\BK_{p_-}) =
(\sqrt q - 1/{\sqrt q}) J_q(\BK_{p_0}). \tag 2a $$
which, via Figure 6 and layered diagrams, may be used, in
conjunction
with the initial data
$$ J_q(\bold O) = 1, \tag 2b $$
to compute $J_q(\BK)$ for all knots and links. Motivated
by the
similarity between (1a)--(1b) and (2a)--(2b), several
authors \cite {LM,
Ho, PT} were led to consider a more general
crossing-change formula, which for our purposes may be
described as an infinite
sequence of crossing-change formulas:
$$ q^{-n}H_{q,n}(\BK_{p_+}) - q^nH_{q,n}(\BK_{p_-}) =
(\sqrt q - 1/{\sqrt q}) H_{q,n}(\BK_{p_0}), \tag 3a $$
$$ H_{q,n}(\bold O) = 1, \tag 3b $$
where $n \in \roman{Z}$. It turns out that (3a)--(3b)
determine an infinite
sequence of one-variable polynomials which in turn extend
uniquely to give a
two-variable invariant which has since become known as the
HOMFLY polynomial
\cite {FHL}. Later, (1a)--(3b) were replaced by a more
complicated family of
crossing-change formulas, yielding the Kauffman polynomial
invariant of knots
and links \cite {Kau1}. A unifying principle was
discovered (see \cite {Re2})
which yielded still further invariants, for example, the
$G_2$ invariant of
embedded knots, links, and graphs \cite {Ku}. Later,
crossing-change formulas
were used to determine other polynomial invariants of
knotted graphs \cite
{Y1} as well.
At this time we know many other polynomial invariants of
knots, links, and
graphs. In principle, all of them can be defined via
generalized
crossing-changes, together with initial data. In general,
a particular
polynomial will be defined by a family of equations which
are like (2a) and
(3a). It is to be expected that each such equation will
relate the invariants
of knots which are defined by diagrams which differ in
some specified way, in a
region which has a fixed number of incoming and outgoing
arcs. The ways in
which they differ will be more complicated than a simple
change in a crossing
and ``surgery" of the crossing. Thus we have a conundrum:
on the one hand,
knot and\ link diagrams and crossing-change formulas
clearly have much to do
with the subject; on the other hand, their role is in
many ways puzzling,
because we do not seem to be learning as much as we might
expect to learn about
diagram-related invariants from the polynomials.
We make this last assertion explicit. First, let us define
three diagram-related
knot invariants:
(i) the minimum crossing number $c(\BK)$ of a knot,
(ii) the minimum number of crossing changes $u(\BK)$,
i.e., to the unknot, and
(iii) the minimum number of Seifert circles $s(\BK)$,
\np where each of these invariants is to be minimized over
all possible knot
diagrams. ({\it Aside\/}. For the reader who is unfamiliar
with the concept of
Seifert circles, we note that by \cite {Y2} the integer
$s(\BK)$ may also be
defined to be the braid index of a knot or link, i.e., the
smallest integer $s$
such that $\BK$ may be represented as a closed $s$-braid.
If the reader is also
not familiar with closed braids, he or she might wish to
peek ahead to \S3.)
All of them satisfy inequalities which are detected by
knot polynomials. For
example, the Morton-Franks-Williams inequality places
upper and lower bounds
on $s(\BK)$ \cite {Mo2, FW}. Also, the Bennequin
inequality, recently
proved by Menasco \cite {Me}, gives a lower bound for
$u(\BK)$ which can be
detected by the one-variable Jones polynomial. As another
example, the
one-variable Jones polynomial was a major tool in the
proof of the Tait
conjecture (see \cite {Kau2, Mu}), which relates to the
definitive
determination of $c(\BK)$ for the special case of
alternating knots which are
defined by alternating diagrams. On the other hand, there
are examples which
show that none of the inequalities mentioned above can be
equalities for all
knots. Indeed, at this writing we do not have a
definitive method for
computing any of these very intuitive diagram-related knot
and link-type
invariants.
\heading 3. R-Matrix representations of the braid group
\endheading
One of the very striking successes of the past eight years
is that, after a
period during which new polynomial invariants of knots and
links were being
discovered at an alarming rate (e.g., see \cite {WAD}),
order came out of chaos
and a unifying principle emerged which gave a fairly
complete description of
the new invariants; we study it in this section. We remark
that this
description may be given in at least two mutually
equivalent ways: the first
is via the algebras of \cite {Jo4, BW, Kal} and the
cabling construction of
\cite {Mu, We2}; the second is via the theory of
R-matrices, as we shall do
here.
\topspace{16.5pc}
\caption{{\smc Figure} 7. {\rm Braids.}}
Our story begins with the by-now familiar notion of an
$n$-braid \cite {Art}.
See Figure 7(a) for a picture, when $n=3$.
Our $n$-braid is to be regarded as living in a slab of
3-space $ \Bbb{R}^2
\times I \subset \Bbb{R}^3.$ It consists of $n$
interwoven oriented strings
which join $n$ points, labeled $1,2,\dots,n$, in the plane
$ \Bbb{R}^2 \times
\lbrace 0 \rbrace$ with corresponding points in $
\Bbb{R}^2 \times \lbrace 1
\rbrace$, intersecting each intermediate plane $
\Bbb{R}^2 \times \lbrace t
\rbrace$ in exactly $n$ points. Two braids are equivalent
if there is an
isotopy of one to the other which preserves the initial
and end point of each
string, fixes each plane $ \Bbb{R}^2 \times \lbrace t
\rbrace$ setwise, and
never allows two strings to intersect. Multiplication is
by juxtaposition,
erasure of the middle plane, and rescaling. Closed braids
are obtained from
(open) braids by joining the initial point of each strand,
in $ \Bbb{R}^3$, to
the corresponding end point, in the manner illustrated in
Figure 7(b), so that
if one thinks of the closed braid as wrapping around the
$z$-axis, it meets
each plane $\theta=$constant in exactly $n$ points. A
famous theorem of Alexander
\cite {Al1} asserts that every knot or link may be so
represented, for some
$n$. ({\it Remark\/}. Lemma 1 of \S6 sketches a
generalization of Alexander's
original proof.) In fact, if $K$ is our oriented link, we
may choose any
oriented $ \Bbb{R}^1 \subset \Bbb{R}^3$ which is disjoint
from $K$ and modify
$K$ by isotopy so that this copy of $\Bbb{R}^1$ is the
braid axis.
The braid group $B_n$ is generated by the elementary braids
$\sigma_1,\dots,\sigma_{n-1}$ illustrated in Figure 8. For
example, the braid in Figure 7(a) may be described, using
the generators given in Figure 8, by the {\it word}\ \
$\sigma_1^{-2} \sigma_2 \sigma_1^{-1}$.
Defining relations in $B_n$ are:
$$ \sigma_i \sigma_j = \sigma_j \sigma_i \quad \text{if }
\mid i-j \mid \geq 2, \tag 4a$$
$$ \sigma_i \sigma_j \sigma_i = \sigma_j \sigma_i \sigma_j
\quad \text{if } \mid i-j \mid = 1, \tag 4b$$
\np We will refer to these as the {\it braid relations}.
Let $B_{\infty}$ be the disjoint union of the braid groups
$ B_1, B_2,\dots.$
Define $\beta, \beta^{\ast} \in B_{\infty}$ to be {\it
Markov-equivalent \/} if the closed braids $\hat {\beta},
\hat {\beta}^{\ast}$
which they define represent the same link type.
Markov's theorem, announced\vadjust{\fighere{9.5pc}
\caption{{\smc Figure 8}. {\rm Elementary braids, singular
braids, and
tangles.}}}
in \cite {Mar} and proved forty years later in \cite {Bi},
asserts that Markov
equivalence is generated by conjugacy in each $B_n$ and
the map $B_n
\rightarrow B_{n+1}$ which takes a word
$W(\sigma_1,\dots\sigma_{n-1})$ to
$W(\sigma_1,\dots\sigma_{n-1}) \sigma_n^{\pm 1}$. We call
the latter {\it
Markov's second move \/}. The examples in Figures 7(a),
7(b) may be used to
illustrate Markov equivalence. The 3-braid $\sigma_1^{-2}
\sigma_2
\sigma_1^{-1}$ shown there is conjugate in $B_3$ to
$\sigma_1^{-3} \sigma_2$,
which may be modified to the 2-braid $\sigma_1^{-3}$ using
Markov's second
move, so that $\sigma_1^{-3} \in B_2$ is Markov
equivalent to $\sigma_1^{-2}
\sigma_2 \sigma_1^{-1} \in B_3$.
Let $\{ \rho_n:B_n \rightarrow \GL _{m_n}(\Cal{E});
n=1,2,3,\dots\}$ be a
family of matrix representations of $B_n$ over a ring
$\Cal{E}$ with 1. Let
$i_n:B_n \rightarrow B_{n+1}$ be the natural inclusion map
which takes $B_n$ to
the subgroup of $B_{n+1}$ of braids on the first $n$
strings. A linear function
$\TR:\rho_n (B_n) \rightarrow \Cal{E}$ is said to be a
{\it Markov trace \/}
if:
(i) $\TR(1)=1$,
(ii) $\TR(\rho_n (\alpha \beta)) = \TR(\rho_n (\beta
\alpha))\ \ \
\forall \alpha, \beta \in B_n$,
(iii) $\exists z \in \Cal{E}$ such that if $\beta \in
i_n(B_n)$, then
$\TR(\rho_{n+1}(\beta \sigma_n^{\pm 1})) =
(z)(\TR(\rho_{n}(\beta))$.
\
\np In particular, by setting $\beta = 1$ in (iii) we see
that $z =
\TR(\rho_n(\sigma_{n-1}^{\pm 1}))$. In view of Markov's
theorem, it is
immediate that every family of representations of $B_n$
which supports a Markov
trace determines a link type invariant
$F(\bold{K}_{\beta})$ of the link type
$\bold{K}_{\beta}$ of the closed braid $\hat \beta$,
defined by
$$F(\bold{K}_{\beta}) = z^{1-n}\TR(\rho_n(\beta)). \leqno
(5)$$
Later, we will need to ask what happens to (5) if we
rescale the
representation, so we set the stage for the modifications
now. Suppose that
$\rho = \rho_n$ is a representation of $B_n$ in $\GL
_m(\Cal{E})$ which
supports a function $\TR$ which satisfies (i)--(iii). Let
$\gamma$ be any
invertible element in $\Cal{E}$. Then we may define a new
representation
$\rho^{\prime}$ by the rule $\rho^{\prime}(\sigma_i) =
\gamma \rho(\sigma_i)$
for each $i = 1,\dots,n$. It is easy to see that
$\rho^{\prime}$ is a
representation if and only if $\rho$ is, because (4a) and
(4b) are satisfied
for one if and only if they are satisfied for the other.
Properties (i) and
(ii) will continue to hold if we replace $\rho$ by
$\rho^{\prime}$, but (iii)
will be modified because the traces of $\sigma_i$ and
$\sigma_i^{-1}$ will not
be the same in the new representation. To determine the
effect of the change on
equation (5), let $z^{\prime} =
\TR(\rho^{\prime}(\sigma_i)) = \gamma \
\TR(\rho(\sigma_i)) = \gamma z$, so $z = \gamma ^{-1}
z^{\prime}$. Choose any
$\beta \in B_n$. Then $\beta$ may be expressed as a word
$\sigma_{\mu_1}^{\varepsilon_1}
\sigma_{\mu_2}^{\varepsilon_2} \cdots
\sigma_{\mu_r}^{\varepsilon_r}$ in the generators, where
each $\varepsilon_j =
\pm 1$. Let $\varepsilon (\beta) = \sum_i^r
\varepsilon_i$. Then equation (5)
will be replaced by
$$F(\bold{K}_{\beta}) =
\gamma^{-\varepsilon(\beta)}(z^{\prime})^{1-n}\TR(\rho_n^{%
\prime}(\beta)).
\leqno (5^{\prime})$$
Thus the existence of a link type invariant will not be
changed
if we rescale the representation by introducing the
invertible element
$\gamma$. Similar considerations apply if we rescale the
trace function. For
example, instead of requiring that $\TR(1) = 1$, we could
require that if $1_n$
denotes the identity element in $B_n$, then $\TR(1_n) =
z^{n-1}$, in which case
the factor $z^{1-n}$ in equation (5) would vanish.
The invariant which is defined by equation (5) or
(5$^{\prime}$) depends
directly on the traces of matrices in a finite-dimensional
matrix
representation $\rho_n$ of the braid group $B_n$. Any
such representation is
determined by its values on the generators $\sigma _i$,
and since these are
finite-dimensional matrices they satisfy their
characteristic polynomials,
yielding polynomial identities. From the point of view of
this section, such
identities are one source of the crossing-change formulas
we mentioned
earlier, in \S2. Another source will be trace identities,
which always exist in
matrix groups. (See, e.g., \cite {Pr}.) In general one
needs many such
identities (i.e., polynomial equations satisfied by the
images of various
special braid words) to obtain axioms which suffice to
determine a link type
invariant.
In the summer of 1984, almost eight years ago to the day
from this writing, this
author met with Vaughan Jones to discuss possible
ramifications of a discovery
Jones had made in \cite {Jo2} of a new family of matrix
representations of
$B_n$, in conjunction with his earlier studies of type
II$_1$ factors and their
subfactors in von Neumann algebras. Before that time,
there was essentially
only one matrix representation of the braid group which
had a chance of being
faithful and had been studied in any detail---the Burau
representation (see
\cite {Bi}). The knot invariant which was associated to
that representation was
the Alexander polynomial. Jones had shown that his
representations contained
the Burau representation as a proper summand. Thus his
representations were new
and interesting. Also, they supported an interesting trace
function. As it
developed \cite {Jo3}, his trace functions were Markov
traces. If $\rho _n$ is
taken to be the Jones's representation of $B_n$, the
invariant we called
$F(\BK_{\beta})$ in (5) is the one-variable Jones
polynomial $J_q(\BK)$.
\subheading{Wandering along a bypath}
The reader who is pressed for time may wish to omit this
detour. We interrupt
our main argument to discuss how the Jones representations
were discovered,
because it is very interesting to see how an almost chance
discovery of an
unexpected relationship between two widely separated areas
of mathematics had
ramifications which promise to keep mathematicians busy at
work for years to
come! Let $M$ denote a von Neumann algebra. Thus $M$ is an
algebra of bounded
operators acting on a Hilbert space $\Cal{H}$. The
algebra $M$ is called a
factor if its center consists only of scalar multiples
of the identity. The
factor is type II$_1$ if it admits a linear functional
$\TR:M \rightarrow
\Bbb{C}$, which satisfies:
(i) $\TR(1) = 1$,
(ii) $\TR(xy) = \TR(yx) \ \ \ \forall x,y \in M$,
\np and a positivity condition which shall not concern us
here. It is known
that the trace is unique, in the sense that it is the
only linear form
satisfying (i) and (ii). An old discovery of Murray and
von Neumann was that
factors of type II$_1$ provide a type of ``scale" by
which one can measure the
dimension $\roman{dim}_M\Cal{H}$ of the Hilbert space
$\Cal{H}$. The notion of
dimension which occurs here generalizes the familiar
notion of integer-valued
dimensions, because for appropriate $M$ and $\Cal{H}$ it
can be any
nonnegative real number or infinity. The starting point
of Jones's work was
the following question: if $M_1$ is a type II$_1$ factor
and if $M_0 \subset
M_1$ is a subfactor, is there any restriction on the
real numbers $\lambda$
which occur as the ratio $\lambda =
\roman{dim}_{M_0}\Cal{H}/\roman{dim}_{M_1}\Cal{H}$?
The question has the flavor of questions one studies in
Galois theory. On the
face of it, there was no reason to think that $\lambda$
could not take on any
value in [1,$\infty$], so Jones's answer came as a
complete surprise. He called
$\lambda$ the {\it index \/} [$M_1:M_0$] of $M_0$ in
$M_1$ and proved a
type of rigidity theorem about it:
\proclaim {The Jones Index Theorem} If $M_1$ is a
$\roman{II}_1$ factor and
$M_0$ a subfactor, then the possible values of the index
$\lambda$ are
restricted to $[4,\infty] \cup [4\cos^2(\pi /p)]$, where
$p \geq 3$ is a
natural number. Moreover, each such real number occurs for
some pair $M_0,
M_1$.\endproclaim
We now sketch the idea of Jones's proof, which is to be
found in \cite {Jo1}.
Jones begins with the type II$_1$ factor $M_1$ and a
subfactor $M_0$. There is
also a tiny bit of additional structure: In this setting
there exists a map
$e_1:M_1 \rightarrow M_0,$ known as the {\it conditional
expectation \/} of
$M_1$ on $M_0$. The map $e_1$ is a projection, i.e.,
$e_1^2 = e_1$. His first
step is to prove that the ratio $\lambda$ is independent
of the choice of the
Hilbert space $\Cal{H}$. This allows him to choose an
appropriate $\Cal{H}$ so
that the algebra $M_2 = \langle M_1,e_1\rangle$ generated
by $M_1$ and $e_1$
makes sense. He then investigates $M_2$ and proves that it
is another type
II$_1$ factor, which contains $M_1$ as a subfactor;
moreover, $|M_2:M_1| =
|M_1:M_0| = \lambda$. Having in hand another type II$_1$
factor, i.e., $M_2$
and its subfactor $M_1$, there is also a trace on $M_2$
which (by the
uniqueness of the trace) coincides with the trace on
$M_1$ when it is
restricted to $M_1$ and another conditional expectation
$e_2:M_2 \rightarrow
M_1$. This allows Jones to iterate the construction and
to build algebras
$M_1,M_2, \dots$ and from them a family of algebras:
$$ \Cal{A}_n = \langle 1,e_1,...,e_{n-1} \rangle \subset
M_n,\qquad
n=1,2,3,\dots\.$$
We now replace the $e_k$\<'s by a new set of generators
which
are units, defining
$g_k = qe_k - 1 + e_k$, where
$(1-q)(1-q^{-1}) = 1/\lambda$.
The $g_k$\<'s generate $\Cal{A}_n$, because the $e_k$\<'s
do,
and we can solve for the $e_k$\<'s in terms of the
$g_k$\<'s. Thus
$$\Cal{A}_n = \Cal{A}_n(q) = \langle
1,g_1,\dots,g_{n-1}\rangle, $$
and we have a tower of algebras, ordered by inclusion:
$$\Cal{A}_1(q) \subset \Cal{A}_2(q) \subset \Cal{A}_3(q)
\subset \cdots.$$
The parameter $q$, which replaces the index $\lambda$, is
the quantity now
under investigation.
The $g_i$\<'s turn out to be
invertible and to satisfy the braid relations (4a)--(4b), so
that there is a homomorphism from $B_n$ to $\Cal{A} _n$,
defined by mapping
the elementary braid $\sigma_i$ to $g_i$. The parameter
$q$ is woven into the
construction of the tower. Defining relations in
$\Cal{A}_n(q)$
depend upon $q$, for example, the relation $g_i^2 =
(q-1)g_i + q$ holds.
Recall that, since $M_n$ is type II$_1$, it supports a
unique trace,
and, since $\Cal{A}_n$ is a subalgebra,
it does too, by restriction. This trace is a
Markov trace! Jones's proof of the Index Theorem is
concluded when he shows that the
infinite sequence of algebras $\Cal{A}_n(q)$, with the
given trace, could not exist
if $q$ did not satisfy the stated restrictions.
Thus the ``independent variable" in the Jones polynomial
is essentially the
index of a type II$_1$ subfactor in a type II$_1$ factor!
Its discovery opened
a new chapter in knot and link theory.
\subheading{Back to the main road} We now describe a
method, discovered by
Jones (see the discussion of vertex models in \cite
{Jo5}) but first worked
out in full detail by Turaev in \cite {Tu}, which can be
applied to give, in
a unified setting, every generalized Jones invariant via a
Markov trace on an
appropriate matrix representation of $B_n$. As before,
$\Cal{E}$ is a ring with
1. Let $V$ be a free $\Cal{E}$-module of rank $m \geq 1$.
For each $n \geq 1$
let $V^{\otimes n}$ denote the $n$-fold tensor product $V
\otimes_{\Cal{E}} \dots
\otimes_{\Cal{E}}V$. Choose a basis $v_1,\dots,v_m$ for
$V$, and choose a
corresponding basis $\lbrace v_{i_1} \otimes \dots \otimes
v_{i_n} ; 1
\leq i_1,\dots ,i_n \leq m \rbrace$ for $V^{\otimes n}$.
An $\Cal{E}$-linear
isomorphism $f$ of $V^{\otimes n}$ may then be represented
by an
$m^n$-dimensional matrix $(f_{i_1 \cdots i_n}^{j_1 \cdots
j_n})$ over
$\Cal{E}$, where the $i_k$\<'s (resp. $j_k$\<'s) are row
(resp.\ column)
indices.
The family of representations of $B_n$ which we now
describe have a very special form.
They are completely determined by an $\Cal{E}$-linear
isomorphism $\roman{R}:V^{\otimes 2}
\rightarrow V^{\otimes 2}$ with matrix $\lbrack
\roman{R}_{i_1i_2}^{j_1j_1} \rbrack$ as above.
Let $I_V$ denote the identity map on the vector space $V$.
The representation
$\rho _{n,\roman{R}}:B_n \rightarrow \GL _{m^n}(\Cal{E})$
that we need is defined by
$$\rho _{n,\roman{R}}(\sigma _i) = I_V \otimes \dots
\otimes I_V
\otimes \roman{R}
\otimes I_V \otimes \dots \otimes I_V \leqno (6)$$
where $\roman{R}$ acts on the $i$\ i. \tag 10b$$
The smallest such $i$ is the {\it orde \/r} of $v$. To
stress it, we call our invariant
$v_i$ from now on.
\subheading{Initial data}
In addition to (10a) and (10b) we need initial data. The
first
piece of initial data relates to the normalization
mentioned earlier:
$$v_i(\bold{O}) = 0 \.\tag 10c$$
To describe the second piece of initial data, we need a
definition. A singular point on a
knot diagram will be called {\it nugatory \/} if its
positive and negative resolutions define
the same knot type, in the obvious manner indicated in
Figure 10. It is clear that if we are to
obtain a true knot invariant, its value on $v_i(\BK
^{j-1}_{p_+})$ and $v_i(\BK ^{j-1}_{p_-})$
must agree when $p$ is a nugatory crossing, so by (10a)
the initial
data must satisfy
$$v_i(\BK ^j_p) = 0 \text{ if } p \text{ is a nugatory
crossing}. \tag 10d$$
\topspace{16pc}
\caption{{\smc Figure} 9. {\rm Actuality table for $i=2$.}}
\np The final piece of initial data is in the form of a
table, but before we
can describe it we need to discuss a point that we avoided
earlier: The space
$\Cal{M}_j - \Sigma_j$ has a natural decomposition into
components, such that
two singular knots cannot define the same singular knot
type if they belong to
distinct components. To make this assertion precise, let
$\BK ^j$ be a singular
knot of order $j$, i.e., the image of $S^1$ under a
$j$-embedding $\phi \in
\Cal{M}_j - \Sigma_j$. Then $\phi^{-1}(\BK ^j)$ is a
circle with $2j$
distinguished points, arranged in pairs, where two
distinguished points are
paired if they are mapped to the same double point on $\BK
^j$. The $[j]$-{\it
configuration \/} which $\BK ^j$ {\it respects \/} is the
cyclically ordered
collection of point pairs. We will use a picture to define
it, i.e., a circle
with arcs joining the paired points, as in the top row of
Figure 9, where we
show the two possible [2]-configurations, together with a
choice of a singular
knot which respect each. The initial data must take
account of the following
(see \cite {ST 1} for a proof):
\proclaim{Lemma 1} Two singular knots $K^j_1, K^j_2$
become equivalent after an appropriate series of
crossing changes if and only if they respect the same
$[j]$-configuration.
\endproclaim
\fighere{5pc}
\caption{{\smc Figure 10}. {\rm Nugatory crossing.}}
The table which we now construct to complete the initial
data is called an {\it
actuality table \/}. Figure 9 is an example, when $i=2$.
It gives the values
of $v_i(\BK ^j)$ for a representative collection of
singular knots of order $j
\leq i$. The table contains a choice of a singular knot
$K^j$ which respects
each $[j]$-configuration, for $j=1,2,\dots,i$. The choice
is arbitrary (however,
the work in completing the table will be increased if poor
choices are made).
Next to each $K^j$ in the table is the configuration it
respects, and below it
is the value of $v_i(\BK ^j)$. These values are, of
course, far from arbitrary,
and the heart of the work in \cite {V} is the discovery of
a finite set of
rules which suffice to determine them. The rules turn out
to be in the form of
a system of linear equations. The unknowns are the values
of the functionals on
the finite set of singular knots in the actuality table.
The linear equations
which hold between these unknowns are consequences of the
local equations
(which may be thought of as crossing-change formulas)
illustrated in Figure 11.
These equations are not difficult to understand: use (10a)
to resolve each
double point into a sum of two crossing points. Then each
local picture in
Figure 11 will be replaced by a linear combination of four
pictures. The
equations in Figure 11 will then be seen to reduce to a
sequence of
applications of Reidemeister's third move. See \S3 of
\cite {BL} for a
description of the method that allows one to write down
the full set of
equations. See \S 2.4 of \cite {BN} for a proof that
solutions to the
equations, in the special case when $j=i$, may be
constructed out of
information about the irreducible representations of
simple Lie algebras. It
is not known whether the methods of \cite {BN} yield {\it
all\/} solutions in the
case $j=i$. The extension of the solutions for the case
$j=i$ to the cases
$2\le j\le i-1$ must be handled by the less routine
methods described in \cite
{BL}, at this time.
An example should suffice to illustrate that (10a)--(10d)
and the actuality table allow one to
compute $v_i(\BK)$ on all knots. For our example we
compute $v_2(\BK)$ when $\BK$ is the
trefoil knot. The\vadjust{\fighere{14.5pc}
\caption{{\smc Figure 11}. {\rm Crossing change formula
for Vassiliev invariants of
related singular knots.}}}
first
picture in Figure 12 shows our representative of the
trefoil, with a
crossing which is marked. Changing it, we will obtain the
unknot $\bold{O}$.
\topspace{19.5pc}
\caption{{\smc Figure 12}. {\rm Computing $v_2(\BK)$ for
the trefoil knot.}}
The crossing we selected is positive and so (10a) yields
$$v_2(K) =v_2(O) + v_2(N^1)$$
where $N^1$ is the indicated singular knot. It does
not have the same singular knot type as the singular knot
$K^1$ in the table, so (using the
lemma) we introduce another crossing
change to modify it to the singular knot in the
table which respects the unique [1]-configuration. In so
doing, we obtain a singular knot $N^2$
with two singular points. It does not have the same
singular knot type as the representative in
the table which respects the same [2]-configuration, but
by (10b) that does not matter. Thus,
using (10b), the calculation comes to an end in finitely
many steps.
\heading 6. Singular braids
\endheading
In \S 3 we showed that any generalized Jones invariant may
be obtained from a Markov
trace on an appropriate family of finite-dimensional
matrix representations of the braid
groups. Up to now, braids have not entered the picture as
regards Vassiliev invariants,
but that is easy to rectify. To do so, we need to extend
the usual notions of braids and
closed braids to singular braids and closed singular
braids.
A representative $K ^j$ of a singular knot or link $\BK
^j$ will be said to be
a {\it closed singular braid \/} if there is an axis $A$
in $\Bbb{R}^3$ (think
of it as the $z$-axis) such that if $K ^j$ is
parametrized by cylindrical
coordinates $(z, \theta)$ relative to $A$, the polar angle
function restricted
to $K ^j$ is monotonic increasing. This implies that $K^j$
meets each
half-plane $\theta = \theta{_0}$ in exactly $n$ points,
for some $n$. In
\cite {Al1} Alexander proved the well-known\ fact that
every knot or link $\BK$ may be
so represented, and we begin our work by extending his
theorem to singular
knots and links.
\proclaim{Lemma 2} Let $K^j$ be an arbitrary
representative of a singular knot or link
$\BK ^j$. Choose any copy $A$ of $\Bbb{R}^1$ in $\Bbb{R}^3
- K^j$. Then $K^j$ may be deformed to a
closed singular n-braid, for some n, with $A$ \ as
axis.\endproclaim
\demo{Proof} Regard $A$ as the $z$-axis in $\Bbb{R}^3$.
After an isotopy of
$K^j$ in \ $\Bbb{R}^3 - A$ \ we may assume that $K^j$ is
defined by a diagram
in the $(r, \theta)$-plane. By a further isotopy we may
also arrange that each
singular point $p_k$ has a neighborhood $N(p_k) \in K^j$
such that the polar
angle function restricted to $\bigcup_{k=1}^{j} N(p_k)$ is
monotonic
increasing. The proof then proceeds exactly as in \cite
{Al1}, vis: Modify \
$K^j - \bigcup_{k=1}^{j} N(p_k)$ \ to a piecewise linear
family of arcs
$\Cal{A}$, subdividing the collection if necessary so that
each $\alpha \in
\Cal{A}$ contains at most one undercrossing or
overcrossing of the knot
diagram. After a small isotopy we may assume that the
polar angle function is
nonconstant on each $\alpha \in \Cal{A}$. Call an arc
$\alpha \in \Cal{A}$ {\it
bad \/} or {\it good \/} accordingly as the polar angle
function is increasing
or decreasing on $\alpha$. If there are no bad arcs, we
will have a closed
braid, so we may assume there is at least one, say
$\beta$. Modify $K^j$ by
replacing $\beta$ by two good edges $\beta _1 \cup \beta
_2$ as in Figure 13.
The only possible obstruction is if the interior of the
triangle which is
bounded by $\beta \cup \beta{_1} \cup \beta{_2}$ is
pierced by the rest of
$K^j$, but that may always be avoided by choosing the new
vertex $\beta{_1}
\cap \beta{_2}$ so that it lies very far above (resp.\
below) the rest of
$K^j$ if the arc $\beta$ contains an under- (resp.\ over-)
crossing of the
diagram. Induction on the number of bad edges completes
the proof.
\qed\enddemo
In view of Lemma 2, every singular knot in the actuality
table of \S5 may be
chosen to be a closed singular braid. To continue, we
split these closed
singular braids open along a plane $\theta = \theta{_0}$
to ``open" singular
braids, which we now define. In \S3 we described a
geometric braid as a pattern
of $n$ interwoven strings in $\Bbb{R}^2 \times I \subset
\Bbb{R}^3$ which join
$n$ points, labeled $1,2,\dots,n$ in $\Bbb{R}^2 \times
\{0\}$ to the
corresponding points in $\Bbb{R}^2 \times \{1\}$,
intersecting each
intermediate plane $\Bbb{R}^2 \times \{t\}$ in exactly $n$
points. To extend
to singular braids, it is only necessary to weaken the
last condition to allow
finitely many values of $t$ at which the braid meets
$\Bbb{R}^2 \times \{t\}$
in $n-1$ points instead of $n$ points. Two singular braids
are equivalent if
they are isotopic through a sequence of singular braids,
the isotopy fixing
the initial and end points of each singular braid strand.
Singular braids
compose like ordinary braids: concatenate two patterns,
erase the middle
plane, and rescale.
\midspace{7pc}
\caption{{\smc Figure 13}. {\rm Replacing a bad arc by two
good arcs.}}
Choose any representative of an element of $SB_n$. After
an isotopy we may
assume that distinct double points occur at distinct
$t$-levels. From this it
follows that $SB_n$ is generated by the elementary braids
$\sigma_1,\dots,\sigma_{n-1}$ and the elementary singular
braids $\tau_1,\dots,
\tau_{n-1}$ of Figure 8. We distinguish between the
$\sigma_i$\<'s and the
$\tau_i$\<'s by calling them {\it crossing points \/} and
{\it double points
\/} respectively in the singular braid diagram. Both
determine double points in
the projection.
The manuscript \cite {Bae} lists defining relations in
$SB_n$ as:
$$[\sigma_i, \sigma_j] = [\sigma_i, \tau_j] = [\tau_i,
\tau_j] = 0\quad
\text{if } |i-j| \geq 2,
\tag 11a$$
$$[\sigma_i, \tau_i] = 0, \tag 11b$$
$$ \sigma_i\sigma_j\sigma_i = \sigma_j\sigma_i\sigma_j
\quad \text{if } |i-j| =
1, \tag 11c$$
$$ \sigma_i\sigma_j\tau_i = \tau_j\sigma_i\sigma_j \quad
\text{if } |i-j| = 1,
\tag 11d$$
where in all cases $1 \leq i, j \leq n-1.$ The same set of
relations also
occurs
in \cite {KV} as generalized Reidemeister moves.The
validity of these relations is easily
established via pictures; for example,
see Figure 14 for special cases of (11a)--(11d).
To the best of our knowledge,
however, there is not even a sketch of a proof that they
suffice
in the
literature, so we sketch one now, as it will be important
for us that no additional relations are
needed.
\proclaim{Lemma 3} The monoid $SB_n$ is generated by
$\{\sigma_i,\tau_i ; \ 1 \leq i \leq n-1\}$. Defining
relations are
{\rm (11a)--(11d)}.
\endproclaim
\demo{Proof} We have already indicated a proof that the
$\sigma_i$\<'s and the
$\tau_i$\<'s generate $SB_n$, so the only question is
whether every relation is
a consequence of (11a)--(11d).
We regard our braids as being defined by diagrams. Let
$\overline{z},
\overline{z}'$ be singular braids which represent the same
element of $SB_n$,
and let $\{ \overline{z}_s; \ s \in I \}$ denote the
family of singular braids
which join them. The fact that the intermediate braid
diagrams $\overline{z}_s$
have no\vadjust{\fighere{14.5pc}
\caption{{\smc Figure 14}. {\rm Relations in $SB_n$.}}}
triple points implies that
there is a well-defined order of the
singularities along each braid strand which is preserved
during the isotopy,
and this allows us to set up a 1-1 correspondence between
double points in
$\overline{z}$ and $\overline{z}'$. We now examine the
other changes which
occur during the isotopy. Divide the $s$-interval [0,1]
into small
subintervals, during which exactly one of the following
changes occurs in the
sequence of braid diagrams:
(i) Two double points in the braid projection interchange
their $t$-levels. See
relations (11a) and (11b) and Figure 14.
(ii) A triple point in the projection is created
momentarily as a ``free"
strand crosses a double point or a crossing point in the
projection. See (11d)
and Figure 14.
(iii) New crossing points in the knot diagram are created
or destroyed. See
Reidemeister's second move in Figure 2.
\np All possible cases of (i) are described by relations
(11a) and (11b).
Noting that the $\sigma_i$\<'s are invertible and that the
mirror image of
$\sigma_i$ (resp.\ $\sigma_i^{-1}, \tau_i)$ is
$\sigma_i^{-1}$ (resp.\
$\sigma_i, \tau_i)$, it is a simple exercise to see that
consequences of (11c)
and (11d) cover all possible cases of (ii). As for (iii),
if we restrict to
small $s$-intervals about the instant of creation or
destruction, these will
occur in pairs and be described by the trivial relation
$\sigma_i \sigma_i^{-1}
= \sigma_i^{-1} \sigma_i = 1$. Outside of these special
$s$-intervals the
singular braid diagram will be modified by isotopy, which
contributes no new
relations. Thus relations (11a)--(11d) are defining
relations for $SB_n$.
\qed\enddemo
Now something really interesting happens. Let
$\widetilde{\sigma}_i$ denote the
image of the elementary braid $\sigma_i$ under the natural
map from the braid
group $B_n$ to its group algebra $\Bbb{C}B_n$.
\proclaim{Theorem 2 {\rm (cf.\ \cite {Bae, Lemma 1})}\rm}
The map $\eta : SB_n \to
\Bbb{C}B_n$ which is defined by $\eta (\sigma_i) =
\widetilde{\sigma}_i, \eta
(\tau_i) = \widetilde{\sigma}_i
- \widetilde{\sigma}_i^{-1}$ is a monoid homomorphism.
\endproclaim
\demo{Proof} Check to see that relations (11a)--(11d) are
consequences of the
braid relations in $\Bbb{C}B_n$.
\qed\enddemo
\proclaim{Corollary 1} Every finite-dimensional matrix
representation $\rho _n:
B_n \to \GL _n(\Cal{E})$ extends to a representation
$\tilde{\rho}_n$ of
$SB_n$, defined by setting $\tilde{\rho}_n(\tau_i) =
\rho_n(\sigma_i) -
\rho_n(\sigma_i^{-1}).$
\endproclaim
\demo{Proof}
Clear.
\qed\enddemo
\np Thus, in particular, all R-matrix representations of
$B_n$ extend to
representations of the singular braid monoid $SB_n$.
\rem{Remark \RM1} Recall that in Figure 11 we gave picture
examples of some of
the relations which need to be satisfied by a functional
on the knot space in
order for the indices in an actuality table to determine a
knot type invariant.
Unlike the relations which we depicted in Figure 14, the
ones in Figure 11 are,
initially, somewhat mysterious. However, if one passes via
Theorem 2 to the
group algebra $\Bbb{C}B_n$ of the braid group, replacing
each $\tau _i$ by
$\sigma _i - \sigma _i^{-1}$, it will be seen that these
relations actually
hold in the algebra, not simply in the space of
VBL-functionals. This fact is
additional evidence of the naturality of the map $\eta$ of
Theorem 2 and indeed
of the Vassiliev construction. We conjecture that the
kernel of $\eta$ is
trivial, i.e., that a nontrivial singular braid in the
monoid $SB_n$ never maps
to zero in the group algebra $\Bbb{C}B_n$.\endrem
\rem{Remark \RM2} Various investigators, for example,
Kauffman in \cite
{Kau1}, have considered a somewhat different monoid which
we shall call the
{\it tangle monoid \/}, obtained by adding the
``elementary tangle"
$\varepsilon _i$ of Figure 8 to the braid group. The
tangle monoid, however,
does not map homomorphically to $\Bbb{C}B_n$, and one
must pass to quotients
of $\Bbb{C}B_n$, for example, the so-called Birman-Wenzl
algebra \cite {BW,
We1}, to give it algebraic meaning. In that sense the
tangle monoid appears to
be less fundamental than the singular braid monoid.\endrem
\heading 7. The proof of Theorem 1
\endheading
We are now ready to prove Theorem 1. We refer the reader
to the introduction
for its statement.
Theorem 1 was first proved by the author and Lin in \cite
{BL} for the special
cases of the HOMFLY and Kauffman polynomials and then in
full generality for
all quantum group invariants in \cite {L}. The proof we
give here is new. It is
modeled on the proof for the special cases in \cite {BL}.
We like it because it
is simple and because it gives us an opportunity to show
that the braid groups,
which are central to the study of the Jones invariants,
are equally useful in
Vassiliev's setting. The tools in our proof are the
R-matrix representations of
\S 3, the axioms and initial data of \S 5, and the
singular braid monoid of \S
6. Theorem 1 also is implicit in \cite {Bae}, which was
written simultaneously
with and independently of this manuscript. The techniques
used there are very
similar to ours, but the goal is different.
\demo{Proof of Theorem \RM1} By hypothesis, we are given a
quantum group
invariant $\Cal{J}_q: \Cal{M}-\Sigma \to \Cal{E}$, where
$\Cal{E}$ denotes a
ring of Laurent polynomials over the integers in powers of
$q$ (or in certain
cases powers of roots of $q$). Using Lemma 2, we find a
closed braid
representative $K_{\beta}$ of $\BK$, $\beta \in B_n$. We
then pass to the
R-matrix representation $\rho _{n,R}: B_n \to \GL
_{m_n}(\Cal{E})$ associated
to $\Cal{J}_q$. By Corollary 1 that representation extends
to a representation
$\tilde{\rho}_{n,R}$ of $SB_n$. By equation (9) the
Laurent polynomial
$\Cal{J}_q(\BK)$ is the trace of $\rho_{n,R}(\beta)
\cdot\mu$, where $\mu$ is
the enhancement of $\rho_{n,R}$.\enddemo
As was discussed in \S3, our representation $\rho_{n,R}$
is determined by the
choice of a matrix $R$ which acts on the vector space
$V^{\otimes 2}$. Lin
notes in Lemma 1.3 of \cite {L} that on setting $q=1$ the
matrix $R =
(R^{j_1j_2}_{i_1i_2})$ goes over to $\ID_V \otimes
\ID_V$. From this it
follows that $\rho_{n,R}(\sigma_i)$ (which acts on
$V^{\otimes n}$) has order 2
at $q=1$. Hence, we conclude that $\rho_{n,R}$ goes over
to a representation of
the symmetric group if we set $q=1$, with $\sigma_i$ going
to the transposition
$(i,i+1)$. In particular, this means that at $q=1$ the
images under
$\rho_{n,R}$ of $\sigma_i$ and $\sigma_i^{-1}$ will be
identical, which, in
turn, means that the images of $\sigma_i$ and
$\sigma_i^{-1}$ under
$\rho_{n,R}$ coincide at $q=1$.
Armed with this knowledge, we change variables, as in the
statement of Theorem 1,
replacing $q$ by $e^x$. Expanding the powers of $e^x$ in
its Taylor series, the image of an
arbitrary element $\beta$ under $\rho_{n,R}$ will be a
matrix power series
$$\rho_{n,R}(\beta) = M_0(\beta)
+ M_1(\beta) + M_2(\beta) + \cdots \leqno (12)$$
where each $M_i(\beta) \in \GL _{m_n}(Q)$.
\proclaim{Lemma 5 {\rm (cf.\ \cite {Bae, Corollary 1})}\rm}
In the extended representation,
$M_0(\tau_i) = 0.$ \endproclaim
\demo{Proof} Since $M_0(\sigma_i) = M_0(\sigma_i^{-1})$,
the assertion follows.
\qed\enddemo
Now let us turn our attention to the power series
expansion of $\Cal{J}_q$,
i.e., to $\Cal{J}_x(\BK) = \sum_{i=0}^{\infty} u_i(\BK)
x^i.$ The coefficients
$u_i(\BK)$ in this series, as $\BK$ ranges over all knot
types, determine a
functional $u_i:\Cal{M} - \Sigma \to Q$. We wish to prove
that $u_i$ is a
Vassiliev invariant of order $i$.
By \S5 it suffices to prove that, if we use (10a) to
extend the definition of
$u_i$ to singular knots, then (10b)--(10d) will be
satisfied and a consistent
actuality table exists. The first thing to notice is
that, since we began with
a knot-type invariant $\Cal{J}_q$, the functional $u_i$ is
also a knot-type
invariant. From this it follows that its extension to
singular knots is also
well defined, so using our knowledge of $u_i$ on knots we
can fill in the
actuality table.
The second observation is that (10c) is satisfied,
because every generalized
Jones invariant satisfies $\Cal{J}_q(\bold{O}) \equiv 1$,
and from this it
follows that $\Cal{P}_x(\bold{O}) \equiv 1$. As for (10d),
it is also
satisfied, for if not $u_i$ could not be a knot-type
invariant. Thus the only
problem which remains is to prove that $u_i$ satisfies
axiom (10b). However,
notice that by Lemma 5 we have
$$\tilde{\rho}_{n,R}(\tau_i) =
M_1(\tau_i)x + M_2(\tau_i)x^2 + M_3(\tau_i)x^3 +\cdots,$$
and from this it follows that if $\BK^j$ is a singular
knot which has $j$
singular points, then any singular closed braid
$K_{\gamma}^j, \ \gamma \in
SB_r$, which represents $\BK^j$ will also have $j$
singularities. The singular
braid word $\gamma$ will then contain $j$ elementary
singular braids. From this
it follows that
$$\tilde{\rho}_{n,R}(\gamma) =
M_{j}(\gamma )x^j + M_{j+1}(\gamma )x^{j+1} + \cdots.$$
The coefficient of $x^i$ in this power series is $u_i(\bold
K^j)$. But then, $u_i(\bold K^j) = 0$
if $i 2$. This seems to be a
deep and difficult combinatorial problem for arbitrary i,
and at this writing
the best we can do is to compute, the issue being the
construction of the
actuality table for an invariant of order $i$. This
problem divides naturally
into two parts: the first is to determine the Vassiliev
invariants of the
singular knots $K^i$ in the top row. By axiom (10b) they
only depend upon
the $[i]$-configuration which they respect, so their
determination is easier than
the corresponding problem for the remaining rows. The
former problem is solved
by setting up and solving the system of linear equations
(3.11) of \cite {BL}.
The dimension of the space of solutions is the sought-for
integer $m_i$. One
year ago Bar Natan wrote a computer program which listed
the distinct
$[i]$-configurations and calculated the dimensions (top
row only) for $i \leq 7$.
Very recently Stanford developed a different computer
program which checked
that all of Bar Natan's ``top row" solutions, for $i \leq
7$, extend to the
remaining rows of the actuality table, i.e., to solutions
to the somewhat more
complicated set of equations (3.17) of \cite {BL}, so Bar
Natan's numbers
are actually the dimensions $m_i$ that we seek. The
results of the two
calculations are:
\midinsert
\noindent$$
\table\nobox
i & 1 & 2 & 3 & 4 & 5 & 6 & 7\\
m_i & 0 & 1 & 1 & 3 & 4 & 9 & 14\endtable$$
\endinsert
%\ \ \ \ \ \ \ \ \ \ ${i}$ \ \ \ $m_i$
%\smallskip
%\ \ \ \ \ \ \ \ \ \ 1 \ \ \ 0
%\ \ \ \ \ \ \ \ \ \ 2 \ \ \ 1
%\ \ \ \ \ \ \ \ \ \ 3 \ \ \ 1
%\ \ \ \ \ \ \ \ \ \ 4 \ \ \ 3
%\ \ \ \ \ \ \ \ \ \ 5 \ \ \ 4
%\ \ \ \ \ \ \ \ \ \ 6 \ \ \ 9
%\ \ \ \ \ \ \ \ \ \ 7 \ \ \ 14
The data we just gave leads us to an important question.
We proved, in Theorem
1, that
$$\{\text{Quantum\ group\ invariants}\} \subseteq \{
\text{Vassiliev\ invariants}\}.$$
The question is: Is the inclusion proper?
The data is relevant because it was thought that the
question might be answered
by showing that for some fixed $i$ the quantum group
invariants spanned a
vector space of dimension $d_i < m_i$; however, a
dimension count to $i=6$
shows that there are enough linearly independent
invariants coming from quantum
groups to span the vector spaces $V_i$. Bar Natan's
calculations showed that
$d_7$ is at least 12, whereas $m_7 = 14$; however, the
data on the quantum group
invariants is imprecise because the invariants which come
from the
nonexceptional Lie algebras begin to make their presence
felt as $i$
increases. The only one of those which has been
investigated, to date,
is $G_2$.
For $i=8$ the computations themselves create difficulties.
Bar Natan's
calculation is close to the edge of what one can do,
because to determine $m_8$
he found he had to solve a linear system consisting of
334,908 equations in
41,874 unknowns (the number of distinct [8]-configurations
with no separating
arcs). His approximate calculation shows that the
solution space is of
dimension 27. However, even if his answer is correct, we
would still need to
compute the rest of the actuality table before we could be
sure that $m_8 = 27$
rather than $m_8 \leq 27$.
Vassiliev invariants actually form an algebra, not simply
a sequence of vector
spaces, because the product of a Vassiliev invariant of
order $p$ and one of
$q$ is a Vassiliev invariant of order $p+q$. This was
proved by Lin
(unpublished), using straightforward methods, and more
indirectly by Bar Natan
\cite {BN}. Thus the dimension $\hat{m}_i$ of new
invariants is in general
smaller than $m_i$, because the data in our table includes
invariants which
are products of ones of smaller order. It is a simple
matter to correct the
given data to find $\hat {m}_i$. For example, an invariant
of order $4$ could
be the product of two invariants of order $2$; so when we
correct for the fact
that there is a one-dimensional space of invariants of
order $2$, we see that
$\hat{m}_4 = 3 - 1 = 2$. For $i=1,2,3,4,5,6,7$ we find
that $\hat{m}_i =
0,1,1,2,3,5,8$, i.e., the beginning of the Fibbonaci
sequence! This caused some
excitement until Bar Natan's computation of $m_8$ showed
that $\hat{m}_8$ was
at most 12, not 13. The asymptotic behavior of $m_i$ as $i
\to \infty$ is a
very interesting problem indeed.
We can approach the question of whether the inclusion is
proper from a
different point of view. One of the earliest problems in
knot theory concerned
the fundamental symmetries which are always present in the
definition of knot
type. We defined knot type to be the topological
equivalence class of the pair
$(S^3, K)$ under homeomorphisms which preserve the given
orientation on both
$S^3$ and $K$. A knot type is called {\it amphicheiral \/}
if it is equivalent
to the knot type obtained by reversing the orientation of
$S^3$ (but not $K$)
and {\it invertible \/} if it is equivalent to the knot
type obtained by
reversing the orientation on $K$ (but not $S^3$). As noted
earlier, Max Dehn
proved in 1913 that nonamphicheiral knots exist \cite
{De}, but remarkably, it
took over forty years before it became known that
noninvertible knots exist \cite
{Tr}. The relevance of this matter to our question is:
While the quantum group
invariants detect nonamphicheirality of knots, they do not
detect
noninvertibility of knots. So, if we could prove that
Vassiliev invariants
distinguished a single noninvertible knot from its
inverse, the answer to our
question, Is the inclusion proper? would be yes. We know
we cannot
answer the question this way for $i \leq 7$, and as noted
above $i=8$ presents
serious computational difficulties. On the other hand, the
theoretical problem
seems to be unexpectedly subtle. Thus, at this moment, the
matter of whether
Vassiliev invariants ever detect noninvertibility remains
open.
Setting aside empirical evidence and unsolved problems, we
can ask some easy
questions which will allow us to sharpen the question of
whether Jones or
Vassiliev invariants determine knot type. As was noted in
\S2, there are three
very intuitive invariants which have, to date, proved to
be elusive: the
crossing number $c(\BK)$, the unknotting number $u(\BK)$,
and the
braid index
$s(\BK)$. Clearly these determine functionals on the space
$\Cal{M}- \Sigma$,
and so they determine elements in the group $\widetilde
{H}^0(\Cal{M}-
\Sigma)$. Vassiliev invariants lie in a sequence of
approximations to
$\widetilde {H}^0(\Cal{M}- \Sigma)$. So a reasonable
question to ask is, Are
$c(\BK), \ u(\BK)$, and $s(\BK)$ Vassiliev invariants?
Theorem 5.1 of \cite
{BL} shows that $u(\BK)$ is not, and the proof given there
is easily modified
to show that $c(\BK)$ and $s(\BK)$ are not either. So, at
the very least, we
have learned that there are integer-valued functionals on
Vassiliev's space of
all knots which are not Vassiliev invariants. This leaves
open the question of
whether there are sequences of Vassiliev invariants which
converge to these
invariants.
Another question which has been asked is, How powerful are
the Vassiliev
invariants, if we restrict our attention to invariants of
bounded order? The
answer to that question is, not very good, based on
examples which were
discovered, simultaneously and independently, by Lin \cite
{Li2} and by
Stanford \cite {St2}. We now describe Stanford's
construction, which is
particularly interesting from our point of view because it
is based on the
closed-braid approach to knots and links. See Remark (ii)
for a description of
Lin's construction.
To state his theorem, we return to braids. Let $P_n$ be
the pure braid group,
i.e., the kernel of the natural homomorphism from $B_n$ to
the symmetric group
$S_n$. \ The groups of the lower central series $\{
P_n^k;\ k=1,2,\dots\}$ of
$P_n$ are defined inductively by $P_n^1 = P_n,\ P^k_n =
\lbrack P_n, P_n^{k-1}
\rbrack$. Notice that if $\beta \in B_n$, with the closed
braid
$\widehat{\beta}$ a knot, then $\widehat{\alpha \beta}$
\ will also be a knot
for every $\alpha \in P_n$.
\proclaim{Theorem 3 \cite {St2}\rm} Let $\BK$ be any knot
type, and let
$K_{\beta}$, where $\beta \in B_n$, be any closed braid
representative of
$\BK$. Choose any $\alpha \in P_n^k$. Then the Vassiliev
invariants of order
$\leq k$ of the knots $K_{\beta}$ and $K_{\alpha \beta}$
coincide.
\endproclaim
\rem{Remarks} (i) Using the results in \cite {BM1},
Stanford has constructed
sequences $\alpha_1, \alpha_2,\dots$ of 3-braids such that
the knot types
obtained via Theorem 3 are all distinct and prime.
Intuition suggests that
distinct $\alpha _j$\<'s will always give distinct knot
types, but at this
writing that has not been proved.
(ii) One may choose $K_{\beta}$ in Theorem 3 to represent
the unknot and
obtain infinitely many distinct knots all of whose
Vassiliev invariants of
order $\leq k$, for any $k$, are zero. Lin's construction
gives other such
examples. In particular, he proves that, if $K$ is any
knot and if $K(m)$ is
its $m$\