NEW ZEALAND JOURNAL OF MATHEMATICS Volume 21 (1992), 1-16
KNOTS, BRAIDS, STATISTICAL MECHANICS AND VON NEUMANN ALGEBRAS
V a u g h a n J o n e s (Received June 1991)
Abstract. In the first lecture I will define several invariants of knots including the classical Alexander polynomial. I will show how they are related to statistical me chanics and quantum field theory. In the second lecture I will trace the origin of the new knot invariants through braids and von Neumann algebras and show how totally new methods in knot theory seem to be necessary for understanding and manipulating these invariants.
1. Lecture 1: Knot Polynomials
1.1 Definitions A knot will be a smooth curve in the three sphere S 3 (or R 3) which is the image of an embedding of S'1. A link will be a disjoint union of knots. Links may or may not be oriented, they will be considered up to diffeomorphisms of S3 which are generally required to preserve the orientation of S3 (this is the same as isotopy in S3). Links may always be projected onto some plane so that the only singularities are double points. If one records which part of the knot is closest to the plane of projection one obtains “link diagrams”. Here are some examples. The double points of the projection are called crossings for obvious reasons.
the unknot v V / the figure eight knot
the right trefoil Conway’s knot
QO the left trefoil
Knots have been tabulated up to 13 crossings [Thl]. Forgetting obvious symme tries and composite knots (e.g. Q>! is a composite knot made up of two trefoils), there are about 12^ thousand in the tables. The number of different knots grows exponentially with crossing number. 9 VAUGHAN JONES
The relation of equality of links is translated into the equivalence relation on link diagrams generated by (planar diffeomorphisms and) the Reidemeister moves of types I, II and III shown below
Type I
These moves are applied locally, affecting only that part of the link seen in the drawings above. An invariant is a quantity associated to a knot or link which depends only on the knot in three dimensions. If this invariant is calculated from diagrams, this just means that the answer is invariant under the Reidemeister moves. A very elementary but important example of an invariant is the linking number between two oriented curves. We will call ^ ( a positive crossing and a negative one. Given two nonintersecting smooth curvesC\ and C 2, lk(C \,C 2 ) is defined to be the signed sum of the crossings between C\ and C2 in some generic projection. It is simple to verify invariance under the type II and III moves, the only ones that can involve more than one component. In mathematics, knots and links have been most important in the study of three- manifolds via a process called surgery. Given a link L in a 3-manifold M, one removes a tubular neighbourhood ofL (a solid torus per component ofL ) then one glues it back with some new identification on the boundary. A good exercise is to see that S l x S 2 may be obtained by surgery on the unknot in S 3 by a diffeomorphism of the boundary which reverses the meridian and longitude of the torus. A basic theorem of Lickorish and Wallace shows that any closed oriented 3-manifold may be obtained from S 3 by surgery on a link (see [Ro]).
1.2 Five polynomial invariants in order of appearance
1.2.1 The Alexander polynomial Given an oriented link L in S3, one may realise elements of (S'1 — L) as smooth oriented curves. Taking the sum of the linking numbers with the components ofL gives a homomorphism from 7Ti (S3 —L) —► 7Z . This in turn defines an infinite KNOTS, BRAIDS. VON NEUMANN ALGEBRAS 3
cyclic covering space S 3~L whose first homology becomes aZZ[t,t x] module, t acting by the generator of the deck transformations. Using van Kampen’s theorem one sees that H i(S 3-L ) has a square presentation matrix as a 2 Z [t,t~ l] module. The principal ideal generated by this polynomial is then an invariant of the link L. It is called the Alexander polynomial A i{t) of L. As defined above A i s ambiguous up to multiplication by ± a power of t. It can be shown (see [Co]) that Al(£) may be normalized in such a way that it has the following (skein) properties: 1. A K{t) = 1 if K is the unknot. 2. If L +, L - and L q admit diagrams identical except near one crossing where they are as below, then
then AL+(t) - AL_(t) = \^ /t-
It is easy to see that the skein relation provides an algorithm for the calculation of Az,(t). Examples of Alexander polynomials are: • The trefoils have A^(£) =t~ l — 1 + 1. • The figure 8 has A Lit) = 3 — t — t -1 . • Conway’s knot has A^(i) = 1. In general a split link (one with two or more non empty components separated by a 2-sphere) has Alexander polynomial zero. The Alexander polynomial first appeared in 1928 in [A] and has been a powerful tool in knot theory. It may be calculated in polynomial time with respect to the number of crossings. Since Ai(f) is defined in terms of S 3-L, it may be calculated from any construc tion of this manifold. One way to do this is to glue together infinitely many copies of S 3 - L. indexed by 7 Z . each one joined to the next by identifying along an oriented surface in S 3 whose oriented boundary is L. By considering surfaces of the type shown below, it is possible to
gain complete control over (t).A For instance using a companion matrix one may easily show that, if p{t) is any given polynomial with • P(t)=p(t~1), • P{ 1) = 1 then there is a knot K with A x{t) = p(t). This is due to Seifert in [Se]. 4 VAUGHAN JONES
1.2.2 The polynomial Vi,{t)
It was shown in [J 1] that there is another polynomialV l (£) of oriented links defined by the skein relation
\ v L+(t) - tVL_(t) = fr f- ± ) VLo(t)
(and Kinknot = 1)> where L +, L _ and L q are as for the Alexander polynomial. Examples are:
• The Right trefoil has Vl (t) —t + t3 — t4. • The figure 8 has Vi,{t) = t ~2 — t~ 1 + 1 — t + t2. • Conway’s knot has Vl(£) = —t~ 4 + 2t~3 — 2t~2 + 2t_1 + t2 — 2t3+ 2t4 — 2t5 + t6 Mirror image symmetry is reflected in the polynomial by t —> \ so that the left trefoil has Vl(£) =t ^ 1 -f t~ 3 — t~ 4. Note that the Alexander polynomial of a knot is insensitive to this operation. The ^-polynomial of a distant union of two links is the product of theirV- polynomials multiplied by the factor — ^s/t + Thus split links have non-zero F-polynomial and in fact no link can have F-polynomial equal to zero. It is not known at this stage whether there is a non-trivial knot whoseV- polynomial is that of the unknot, nor is there any idea of how to characterize polynomials of the formV (t). It is unlikely that V can be calculated in polynomial time. In marked contrast to A(£), V{t) is essentially insensitive to the orientation of a link (it changes by a power of t under orientation changes). There is no known interpretation of Vl(£) in terms of the algebraic topology of S^-L.
1.2.3 The skein (or HOMFLY or FLYPTHOM) polynomialP l (1,tti) The similarity between the skein relations of A and V prompted many people ([F+], [PT]) to prove that the coefficients in the skein relation can be chosen arbitrarily. A standard choice is to define Pi,(l,m ) by
l~1PL+(l,m ) + lPL_(l,m ) + m PLo(l,m ) = 0 and .Punknot = 1- Examples are
• Pright trefoil = I 2 / + I Ul .
• P figure 8 = {~l~2 ~ 1 ~ I2) + m2. • PConway1~2 += 7 (2+ 6l2 + 214) + (-31~2 - 11 - ll/2 -l4)m2 3 + ( r 2 + 6 + 612 + l4)m4 + (— 1 —l2)m6. The polynomial P (l,m ) is known to contain more information than just the combination ofV and A (see [LM]). KNOTS, BRAIDS, VON NEUMANN ALGEBRAS 5
1.2.4 The absolute polynomial of Ql (x ) [BLM], [Ho] Considering unoriented links led to the discovery of a polynomial invariant Q l {x ) obeying the following generalized skein relation.
If L +, L_, L q and Loo are unoriented links admitting diagrams identical except near one crossing where they are as below,
then Q l + + Ql _ = x (Ql 0 + Ql J
Examples are: • Qtrefoil = ~ 3 +2x + 2x2. • Qfigure 8 = - 3 - 2X + 4x2 + 2x3. • QConway = 17 - 24x - 52x2 + 54x3 + 76a:4 - 28x5 - 48x6 - 4x7 + 8 x 8 + 2x9. This polynomial is understood no better than V ^ t).
1.2.5 The Kauffman polynomial F i,(a , x ) ([K al]) Regular isotopy is defined to be the relation on link diagrams generated by Rei demeister moves of type II and III. It is very little different from isotopy. Kauffman first defines an invariant F(a, x) of regular isotopy of unoriented link diagrams by
F l + + Fl _ = x (Fl 0 + Fl ^ ) as in section 1.2.4 (.Funknot = 1) and Fl ±i = cl± xFl if L ±l differs from L by a type I Reidemeister move. Then the Kauffman polynomial F-^(a,x) of the oriented link
L is defined as a~wr< These specializations will be understood in terms of simple Lie algebras. All these polynomials can be made considerably more powerful by combining them with the geometric operation of cabling, which means essentially that one string is replaced by several. The Lie algebra based theory will be closed under this operation. 1.3 The statistical mechanics approach An n-spin vertex model on a link diagram will be defined by a pair of functions w±(a, b | x , y){A) where A G [0,27r] and a, b ,x ,y £ {1,2,... , n}. A state of the link diagram will be the assignment of a spin to each edge of the link diagram viewed as a planar graph. Thus the figure below represents a state of the link diagram depicted. Given a state, any crossing i will look like either of the two figures below where A is the angle between the ingoing strings. (+) (-) The partition function Z l of the model defined by w± on the link diagram L is defined to be ^ 2 I I w±(ai b \ x >y)(x )e~ JLade states crossings where a : {1,2,... , n} —* II is some function and dd is the variation in the angle ( ! ^2 w ± (a ,b | x,y)wT(y:z \ 6,c) = 6a,c6x,z b,y J ^ w ± ( y ,x | a,b)wT {b,c \ z,y )e n(a{a)~a{b)) = 6a,c6x,z b,y III: £ w+(a,b | x,y)w+{b,c \ r,s)w+(y,z \ s,t ) b,y,s = ^2 w+(a,b I s,t)w+(x,y \ r,s)w+(b,c \ y,z). b,y,s KNOTS, BRAIDS, VON NEUMANN ALGEBRAS 7 The last condition is a simplified version of the Yang-Baxter equation, used in the theory of soluble models, see [Fa]. The simplest solution of all the equations is: 0 {a,x} ± {y,b} or a < b a = b = x — y w+(a,b | x,y)(\,h) = < e x p u 1 a = b,x = y,a ^ x / A h(b — a) 2 sinh exp a > b,a = y,b = x v 27r (where a, 6, x ,y € { — 1,1} and a (a) — a) for which the partition functionZ i{e h) — VL(eh). In the next section we will see how many solutions of the equations arise. Note that the formalism can easily handle links with coloured components pro vided one is given w± functions for each pair of colours. 1.4 Quantum groups The following relations occurred in the theory of solvable models: ([KR], [S]) [H, E] = 2 E [H, F] = - 2 F , sinh (if) [£ ’ F 1 = ----- 2 In the limit h —> 0 these become the usual defining relations of the Lie algebra 5(2- Now representations of the above commutation relations admit a tensor product: if Hi, Ei, Fi satisfy them on Vi for i = 1,2 then if we define, on V\ H 1 + 1 h 2 hH hH E E\ + e 1 E 2 F2 then H, E and F again satisfy them. Thus we are tempted to look for an object which “quantizes” the Lie algebra h being a deformation parameter. This object cannot be a Lie algebra because of the term sinh. So we work on the level of the enveloping algebra. A further difficulty occurs in interpreting the sinh term in terms of generators and relations. This problem was solved differently by Jimbo [Ji] and Drinfeld [Dr]. Jimbo defines, for every complex value of q = eh a Hopf algebra with “group like” elements K = ehH and “Lie algebra like” elements E and F. Drinfeld does everything in terms of formal power series in the variable h. Either way one obtains a Hopf algebra Uh(o) deforming each simple Lie algebra g (over (D) defined by a presentation deforming the Serre presentation of the Lie algebra. The finite dimensional representations also deform ([Rosl], [Lu]) and admit a tensor product structure coming from Hopf algebra comultiplication. 8 VAUGHAN JONES The key new ingredient is that the Hopf algebra structure is not cocommuta- tive so that the tensor product is not naturally commutative. Once again Jimbo and Drinfeld differ in their approaches. Drinfeld postulates an element R in Uh(&) 0 Uh(&) which corrects the non-commutativity in thata A (x) = R A (x )R ~ 1 where a(x 0 y) = y 0 x is an automorphism and A : Uh(&) —»► Uh{o) 0 Uh{$) is comultiplication. Drinfeld further imposes some natural properties onR which make it unique and gives a beautiful method for calculating it known as the “quan tum double”. It satisfies the (weak) universal Yang-Baxter equation R 12R 13R 23 = ^23-^13-^12 in Uh(o) 0 Uh{9) 0 Uh{$). This means that if a finite dimensional representation V of Uh(g) is chosen, R is defined on V 0 V and may be written R(va 0 vx) = ^2 w+(a, b | x , y)vb® vy given a basis {i;a} of V. The universal (weak) Yang-Baxter equation for R then implies the weak Yang-Baxter equation for w+. In all cases it has been shown ([Ros2], [Re]) that w+ may be extended to iu±(A) so that the corresponding vertex model gives a link invariant in the variable t = eh. If the Lie algebra is sln and the representation chosen is the defining one (on (Dn), one obtains a sequence of specializations of the skein polynomialPl sufficient to determine it. The polynomial Vl corresponds to5 (2- If the Lie algebra is orthogonal or symplectic and the representation is the defining one, one obtains a sequence of specializations of the Kauffman polynomialF l sufficient to determine it. The coincidence F i(a 3, —(a 4- a-1)) = Vi,(a4) is explained by the coincidence s p (l,l) = s l2. One could equally choose different representations per component of the link. This gives a theory for coloured links. The geometric operation of cabling (which can be used for colouring links) then corresponds to a tensor product of represen tations. Returning to Jimbo’s approach, he does not produce a universal .R-matrix but instead works with the affine Lie algebra 9 corresponding to9 . The representations of the deformed Uq(g) have a complex parameter A and Jimbo finds solutions of the full Yang-Baxter equation as Uq(g) intertwiners for the tensor productV\ 0 . The equations to be solved are linear but imply the Yang-Baxter equation. The advantage of this approach for knot theory is thatq may be any complex number, in particular a root of unity, where many interesting phenomena occur. 2. Lecture 2 : Sources and Recent Developments 2.1 Braids A braid is a way of joining hooks on a horizontal stick to hooks on another one directly below, in such a way that the tangent vector to a string always has a non-zero vertical component. Braids are adequately represented by diagrams as below. a 4 string braid KNOTS, BRAIDS, VON NEUMANN ALGEBRAS 9 Oriented links may be obtained from braids by “closing” them as below The braid a Its closure a Braids on a fixed number n of strings form a group denoted B n. The group operation is simply concatenation. A natural set of generators forB n is the set {cri, cr2, • • • , K L I I I I ! ...... II <*1 1 They satisfy the relations (JiOj = aj(Ji if \i — j\ > 2. These relations were shown by E. Artin to give a presentation ofB n (see [Bi]). The relation of isotopy of the closure of the braid is translated into the equiv alence relation on the disjoint union of the braid groups given by the “Markov moves” of types I and II Type I: a <-> /3aP~1, a,/3eB n Type II: a <->■ a a ^ 1, a G B n. The groups B n may be studied by means of their representations. A very simple but quite powerful family of such representations was discovered by Lascoux and Schutzenberger. The idea is to consider the symmetric group acting in the obvious way on a space of functions of the variablesx \, X2 , ... ,x n, and try to perturb it, maintaining the braid relations. Thus one might look for functionsP (x,y) and Q(x, y ) such that if we define (<*if)(x i,... ,x„) = P ( X i, Xi-^-i) f (xi, . .X n). -\- , Q ^ X i, Xi-^-i^f (x^, . . . , X2^-f-l, i, . .X .n) , then OjCTi-i-icrj = 0-j+i<7j<7j+i. It obviously suffices to consider the case n = 3. Lascoux and Schutzenberger found the most general functions P and Q and dis covered the remarkable fact that if the a^s preserve the space of polynomials then they must satisfy a quadratic equation. If moreover they preserve the degree then there is a unique (up to scaling) solution which we write W )(* . ») = / ( * * ) + ( « - x - y 10 VAUGHAN JONES (or P {x ,y ) = {'qx}yX-', Q (x,y) = 1 — ^ = { q - 1 )°i + Q (*) and one recognises this, together with the braid relations, as the defining relations of the Iwahori-Hecke algebra of type A n- \. Another powerful way to get representations of the braid groups is to use the /^-matrices of the last section. For ifR G End(l/ 2.2 Index for subfactors A von Neumann algebra is a *-closed algebra of bounded operators on a Hilbert space which contains the identity and which is closed in the topology of pointwise convergence. A factor is a von Neumann algebra whose centre consists only of scalar multiples of the identity and a II\ factor is an infinite dimensional factor admitting a trace tr which is a non-zero linear function to C withtr(ab) — tr(ba). Such a trace is unique provided tr( 1) = 1. Examples of II\ factors come from discrete groups T all of whose (non-identity) conjugacy classes are infinite. The von Neumann algebra vN (T) generated by the left regular representation on /2(r) is then a type II\ factor. The trace is given by tr{a ) = (a£ ,f) where £ G £2(r) is the characteristic function of the identity element. 1For example - the maximum degree of is 3. KNOTS, BRAIDS, VON NEUMANN ALGEBRAS 11 The trace on a II\ factor M can be used to create a one parameter family of representations ofM. First define L2(M, tr) to be the Hilbert space which is just M completed with respect to the inner product (a, b) = tr(b*a). M acts on L 2(M ,tr) by left multiplication. It is a fundamental property of II\ factors that the range of the trace restricted to orthogonal projections is the whole unit interval [0,1]. Now M acts on the right also onL 2(M ,tr ) so given a projectionp € M one may define the representation L2{M ,tr)p. If one assigns this Hilbert space a “dimension” diniM = tr{p) then, on taking direct sums as well, one finds that a II\ factor M admits representations H with dim M ^) equal to any non-negative real number. This number characterizes the representation up to unitary equivalence. A sub factor of a II\ factor is required to have the same identity as the large factor. If N c M is a subfactor we define [M : N] = dim ]y(L2(M ,tr)), the index of N in M. Examples: (i) If r 0 < T is a subgroup and both Tn and T have infinite conjugacy classes then [vN{T) : vN (T 0)} = [T : T0}. (ii) If M is the n x n matrices over N then [M : N] = n 2. (iii) If N is the fixed point algebra for a group G of outer automorphisms ofM then [M :N] = |G|. Theorem [J2]. (a) If [M : N] < 4 then there is an integer n > 3 with [M : N] — 4cos2 (^). (b) For all the numbers in a) and for any real number > 4 there is a pair N C M with [M : N] equal to that number. The proof of the theorem above proceeds as follows: Let ejv be the orthogonal projection fromL 2(M ,tr) onto L 2{N ,tr) and let (M, ejv) be the von Neumann algebra on L 2(M ,tr) generated by M and e ^ . If [M : AT] < oo then (M,c n ) is a I i i factor and \M : N] = tr(ejv)-1 = [(M, e^r) : M]. If one iterates this basic construction one finds a tower M; ofII\ factors with M0 = AT, Mi = M and M i+1 = (Mj, eAfi_i)- Letting = ejv/i one finds: [M : N]eiei±iei = e* etej = e ^ if \i - j\ > 2 2 * (**) ei ~ ei = ei k [M : N]tr{xen+i) =tr(x) if a: is a word on 1, ei, e2, ... , en These relations completely determine the algebra generated by l,e i,e 2, ... , en and part (a) of the theorem follows from the positive definite ness of the Hermitian form (a,b) = tr(b*a) on this algebra. 12 VAUGHAN JONES 2.3 Origin of the polynomialVi,(t ) The relations (**) are reminiscent of Artin’s presentation of the braid group and Markov’s moves. A simple calculation shows that if we definet by 2 + t + t~ l = [M : N] and gi — tei — (1 — e*) then 9i9i+l9i ~ 9i+l9i9i+l 9i9j = 9j9i if\i-j\> 2 9i =(t~ l )9i + t tr(xgn+1) = tr(x)tr(gn+i) if a: is a word on g1,g2,... ,gn (It is significant that 4 cos2 ~ corresponds tot — e±2^ \) Thus one may obtain representations ofB n by sending cr* to gi. Moreover it follows from the last property that ( ^ ) (v^ r(a) (where e is the exponent sum of the braid a) depends only on a and so is an oriented link invariant. The relation gf = (t — l)gi + t (which is just another way of writing ef = e*) then implies that the above invariant satisfies the skein relation for V^(t), and so is equal to it. This circuitous route is actually howV i(t) was discovered in [J1]. The two variable skein polynomial can be obtained in this way also, simply by taking the relation on the gi above and allowing tr(gi) to be some arbitrary complex number. This was Ocneanu’s approach in [F+]. See [J3]. This approach also produced new examples of subfactors as shown in [We]. 2.4 Origin of the connection with statistical mechanics The relations amoung the projections e* had already occurred in the work of Temperley and Lieb in [TL]. In this work the e^’s were certain matrices which combined to give the partition function for the “Potts” model, and another set of matrices which combined in the same way to give the partition function for the “ice-type” model or “six-vertex” model. This is clearly laid out in chapter 12 of Baxter’s book [Ba], But such is the power of the e* relations that the partition function can in principle be calculated from them alone. This established the equivalence between the two models. But in [Ba] an abstract (e^-free) argument for Temperley-Lieb equivalence is given involving an intermediate step which is a kind of non-local model. Kauff man actually discovered this non-local model when he found his beautiful “states model” for V ^ t) in [Ka2]. Thus it was natural to look for formulae using genuine (local) models. For the Potts model this was implicit in the work of Murasugi and Thistlethwaite ([Mu], [Th2]) and for vertex models of a certain type I discovered how to do it in [J4] and eventually it was published in [J5]. Turaev generalized it for the Kauffman polynomial in [Tu]. KNOTS, BRAIDS, VON NEUMANN ALGEBRAS 13 2.5 Witten’s approach With the statistical mechanics approach firmly under control the hints of a con nection with quantum field theory (often obtained as the continuum limit of statisti cal mechanics) were clear. Atiyah suggested that the polynomials were describable in terms of gauge field theory and that there should be a relationship with Don aldson’s theory of 4-manifolds. The latter has not been found but Witten in [Wi] did show how to interpret V i,(e^L). In gauge field theory one takes a (usually compact) Lie group G with (real) Lie algebra g. The configuration space for the theory will then be the quotient of the space of all g-valued one-forms (vector potentials) on some manifold M by the action of the gauge group. The gauge group is the group of all smooth functions g : M —> G, which acts on the space of vector potentials byg(A) = gAg~l + g~l dg. A classical observable will then be some function on the quotient space, involving also derivatives of the vector potential. The physics is then defined by a Lagrangian or Lagrangian density C(A) so that the action of a vector potential fis M C{A). The classical trajectories are those which extremize the action. These will be certain functions A satisfying in general rather complicated differential equations. For electromagnetism G — U{ 1). In general the action will involve a metric onM. Witten proposed a situation where the action is purely topological, namely M = a three manifold, G = a compact simple Lie group [SU (2) for example) 2 C(A) = tr(A A dA -f -A A A A A) (the Chern-Simons action for which 3 C(A) is well defined modulo the gauge group up to an integer). M To quantize this theory there are various options. The first is to postulate the existence of an appropriate measure (VA) on the space Q}{M, A)/Map(M, G) of vector potentials modulo the gauge group. If / is a function on this space, the expected value of the corresponding quantum observable will be [DA]e i Xw£(A) In Witten’s case the function / is obtained from a link L in M with components K \, K 2, ... , K c. The one-formA defines a connection so choosing a finite dimen sional representation V* ofG for each Ki one may evaluate the trace of the parallel transport around Ki. This trace W k ^ A ) is well defined modulo the gauge group Map(M, G) and one may evaluate the expectation of the products: The “coupling constant” A is a parameter of the theory but must be an integer in order for the exponential to be well defined. 14 VAUGHAN JONES By a series of heuristic arguments involving a different kind of quantization and the identification of certain spaces occurring in it with spaces in a model (the Wess- Zumino-Witten model) of conformal quantum field theory, Witten proposed that if M = S 3 (where the 2-dimensional representation ofsu( 2) is chosen for eachi). The great advance from a mathematical point of view is that the theory should work equally well for any three manifold, and W itten gave explicit formulae for calculating the invariant of a link in a 3-manifold. This includes the empty link - i.e., invariants of 3-manifolds themselves. These formulae have been checked (see [RT], [KM], [Li]) and the role of the has been seen to be the same as for the 4 cos2 ^ in the subfactor theory. 2.6 The Vassiliev theory In [V], Vassiliev introduced a beautiful set of invariants for knots in R 3. The idea is to study the whole space 9JT of smooth mappings fromS 1 to R 3. The discriminant E of this space is defined to be the set of such mappings which are not embeddings so that isotopy classes of knots are precisely the connected components of the complement of the discriminant. Thus any integer-valued knot invariant is an element of the cohomologyH°(Wl — E). One then tries to construct elements of this cohomology group by algebraic topology on the infinite dimensional space 9Jt. In order to approach this space one considers on the one hand a filtration according to how many double points the mapping from S 1 to R 3 has, and on the other hand a sequence of finite dimensional approximations by considering Fourier series for the x , y and z components of the knots. The resulting theory is somewhat complicated and not surprisingly involves spectral sequences. But there does emerge a family of numerical invariants of knots, the completeness of which is conjectured by Vassiliev and is shown to be equivalent to the convergence of the spectral sequence. In an interesting development, Birman and Lin in [BL] have shown that, in the simplest case, the polynomial V^(t) is contained in the Vassiliev invariants. To be more precise, if Vk {gx) is expanded as a power series in x , the coefficients are Vassiliev invariants. The same appears to be true of any polynomial invariant coming from quantum groups (work of Lin). Perhaps the most interesting feature of this work is that it ties in with Witten’s functional integral formula! Indeed Bar-Natan has considered Witten’s functional form from the point of view of perturbation theory and Feynmann diagrams (see [B-N]). One should naturally get a power series in powers pf ^ for V (e n+2) and he has indeed found Vassiliev invariants coming out of the Feynmann diagrams of the perturbation series. This is an exciting new direction of research. KNOTS, BRAIDS, VON NEUMANN ALGEBRAS 15 R eferences [A] J. Alexander, Topological invariants of knots and links , Trans. AMS 30 (1928), 275-306. [Ba] R. Baxter, Exactly Solved Models in Statistical Mechanics , Academic Press, London, 1982. [Bi] J. Birman, Braids, Links and mapping class groups , Annals of Math. Studies 82 (1974). [BL] J. Birman and X-S Lin, Knot Polynomials and Vassiliev’s Invariants , Preprint, 1991. [BLW] W. Brandt, W. Lickorish and K. 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