Vaughan F. R. Jones

A polynomial invariant for knots via von Neumann algebras

Thus, the trivial link with n components is represented by the pair (l ,n), and the unknot is represented by (si

On a certain value of the Kauffman polynomial

2 * * • s n i , n) for any n, where si,

Singly generated planar algebras of small dimension

2, • • • > sn_i are the usual generators for Bn. The second example shows that the correspondence of (b, n) with b is many-to-one, and a theorem of A. Markov [15] answers, in theory, the question of when two braids represent the same link. A Markov move of type 1 is the replacement of (6, n) by (gbg~, n) for any element g in Bn, and a Markov move of type 2 is the replacement of (6, n) by (6s J 1 , n-hl). Markovs theorem asserts that (6, n) and (c, ra) represent the same closed braid (up to link isotopy) if and only if they are equivalent for the equivalence relation generated by Markov moves of types 1 and 2 on the disjoint union of the braid groups. Unforunately, although the conjugacy problem has been solved by F. Garside [8] within each braid group, there is no known algorithm to decide when (6, n) and (c, m) are equivalent. For a proof of Markovs theorem see J. Birmans book [4]. The difficulty of applying Markovs theorem has made it difficult to use braids to study links. The main evidence that they might be useful was the existence of a representation of dimension n — 1 of Bn discovered by W. Burau in [5]. The representation has a parameter t, and it turns out that the determinant of 1-(Burau matrix) gives the Alexander polynomial of the closed braid. Even so, the Alexander polynomial occurs with a normalization which seemed difficult to predict.

The classification of subfactors of index at most 5

IfFL(a, x) is the Kauffman polynomial of a linkL we show thatFL(1, 2 cos 2π/5) is determind up to a sign by the rank of the homology of the 2-fold cover of the complement ofL. This value corresponds to a certain Wenzl subfactor defined by the Birman-Wenzl algebra, which we describe in simple terms. There also corresponds a “solvable” model in statistical mechanics similar to the 5-state Potts model. It is the 5-state case of a general model of Fateev and Zamolodchikov.

The embedding theorem for finite depth subfactor planar algebras

0. Introduction. A subfactorN ⊂M gives rise to a powerful set of invariants that can be approached successfully in several ways. (See, for instance, [B2], [EK], [FRS], [GHaJ], [H], [Iz], [JSu], [Lo1], [Lo2], [Oc1], [Oc2], [Po1], [Po2], [Po3], [Po4], [Wa], [We1], and [We2]). A particular approach suggests a particular kind of subfactor as the “simplest.” For instance, in Haagerup’s approach [H], subfactors of small index are the simplest. In [J2], a pictorial language is developed in which the invariants appear as a graded vector space V = (Vn)n≥0 whose elements can be combined in planar, but otherwise quite arbitrary, ways. Thus, for instance, in the diagram

Lissajous knots and billiard knots

The first author was supported by the NSF under Grant No. DMS-0301173 The second author was supported by the Australian Research Council under the Discovery Early Career Researcher Award DE120100232, and Discovery Project DP140100732 The third author was supported by a NSF Postdoctoral Fellowship at Columbia University. All authors were supported by DARPA grants HR0011-11-1-0001 and HR0011-12-1-0009.

Loop models, random matrices and planar algebras

We define a canonical relative commutant planar algebra from a strongly Markov inclusion of finite von Neumann algebras. In the case of a connected unital inclusion of finite dimensional C*-algebras with the Markov trace, we show this planar algebra is isomorphic to the bipartite graph planar algebra of the Bratteli diagram of the inclusion. Finally, we show that a finite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph.

The Jones Polynomial

We show that Lissajous knots are equivalent to billiard knots in a cube. We consider also knots in general 3-dimensional billiard tables. We analyse symmetry of knots in billiard tables and show in particular that the Alexander polynomial of a Lissajous knot is a square modulo 2. 0. Introduction. A Lissajous knot K is a knot in R given by the parametric equations x = cos(ηxt+ φx), y = cos(ηyt+ φy), z = cos(ηzt+ φz), for integers ηx, ηy, ηz. A Lissajous link is a collection of disjoint Lissajous knots. The fundamental question was asked in [BHJS94]: which knots are Lissajous? One defines a billiard knot (or racquetball knot) as the trajectory inside a cube of a ball which leaves a wall at rational angles with respect to the natural frame, and travels in a straight line except for reflecting perfectly off the walls; generically it will miss the corners and edges, and will form a knot. We will show that these knots are precisely the same as the Lissajous knots. We will also speculate about more general billiard knots, e.g. taking another polyhedron instead of the ball, considering a non-Euclidean metric, or considering the trajectory of a ball in the configuration space of a flat billiard. We will illustrate these by various examples. For instance, the trefoil knot is not a Lissajous knot 1991 Mathematics Subject Classification: 57M25, 58F17. This is an extended version of the talk given in August 1995, at the minisemester on Knot Theory at the Banach Center. We would like to acknowledge the support from USAF grant 1-443964-22502. The paper is in final form and no version of it will be published elsewhere.

Singly generated planar algebras of small dimension, Part II

We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar algebra. We apply this construction to compute the generating functions of the Potts model on a random planar map.

On the symmetric enveloping algebra of planar algebra subfactors

A link is a finite family of disjoint, smooth, oriented or unoriented, closed curves in R or equivalently S. A knot is a link with one component. The Jones polynomial VL(t) is a Laurent polynomial in the variable √ t which is defined for every oriented link L but depends on that link only up to orientation preserving diffeomorphism, or equivalently isotopy, of R. Links can be represented by diagrams in the plane and the Jones polynomials of the simplest links are given below.