State Models and the Jones Polynomial

  title={State Models and the Jones Polynomial},
  author={Louis H. Kauffman},
IN THIS PAPER I construct a state model for the (original) Jones polynomial [5]. (In [6] a state model was constructed for the Conway polynomial.) As we shall see, this model for the Jones polynomial arises as a normalization of a regular isotopy invariant of unoriented knots and links, called here the bracket polynomial, and denoted 〈K〉 for a link projectionK . The concept of regular isotopy will be explained below. The bracket polynomial has a very simple state model. In §2 (Theorem 2.10) I… Expand

Figures from this paper

An oriented state model for the Jones polynomial and its applications alternating links
An ambient isotopy invariant, N"L, for oriented knots and links, is defined by multiplying it by a normalizing factor and shown to yield the Jones polynomial and the normalized bracketPolynomial. Expand
Jones and $Q$ polynomials for $2$-bridge knots and links
It is known that the Q polynomial of a 2-bridge knot or link can be obtained from the Jones polynomial. We construct arbitrarily many 2-bridge knots or links with the same Q polynomial but distinctExpand
It is well known that Kauffman constructed a state model of the Jones polynomial based on unoriented link diagrams. In his approach, in order to obtain Jones polynomial one needs to calculate bothExpand
Spin models for link polynomials, strongly regular graphs and Jaeger's Higman-Sims model.
We recall first some known facts on Jones and Kauffman polynomials for links, and on state models for link invariants. We give next an exposition of a recent spin model due to F. Jaeger and whichExpand
A link polynomial via a vertex-edge-face state model
AbstractWe construct a 2-variable link polynomial, called W L , for classicallinks by considering simultaneously the Kauffman state models forthe Alexander and for the Jones polynomials. We conjectureExpand
A study of braids in 3-manifolds
This work provides the topological background and a preliminary study for the analogue of the 2-variable Jones polynomial as an invariant of oriented links in arbitrary 3- manifolds via normalizedExpand
The Colored Jones Polynomial and the A-Polynomial of Two-Bridge Knots
We study relationships between the colored Jones polynomial and the A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ conjecture (of Garoufalidis) that relates theExpand
Some links with non-trivial polynomials and their crossing-numbers
One of the main applications of the Jones polynomial invariant of oriented links has been in understanding links with (reduced, connected) alternating diagrams [2], [8], [9]. The Jones polynomial forExpand
A state sum invariant for regular isotopy of links having a polynomial number of states
The state sum regular isotopy invariant of links which I introduce in this work is a generalization of the Jones Polynomial. So it distinguishes any pair of links which are distinguishable by Jones'.Expand
A 3-Variable Bracket
Kauffman’s bracket is an invariant of regular isotopy of knots and links which since its discovery in 1985 it has been used in many different directions: (a) it implies an easy proof of theExpand


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$2 * * • s n i , n) for any n, where si, $2, • • • > sn_i are the usual generators forExpand
Braids, link polynomials and a new algebra
A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on 2 parameters. The decomposition ofExpand
Jones polynomials and classical conjectures in knot theory
The primeness is necessary in the last statement ofTheorem B, since the connected sum of two figure eight knots is alternating, but it has a minimal non-alternating projection. Note that the figureExpand
Braids, Links, and Mapping Class Groups.
The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned withExpand
Braids : proceedings of the AMS-IMS-SIAM joint summer research conference on Artin's braid group held July 13-26, 1986 at the University of California, Santa Cruz, California
A construction of integrable differential system associated with braid groups by K. Aomoto Mapping class groups of surfaces by J. S. Birman Automorphic sets and braids and singularities by E.Expand
Jones’ braid-plat formula and a new surgery triple
A link Lu(2k, n 2k) is defined by a type (2k, n 2k) pairing of an n-braid Ii if the first 2 k strands are joined up as in a plat and the remaining n 2 k as in a closed braid. The main result is aExpand
A spanning tree expansion of the jones polynomial
A NEW combinatorial formulation of the Jones polynomial of a link is used to establish some basic properties of this polynomial. A striking consequence of these properties is the result that a linkExpand
Exactly Solved Models in Statistical Mechanics
R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved andExpand
Santa Cruz, California
H[...] U.S. Copyright Office Copyright claimant's address: Chicago. Copyright deposit; Geo. R. Lawrence; August 27, 1906.
JONES: A polynomial invariant for knots via Von Neumann algebras
  • Bull. Am. Math. Soc. Vol. 12, Number
  • 1985