# State Models and the Jones Polynomial

```@article{Kauffman1987StateMA,
title={State Models and the Jones Polynomial},
author={Louis H. Kauffman},
journal={Topology},
year={1987},
volume={26},
pages={395-407}
}```
IN THIS PAPER I construct a state model for the (original) Jones polynomial . (In  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
1,239 Citations
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
ORIENTED STATE MODEL OF THE JONES POLYNOMIAL AND ITS CONNECTION TO THE DICHROMATIC POLYNOMIAL
• Mathematics
• 2010
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
• Mathematics
• 1988
One of the main applications of the Jones polynomial invariant of oriented links has been in understanding links with (reduced, connected) alternating diagrams , , . 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