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}
}

Figures from this paper

An oriented state model for the Jones polynomial and its applications alternating links
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 distinct
ORIENTED STATE MODEL OF THE JONES POLYNOMIAL AND ITS CONNECTION TO THE DICHROMATIC POLYNOMIAL
TLDR
This paper succeeds in adding the writhe to the state sum model and need not to compute the writher any more, and shows that Jones polynomial of any link (alternating or not) is a special parametrization of the dichromatic polynometric of a weighted graph with two different edge weights.
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 conjecture
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 the
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 for
Categorical lifting of the Jones polynomial: a survey
This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology. 1. Constructions of the Jones
A pr 2 00 8 A state sum invariant for regular isotopy of links having a polynomial number of states by Sóstenes Lins
TLDR
The state sum regular isotopy invariant of links, denoted VSE-invariant, is a generalization of the Jones Polynomial and is strictly stronger than Jones’: I detected a pair of links which are not distinguished byJones’ but are distinguished by the new invariant.
...
...

References

SHOWING 1-10 OF 41 REFERENCES
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 for
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 of
Jones polynomials and classical conjectures in knot theory. II
  • K. Murasugi
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1987
Let L be an alternating link and be its reduced (or proper) alternating diagram. Let w() denote the writhe of [3], i.e. the number of positive crossings minus the number of negative crossings. Let
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.
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 a
A spanning tree expansion of the jones polynomial
Exactly solved models in statistical mechanics
exactly solved models in statistical mechanics exactly solved models in statistical mechanics rodney j baxter exactly solved models in statistical mechanics exactly solved models in statistical
ON A CERTAIN NUMERICAL INVARIANT OF LINK TYPES
JONES: A polynomial invariant for knots via Von Neumann algebras
  • Bull. Am. Math. Soc. Vol. 12, Number
  • 1985
KAUFFMAN: Formal Knot Theory
  • Lecture Notes No
  • 1983
...
...