The UniversalR-Matrix, Burau Representation, and the Melvin–Morton Expansion of the Colored Jones Polynomial

@article{Rozansky1996TheUB,
  title={The UniversalR-Matrix, Burau Representation, and the Melvin–Morton Expansion of the Colored Jones Polynomial},
  author={Lev Rozansky},
  journal={Advances in Mathematics},
  year={1996},
  volume={134},
  pages={1-31}
}
  • L. Rozansky
  • Published 3 April 1996
  • Mathematics
  • Advances in Mathematics
Abstract P. Melvin and H. Morton [9] studied the expansion of the colored Jones polynomial of a knot in powers of q −1 and color. They conjectured an upper bound on the power of color versus the power of −1. They also conjectured that the bounding line in their expansion generated the inverse Alexander–Conway polynomial. These conjectures were proved by D. Bar-Natan and S. Garoufalidis [1]. We have conjectured [12] that other ‘lines' in the Melvin–Morton expansion are generated by rational… 

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References

SHOWING 1-10 OF 15 REFERENCES

On the Melvin–Morton–Rozansky conjecture

Abstract. We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander–Conway polynomial of a knot can be read from some of the coefficients of the

Higher Order Terms in the Melvin-morton Expansion of the Colored Jones Polynomial

We formulate a conjecture about the structure of 'upper lines' in the expansion of the colored Jones polynomial of a knot in powers of (q−1). The Melvin-Morton conjecture states that the bottom line

Free fermions and the Alexander-Conway polynomial

AbstractWe show how the Conway Alexander polynomial arises from theq deformation of (Z2 graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot

Asymptotic expansions of Witten-Reshetikhin-Turaev invariants for some simple 3-manifolds

the link L. This paper investigates the invariant Z,*_,(M,0) when g=sK, for a simple family of rational homology 3-spheres, obtained by integer surgery around (2, n)-type torus knots. In particular,

A contribution of the trivial connection to the Jones polynomial and Witten's invariant of 3d manifolds, II

We extend the results of our previous paper [1] from knots to links by using a formula for the Jones polynomial of a link derived recently by N. Reshetikhin. We establish a relation between the

On $p$-adic convergence of perturbative invariants of some rational homology spheres

R.~Lawrence has conjectured that for rational homology spheres, the series of Ohtsuki's invariants converges p-adicly to the SO(3) Witten-Reshetikhin-Turaev invariant. We prove this conjecture for

Homological representations of the Hecke algebra

In this paper a topological construction of representations of theAn(1)-series of Hecke algebras, associated with 2-row Young diagrams will be given. This construction gives the representations in

Connections between CFT and topology via Knot theory

In this paper we shall discuss some of the isomorphisms established between the approach to conformal field theory on P1 of [TK], and the topological construction of braid group representations of [L

New points of view in knot theory

In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial in 1984. The focus of our account will be recent glimmerings

Ribbon graphs and their invaraints derived from quantum groups

The generalization of Jones polynomial of links to the case of graphs inR3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum