# 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} }

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…

## 64 Citations

### Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial

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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…

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Abstract
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ANDREA OVERBAY: Perturbative Expansion of the Colored Jones Polynomial (Under the direction of Lev Rozansky) Both the Alexander polynomial ∆K(t) and the colored Jones polynomial Vα(K; q) are…

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S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series FK(x, q), which is the knot complement…

### Asymptotics of the colored Jones function of a knot

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To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose nth term is the nth colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the…

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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…

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