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Inventiones mathematicae

, Volume 125, Issue 1, pp 103–133 | Cite as

On the Melvin–Morton–Rozansky conjecture

  • Dror Bar-Natan
  • Stavros Garoufalidis

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 Jones polynomials of cables of that knot (i.e., coefficients of the “colored” Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the Melvin–Morton–Rozansky (MMR) conjecture to knot invariants coming from arbitrary semi-simple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial.

Keywords

Color General Principle Independent Interest Weight System Jones Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Dror Bar-Natan
    • 1
  • Stavros Garoufalidis
    • 2
  1. 1.Department of Mathematics, Harvard University, Cambridge, MA 02138, USA; e-mail: drorbn@ math.huji.ac.il US
  2. 2.Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA; e-mail: stavros@ math.mit.eduUS

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