A volumish theorem for the Jones polynomial of alternating knots

Abstract

The Volume Conjecture claims that the hyperbolic volume of a knot is determined by the colored Jones polynomial.Here we prove a "Volumish Theorem" for alternating knots in terms of the Jones polynomial, rather than the colored Jones polynomial: The ratio of the volume and certain sums of coefficients of the Jones polynomial is bounded from above and from below by constants.Furthermore, we give experimental data on the relation of the growths of the hyperbolic volume and the coefficients of the Jones polynomial, both for alternating and nonalternating knots.

On the crossing number of semiadequate links

“…Unlike adequate links, it completely contains other interesting classes like positive, Montesinos [39], and 3-braid links [74]. Semiadequacy also plays a central role in recent attempts [14,69] to understand some coefficients of the Jones polynomial. Based on this, it is a basic ingredient in the proof of existence of amphicheiral knots of almost all odd crossing numbers (relating to another old problem of Tait) [74].…”

The Jones polynomial and graphs on surfaces

“…The state surface G(s) provides a concrete connection between the Jones polynomial of a link L and the geometry and topology of the link complement. For example, the papers [10,13] use the checkerboard coloring of this surface to relate the coefficients of the Jones polynomial to the hyperbolic volume of the link. It also leads to a natural knot invariant: Proof.…”

“…For alternating links, our approach coincides with Thistlethwaite's approach: the Jones polynomial of an alternating link is a specialization of the Tutte polynomial of Tait's checkerboard graph of an alternating link projection. The connection between the Tutte polynomial and the Jones polynomial for alternating knots was fruitfully used in [10,11]. The books [3,28] give a good introduction to the interplay between knots and graphs.…”

Tait’s conjectures and odd crossing number amphicheiral knots

“…In an attempt to gain more insight into the appearance of the Jones polynomial, the author [St], and (up to minor interaction, independently) Dasbach and Lin [DL,DL2], initiated a detailed study of some coefficients of V in semiadequate diagrams. Let us remark here that, while adequacy is only a slight extension of alternation, semiadequacy is a rather wide extension of adequacy.…”