A refined upper bound for the hyperbolic volume of alternating links and the colored Jones polynomial

Abstract

Abstract. We give a refined upper bound for the hyperbolic volume of an alternating link in terms of the first three and the last three coefficients of its colored Jones polynomial.

Simplicial volume of links from link diagrams

“…The work of Lackenby, and of Agol and Thurston [Lac04] shows that for a volume bound formulated in terms of the twist number t of a link diagram, 10 is an optimal constant, and the bound 10(t − 1)v 3 is sharp. In [DT15], we refine this bound by involving more parameters into it. It provides an advantage in estimating volumes of many links, and it is used to show that the first and last three coefficients of the colored Jones polynomial correlate with the volume.…”

“…We will briefly recall the argument from [DT15] for alternating links. If K is a hyperbolic alternating link, then N is a link that is called an augmented alternating link.…”

“…Adding a crossing circle to a diagram is called augmenting the crossing. The proofs of several bounds for hyperbolic volume of links [DT15,Lac04,Pur07] are built on passing from an original link to a link for which a suitable polyhedral decomposition can be obtained. This is achieved by augmenting certain crossings and by removing adjacent full twists and half-twists.…”

“…The change of the link diagram that does not change the volume As a corollary, the upper bounds for volume proved by Marc Lackenby, Ian Agol and Dylan Thurston [Lac04] generalize from hyperbolic volume to simplicial volume. However our original motivation comes from a new refined bound, which was originally established only for hyperbolic volume of alternating links in [DT15] (and recently improved, using different techniques, by Colin Adams in [Ada15]). To generalize this bound beyond alternating links, the use of simplicial volume is necessary, even under the assumption that the original link is hyperbolic.…”

Bipyramids and bounds on volumes of hyperbolic links

“…In [10], Dasbach and Tsvietkova refined this bound so that if L is a hyperbolic alternating link in a reduced alternating projection P , then vol(S 3 −L) ≤ (4t 1 (P )+6t 2 (P )+8t 3 (P )+10g 4 (P )−a)v tet where a = 10 when g 4 ≥ 1, a = 7 when g 4 = 0 but t 3 ≥ 1 and a = 6 otherwise. We refer to this as the DT bound on volume.…”