Guts, volume and Skein Modules of 3-manifolds

TL;DR

This paper establishes a lower bound on the volume of hyperbolic links in 3-manifolds using a Kauffman bracket-based polynomial, linking diagram invariants to geometric properties of the link complement.

Contribution

It introduces a new Jones-type polynomial invariant derived from the Kauffman bracket on surfaces, providing bounds on hyperbolic volume and invariants of link projections.

Findings

Coefficients of the polynomial bound hyperbolic volume from above and below.

The twist number of certain link projections is an invariant.

The polynomial is an isotopy invariant for links in thickened surfaces.

Abstract

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. In the case that the 3-manifold is a thickened surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist reduced, checkerboard alternating link projection with disk regions, is an invariant of the link.