A volumish theorem for alternating virtual links

TL;DR

This paper extends the volumish theorem to alternating links on surfaces, linking hyperbolic volume bounds to polynomial coefficients, and explores graph cycle ranks related to the Krushkal polynomial.

Contribution

It introduces a volumish theorem for surface-based alternating links, connecting polynomial invariants to hyperbolic volume bounds, and analyzes Krushkal polynomial coefficients.

Findings

Bounds on hyperbolic volume in terms of polynomial coefficients

Coefficients of the Krushkal polynomial relate to cycle ranks of Tait graphs

Extension of volumish theorem to links on surfaces

Abstract

Dasbach and Lin proved a "volumish theorem" for alternating links. We prove the analogue for alternating link diagrams on surfaces, which provides bounds on the hyperbolic volume of a link in a thickened surface in terms of coefficients of its reduced Jones-Krushkal polynomial. Along the way, we show that certain coefficients of the 4-variable Krushkal polynomial express the cycle rank of the reduced Tait graph on the surface.