Homological Polynomial Coefficients and the Twist Number of Alternating Surface Links

TL;DR

This paper establishes bounds on the twist number and hyperbolic volume of alternating surface links using polynomial invariants derived from a generalized homological Kauffman bracket, linking topological and geometric properties.

Contribution

It introduces a generalized homological Kauffman bracket and uses it to relate polynomial coefficients to the twist number and volume of alternating surface links.

Findings

Bound the twist number using polynomial coefficients.

Bound the hyperbolic volume in terms of polynomial coefficients.

Provides a new link between polynomial invariants and geometric properties.

Abstract

For $D$ a reduced alternating surface link diagram, we bound the twist number of $D$ in terms of the coefficients of a polynomial invariant. To this end, we introduce a generalization of the homological Kauffman bracket defined by Krushkal. Combined with work of Futer, Kalfagianni, and Purcell, this yields a bound for the hyperbolic volume of a class of alternating surface links in terms of these coefficients.