The homological arrow polynomial for virtual links

TL;DR

This paper introduces the homological arrow polynomial, a new invariant for virtual links that extends the arrow polynomial by incorporating homology classes, enabling advanced analysis of virtual link properties.

Contribution

It defines the homological arrow polynomial for virtual links, providing a graphical calculus and applying it to study nullhomologous links and checkerboard colorability.

Findings

Provides a graphical calculus with labeled whiskers

Characterizes checkerboard colorability using the polynomial

Proves a Kauffman-Murasugi-Thistlethwaite type theorem

Abstract

The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface's skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled "whiskers" that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study $(\mathbb{Z}/n\mathbb{Z})$-nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu's characterization of checkerboard colorability of virtual links with up to four crossings. We also prove a version of the Kauffman-Murasugi-Thistlethwaite theorem that the breadth of an evaluation of the homological arrow polynomial for an "h-reduced" diagram $D$ is $4(c(D)-g(D)+1)$.