The homological arrow polynomial for virtual links

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