  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 $4(c(D)-g(D)+1)$.
