Abstract: The Kauffman bracket is a Laurent polynomial invariant of framed unoriented links in the 3-sphere, and it can be calculated via locally defined skein relations. Applying these skein relations to links in other 3-manifolds yields invariants in the Kauffman bracket skein module, which is a relatively complicated object. I will talk about a relatively easy to compute multivariable Laurent polynomial invariant for links in thickened surfaces, obtained as a functional on the skein module, and how an evaluation of this invariant coincides with the arrow polynomial defined by Kauffman and Dye. This homological arrow polynomial is an invariant of virtual links (i.e., it is unchanged under destabilization along vertical annuli in the complement), and it has applications to checkerboard-colorability of virtual links, for example in completing Imabeppu's characterization of checkerboard colorability of virtual knots with at most four crossings.
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