The Gordon-Litherland pairing for knots and links in thickened surfaces

We introduce the Gordon-Litherland pairing for knots and links in thickened surfaces that bound unoriented spanning surfaces. Using the GL pairing, we define signature and determinant invariants for such links. We relate the invariants to those derived from the Tait graph and Goeritz matrices. These invariants depend only on the $S^*$ equivalence class of the spanning surface, and the determinants give a simple criterion to check if the knot or link is minimal genus. We apply the GL pairing to give a topological characterization of alternating links in thickened surfaces, extending the results of J. Greene and J. Howie. This is joint work with M. Chrisman and H. Karimi.