A characterization of alternating links in thickened surfaces

We use an extension of Gordon–Litherland pairing to thickened surfaces to give a topological characterization of alternating links in thickened surfaces. If Σ is a closed oriented surface and F is a compact unoriented surface in Σ × I , then the Gordon–Litherland pairing defines a symmetric bilinear pairing on the first homology of F . A compact surface in Σ × I is called definite if its Gordon–Litherland pairing is a definite form. We prove that a link L in a thickened surface is non-split, alternating, and of minimal genus if and only if it bounds two definite surfaces of opposite sign.