A characterization of alternating links in thickened surfaces

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