A characterization of alternating links in thickened surfaces

TL;DR

This paper provides a topological characterization of alternating links in thickened surfaces using an extended Gordon-Litherland pairing, identifying them via bounding two definite surfaces of opposite sign.

Contribution

It introduces a new topological criterion for alternating links in thickened surfaces based on definite surfaces and the Gordon-Litherland pairing extension.

Findings

Alternating links in thickened surfaces are characterized by bounding two definite surfaces of opposite sign.

The extension of Gordon-Litherland pairing applies to surfaces in thickened surfaces.

Provides a topological criterion for identifying alternating links in complex surfaces.

Abstract

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