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arXiv · 2010.14030

A characterization of alternating links in thickened surfaces

Hans U. Boden·Homayun Karimi
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Abstract

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 $Σ\times I$, then the Gordon-Litherland pairing defines a symmetric bilinear pairing on the first homology of $F$. A compact surface in $Σ\times I$ 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.

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Hans U. Boden, Homayun Karimi. 2021-11-08. A characterization of alternating links in thickened surfaces. https://doi.org/10.1017/prm.2021.78

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