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A characterization of alternating links in thickened surfaces

Published online by Cambridge University Press:  13 December 2021

Hans U. Boden Open the ORCID record for Hans U. Boden [Opens in a new window]  and
Homayun Karimi
Hans U. Boden
Affiliation:
Mathematics & Statistics, McMaster University, Hamilton, Ontario, (boden@mcmaster.ca; karimih@math.mcmaster.ca)
Homayun Karimi
Affiliation:
Mathematics & Statistics, McMaster University, Hamilton, Ontario, (boden@mcmaster.ca; karimih@math.mcmaster.ca)
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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 $\Sigma$ 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 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.

Keywords

Links in thickened surfacesalternating linkspanning surfaceGordon–Litherland pairingdefinite surfacevirtual linkcheckerboard colouring

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Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 1 , February 2023 , pp. 177 - 195
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

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