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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, ([email protected]; [email protected])
Homayun Karimi
Affiliation:
Mathematics & Statistics, McMaster University, Hamilton, Ontario, ([email protected]; [email protected])
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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
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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