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1 results

  • A characterization of alternating links in thickened surfaces

  • Hans U. Boden, Homayun Karimi
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 1 / February 2023
    Published online by Cambridge University Press:
    13 December 2021, pp. 177-195
    Print publication:
    February 2023
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  • 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.