A Levine-Tristram invariant for knotted tori

Daniel Ruberman

doi:10.2140/agt.2022.22.2395

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arXiv · 2010.02355

A Levine-Tristram invariant for knotted tori

Daniel Ruberman

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Abstract

Echeverria recently introduced an invariant for a smoothly embedded torus in a homology $S^1\times S^3$, using gauge theory for singular connections. We define a new topological invariant of such an embedded torus, analogous to the classical Levine-Tristram invariant of a knot. In the 3-dimensional situation, a count of singular connections on a knot complement reproduces the Levine-Tristram invariant. We compute the invariant for a number of examples embedded tori, and show that our topological invariant is the same as what one might expect from Echeverria's invariant. Langte Ma has subsequently shown this in general.

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Daniel Ruberman. 2020-10-05. A Levine-Tristram invariant for knotted tori. https://doi.org/10.2140/agt.2022.22.2395

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