A Levine-Tristram invariant for knotted tori

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.

arXiv comment28 pages, one figure. Revised introduction, amended discussion of one example, and added another example. Version 3 is the final version, to appear in Algebraic & Geometric Topology

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Cited by in corpus (1)

• A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori