A Levine-Tristram invariant for knotted tori

TL;DR

This paper introduces a new topological invariant for knotted tori in homology S^1×S^3, extending classical knot invariants and confirming its consistency with Echeverria's gauge-theoretic invariant through computations and comparisons.

Contribution

The paper defines a novel topological invariant for embedded tori, analogous to the Levine-Tristram knot invariant, and demonstrates its equivalence to Echeverria's gauge-theoretic invariant.

Findings

Invariant computed for various embedded tori.

Invariant matches expectations from Echeverria's gauge theory.

Confirmed equivalence with Echeverria's invariant in general.

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.