A Levine–Tristram invariant for knotted tori

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 embedded tori, and compare with what one might expect from Echeverria's invariant. For the simplest example--the product of an ordinary knot with a circle--the answers coincide. But for more general examples, the invariants are different.