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

TL;DR

This paper generalizes the Tristram-Levine knot signatures to a singular Furuta-Ohta invariant for tori, connecting knot invariants with 4-manifold invariants and Floer homology.

Contribution

It introduces a new invariant for embedded tori in 4-manifolds, extending knot signatures and relating them to Donaldson and Floer invariants.

Findings

Defines a signed count of conjugacy classes for torus complements in SU(2)

Recovers the Casson-Lin-Herald invariant in the product case

Establishes a Floer homology framework and Fr{ exto}yshov invariants for knots and tori

Abstract

Given a knot $K$ inside an integer homology sphere $Y$, the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into $SU(2)$ which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen. Turning things around, given a 4-manifold $X$ with the integral homology of $S^{1}\times S^{3}$, and an embedded torus $T$ inside $X$ such that $H_{1}(T;\mathbb{Z})$ surjects onto $H_{1}(X;\mathbb{Z})$, we define a signed count of conjugacy classes of irreducible representations of the torus complement into $SU(2)$ which satisfy an analogous fixed conjugacy class condition to the one mentioned above for the knot case. Our count recovers the Casson-Lin-Herald invariant of the knot in the product case, thus it can be regarded as implicitly defining a Tristram-Levine signature for tori. This count can also be considered as a singular Furuta-Ohta invariant, and it is a special case of a larger family of Donaldson invariants which we also define. In particular, when $(X,T)$ is obtained from a self-concordance of a knot $(Y,K)$ satisfying an admissibility condition, these Donaldson invariants are related to the Lefschetz number of an Instanton Floer homology for knots which we construct. Moreover, from these Floer groups we obtain Fr{\o}yshov invariants for knots which allows us to assign a Fr{\o}yshov invariant to an embedded torus whenever it arises from such a self-concordance.