Twisted homology cobordism invariants of knots in aspherical manifolds

TL;DR

This paper introduces new invariants based on twisted homology cobordism to distinguish knots in aspherical 3-manifolds, demonstrating their effectiveness through $L^2$-methods.

Contribution

It develops a novel approach using twisted homology cobordism invariants and $L^2$-techniques to differentiate knots that are homotopic but not concordant.

Findings

Constructed an infinite family of knots characteristic to a fixed knot J

Proved these knots are not concordant to J using $L^2$-methods

Established new invariants for knots in aspherical manifolds

Abstract

We fix a null-homologous, homotopically essential knot $J$ in a 3-manifold with PTFA fundamental group and study concordance of knots that are homotopic to $J$. We construct an infinite family of knots that are characteristic to $J$, and then use $L^2$-methods to show that they are not concordant to $J$.