Twisted torsion invariants and link concordance

TL;DR

This paper introduces a new twisted torsion invariant for 3-manifolds that is always non-zero and relates to the intersection form of bounding 4-manifolds, providing new obstructions in link concordance and homology cobordism.

Contribution

It presents a novel twisted torsion invariant that overcomes previous limitations and connects to 4-manifold intersection forms, enhancing tools for studying link concordance.

Findings

New non-zero twisted torsion invariant introduced

Obstructions to homology cobordism and link concordance developed

Detected that the Bing double of the figure eight knot is not slice

Abstract

The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.