On concordances in 3-manifolds

TL;DR

This paper introduces a new framework for understanding knot concordance in 3-manifolds, defining almost-concordance, and demonstrates the existence of complex classes and invariants that distinguish knots in various 3-manifolds.

Contribution

It extends the concept of knot concordance to arbitrary 3-manifolds, introduces almost-concordance, and develops invariants to distinguish knots, including in lens spaces.

Findings

Existence of non-trivial almost-concordance classes in all non-abelian 3-manifolds

Infinite non almost-concordant knots in each lens space L(p,1)

An inequality relating cobordism PL-genus and tau-invariants

Abstract

We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manifolds. Afterwards, we focus the attention on the case of lens spaces, and use a modified version of the Ozsvath-Szabo-Rasmussen's tau-invariant to obstruct almost-concordances and prove that each L(p,1) admits infinitely many nullhomologous non almost-concordant knots. Finally we prove an inequality involving the cobordism PL-genus of a knot and its tau-invariants.