Almost-concordance of knots in aspherical 3-manifolds

TL;DR

This paper investigates the almost-concordance classes of knots in aspherical 3-manifolds, confirming a conjecture that such classes are infinite in many cases using extended Milnor invariants.

Contribution

It develops a new method extending Milnor's invariants to non-simply-connected 3-manifolds and confirms the conjecture for broad classes of knots.

Findings

Confirmed the conjecture for all nontrivial classes in aspherical 3-manifolds.

Extended Milnor invariants to knots in non-simply-connected 3-manifolds.

Identified large families of knots distinguished by these invariants.

Abstract

In this paper, we study topological concordance modulo local knotting, or almost-concordance, of knots in 3-manifolds $M\neq S^3$. A. Levine, Celoria (arXiv:1602.05476v4), and Friedl-Nagel-Orson-Powell (arXiv:1611.09114v2) conjecture that, absent the presence of an embedded dual 2-sphere, any free homotopy class $x$ of knots in $M$ contains infinitely many concordance classes modulo the action of the concordance group of knots in $S^3$ by local knotting. We develop a method for confirming this conjecture for any nontrivial class $x$ in any aspherical $M$ and provide computations that prove the conjecture in a large family of open cases. Our technique employs an extension of Milnor's link invariants to knots and links in non-simply-connected 3-manifolds (arXiv:2310.10918v2). We exhibit a large family of examples where, in a precise sense, we maximize the number of almost-concordance classes distinguished by these invariants.