Stable concordance of knots in 3-manifolds

TL;DR

This paper introduces new intersection invariants for knots and links in 3-manifolds, generalizing classical invariants and providing obstructions to certain concordances, with applications to stable knot and link classifications.

Contribution

It develops a unified theory of intersection invariants for singular concordances, extending classical invariants and applying them to classify stable knots and links in 3-manifolds.

Findings

Classifies stably slice links in orientable 3-manifolds

Describes stable knot concordance in surface products with S^1

Provides stable link concordance results for null-homotopic knots

Abstract

Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orientable surface with the circle, and stable link concordance for many links of null-homotopic knots in orientable 3-manifolds.