An Analogue of Milnor's Invariants for Knots in 3-Manifolds

TL;DR

This paper introduces the Dwyer number, a new invariant for knots in 3-manifolds, generalizing Milnor's invariants, and demonstrates its computational methods and applications in detecting non-trivial knots.

Contribution

It defines the Dwyer number for knots in 3-manifolds, relates it to Massey products, and shows its effectiveness in distinguishing certain non-concordant knots.

Findings

Dwyer number generalizes Milnor's invariants for knots in 3-manifolds.

Provides methods to compute the Dwyer number for null-homologous knots.

Dwyer number detects knots bounding disks that are not concordant to the unknot.

Abstract

Milnor's invariants are some of the more fundamental oriented link concordance invariants; they behave as higher order linking numbers and can be computed using combinatorial group theory (due to Milnor), Massey products (due to Turaev and Porter), and higher order intersections (due to Cochran). In this paper, we generalize the first non-vanishing Milnor's invariants to oriented knots inside a closed, oriented $3$-manifold $M$. We call this the Dwyer number of a knot and show methods to compute it for null-homologous knots inside connected sums of $S^1 \times S^2$. We further show in this case the Dwyer number provides the weight of the first non-vanishing Massey product in the knot complement in the ambient manifold. Additionally, we prove the Dwyer number detects a family of knots K in $\#^{\ell}S^1 \times S^2$ bounding smoothly embedded disks in $\natural^{\ell} D^2 \times S^2$ which are not concordant to the unknot.