An analogue of Milnor’s invariants for knots in 3–manifolds
Year: 2024
Type: article
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.
Source: Algebraic & Geometric Topology
Author: Miriam Kuzbary
Institution: Georgia Institute of Technology
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W2981448660