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

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.

Figures (10)

• Figure 1: A family of knots $L_{2n}$ in $S^1 \times S^2$ with $D(L_{2n})=2n-1$, where $n$ refers to the number of copies of the repeating tangle in the middle of the knot.
• Figure 2: An order 3 half-surface tower.
• Figure 3: A null-homologous knot in $S^1 \times S^2$.
• Figure 4: A family of two-component links $L_{2n}$, $n \geq 5$ which are homotopically unlinked, but have Milnor invariant $\bar{\mu}(1...12211221...1)=(-1)^{n+1}$ where each string $1...1$ has $1$$n-3$ times. The ellipses denote $n-3$ repetitions of the repeated tangle.
• Figure 5: The result of $0$-surgery on one component of $L_{2n}$. By Theorem \ref{['thm:computationthm']}, these knots have Dwyer number $2n$.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.1
• Proposition 1.0
• Proposition 1.0
• Proposition 1.0
• Theorem 1.1
• Proposition 1.0
• Remark 2.1
• Theorem 2.2: Casson casson
• Theorem 2.3: Stallings' Integral Theorem stallings