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Branched covers and rational homology balls

Charles Livingston

Abstract

The concordance group of knots in the three-sphere contains an infinite subgroup generated by elements of order two, each one of which is represented by a knot K with the property that for every n > 0, the n-fold cyclic cover of S^3 branched over K bounds a rational homology ball. This implies that the kernel of the canonical homomorphism from the knot concordance group to the infinite direct sum of rational homology cobordism groups (defined via prime-power branched covers) contains an infinitely generated two-torsion subgroup.

Branched covers and rational homology balls

Abstract

The concordance group of knots in the three-sphere contains an infinite subgroup generated by elements of order two, each one of which is represented by a knot K with the property that for every n > 0, the n-fold cyclic cover of S^3 branched over K bounds a rational homology ball. This implies that the kernel of the canonical homomorphism from the knot concordance group to the infinite direct sum of rational homology cobordism groups (defined via prime-power branched covers) contains an infinitely generated two-torsion subgroup.
Paper Structure (8 sections, 4 theorems, 8 equations, 2 figures)

This paper contains 8 sections, 4 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result.
  3. Proof: $2L_n = 0 \in \mathcal{C}$
  4. Proof $L_n \in \ker \varphi$
  5. Review of twisted polynomials and $8_{17}$
  6. Main Proof
  7. Factoring $f(t^p)$
  8. Twisted polynomials of $8_{17}$

Key Result

Theorem 1

Let $K_n = P_n(8_{17})$. For all odd $n$, the knot $L_n = K_n \mathbin{\#} -K_n^r$ satisfies $2L_n = 0 \in \mathcal{C}$ and $L_n \in \ker \varphi$. There is an infinite set of prime integers $\mathcal{P}$ for which $L_\alpha \ne L_\beta \in \mathcal{C}$ for all $\alpha \ne \beta$ in $\mathcal{P}$. I

Figures (2)

  • Figure 1: The knot $P_n \subset S^1 \times B^2$ , $J_n$, and $J_n^*$.
  • Figure 2: $P_5(K) \mathbin{\#} P_5(K)$.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof