Stabilizing Four-Torsion in Classical Knot Concordance

TL;DR

This paper introduces a new obstruction method using 2-fold branched covers to identify knots of order 4 in the concordance group, revealing infinite families of such knots not detected by previous invariants.

Contribution

It provides a novel obstruction technique based on homology of branched covers to distinguish order 4 elements in knot concordance.

Findings

Detects infinite families of order 4 knots

Distinguishes algebraic from geometric concordance order

Advances understanding of knot concordance structure

Abstract

Let $M_K$ be the 2-fold branched cover of a knot $K in $S^3$. If $H_1(M_K) = {\bf Z}_3 \oplus {\bf Z}_{3^{2i}} \oplus G$ where 3 does not divide the order of $G$ then $K$ is not of order 4 in the concordance group. This obstruction detects infinite new families of knots that represent elements of order 4 in the algebraic concordance group that are not of order 4 in concordance.