Concordance properties of parallel links

TL;DR

This paper studies the concordance properties of parallel links derived from knots, revealing conditions under which these links are concordant to split links and implications for knot invariants.

Contribution

It establishes new obstructions to smooth concordance of parallel links to split links and links these to the vanishing of key knot invariants.

Findings

Vanishing tau and s-invariants imply concordance to split links

Obstructions for (2,2m) cables to be concordant to torus links plus split links

Smooth concordance invariants provide constraints on parallel link concordance

Abstract

We investigate the concordance properties of `parallel links' P(K), given by the (2,0) cable of a knot K. We focus on the question: if P(K) is concordant to a split link, is K necessarily slice? We show that if P(K) is smoothly concordant to a split link, then many smooth concordance invariants of K must vanish, including the tau and s-invariants, and suitably normalized d-invariants of surgeries on K. We also investigate the (2,2m) cables P_m(K), and find obstructions to smooth concordance to the sum of the (2,2m) torus link and a split link.