Smooth concordance of cables of the figure-eight knot

TL;DR

This paper proves that all nontrivial cables of the figure-eight knot have infinite order in the smooth knot concordance group using new invariants derived from branched covers and Seiberg--Witten Floer theory.

Contribution

Introduces a family of concordance invariants $ppa_R^{(k)}$ applicable to all $(2n,1)$-cables of the figure-eight knot, generalizing previous $K$-theoretic invariants.

Findings

All nontrivial cables of the figure-eight knot have infinite order in the smooth concordance group.

The invariants $ppa_R^{(k)}$ effectively distinguish concordance classes of these cables.

The proof provides a uniform approach applicable to an entire family of cables.

Abstract

We prove that every nontrivial cable of the figure-eight knot has infinite order in the smooth knot concordance group. Our main contribution is a uniform proof that applies to all $(2n,1)$-cables of the figure-eight knot. To this end, we introduce a family of concordance invariants $\kappa_R^{(k)}$, defined via $2^k$-fold branched covers and real Seiberg--Witten Floer $K$-theory. These invariants generalize the real $K$-theoretic Fr\o yshov invariant developed by Konno, Miyazawa, and Taniguchi.