Smooth concordance of cables of the figure-eight knot

Sungkyung Kang, JungHwan Park, Masaki Taniguchi

Published 2025-05-06Version 1

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.