Smooth concordance of cables of the figure-eight knot

Sungkyung Kang; JungHwan Park; Masaki Taniguchi

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arXiv · 2505.03720

Smooth concordance of cables of the figure-eight knot

Sungkyung Kang·JungHwan Park·Masaki Taniguchi

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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.

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Sungkyung Kang, JungHwan Park, Masaki Taniguchi. 2025-05-06. Smooth concordance of cables of the figure-eight knot. https://arxiv.org/abs/2505.03720

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