Miyazaki's torsion conjecture for cables

Let KK be a knot, let pp and qq be relatively prime integers with p>1|p|>1, and let Kp,qK_{p,q} denote the (p,q)(p,q)-cable of KK. Let C\mathcal{C} denote the smooth knot concordance group. Miyazaki's torsion conjecture. The (p,q)(p,q)-cable of KK is torsion in C\mathcal{C} if and only if KK is slice and q=1|q|=1. This strengthens the question of whether a (p,1)(p,1)- or (p,1)(p,-1)-cable can be slice only when the companion knot is slice. The paper proves infinite-order and linear-independence results for many odd cables, but the stated equivalence is not resolved here.

Sources & referencesView supporting material

Primary source

Sungkyung Kang and JungHwan Park, “Torsion in the knot concordance group and cabling”, arXiv:2207.11870 (2022).

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