Mathematical Research Letters

Volume 28 (2021)

Number 5

Ribbon knots, cabling, and handle decompositions

Pages: 1441 – 1457



Jennifer Hom (School of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)

Sungkyung Kang (Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, South Korea)

Junghwan Park (Department of Mathematical Sciences, Korea Advanced Institute for Science and Technology, Daejeon, South Korea)


The fusion number of a ribbon knot is the minimal number of $1$‑handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of $2$‑handles in a handle decomposition of a ribbon disk complement. We demonstrate that these invariants behave completely differently under cabling by showing that the $(p, 1)$ ‑cable of any ribbon knot with fusion number one has strong homotopy fusion number one and fusion number $p$. Our main tools are Juhász–Miller–Zemke’s bound on fusion number coming from the torsion order of knot Floer homology and Hanselman–Watson’s cabling formula for immersed curves.

The first author was partially supported by NSF grant DMS-1552285. The second author was supported by the Institute for Basic Science (IBS-R003-D1).

Received 7 June 2020

Accepted 6 October 2020

Published 16 August 2022