Mathematical Research Letters
Volume 28 (2021)
Ribbon knots, cabling, and handle decompositions
Pages: 1441 – 1457
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