The pretzel-knot ribbon number spectrum conjecture

Let p,qZp,q\in\mathbb{Z} be odd, let nNn\in\mathbb{N}, and let KK be the (2n+1)(2n+1)-stranded pretzel knot

K=P(q,p,p,,p,p).K=P(q,p,-p,\ldots,p,-p).

For 0gn0\leq g\leq n, write rg(K)r_g(K) for the higher-genus ribbon number and let r(K)\frak r(K) denote the ribbon number spectrum. Pretzel-knot ribbon spectrum conjecture. The values satisfy

rg(K)=(ng)(p1),r_g(K)=(n-g)(p-1),

so

r(K)=(n(p1),(n1)(p1),(n2)(p1),,p1,0).\frak r(K)=(n(p-1),(n-1)(p-1),(n-2)(p-1),\ldots,p-1,0).

If true, this would show that the ribbon spectrum can have arbitrarily many jumps of arbitrarily large decrease, illustrating behavior beyond the previously understood stair-step spectra.

Sources & referencesView supporting material

Primary source

Xianhao An, Matthew Aronin, David Cates, Ansel Goh, Benjamin Kirn, Josh Krienke, Minyi Liang, Samuel Lowery, Ege Malkoc, Jeffrey Meier, Max Natonson, Veljko Radić, Yavuz Rodoplu, Bhaswati Saha, Evan Scott, Roman Simkins and Alexander Zupan, “Ribbon numbers of 12-crossing knots”, arXiv:2409.12910 (2024).

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