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Ribbon numbers of 12-crossing knots

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arxiv 2409.12910 v1 pith:KYLSB67T submitted 2024-09-19 math.GT

Ribbon numbers of 12-crossing knots

Xianhao An , Matthew Aronin , David Cates , Ansel Goh , Benjamin Kirn , Josh Krienke , Minyi Liang , Samuel Lowery
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Ege Malkoc Jeffrey Meier Max Natonson Veljko Radi\'c Yavuz Rodoplu Bhaswati Saha Evan Scott Roman Simkins Alexander Zupan
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classification math.GT
keywords ribbonknotsnumbersknotnumberalexandercrossingpolynomials
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The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and Zupan. We show that the set of Alexander polynomials of knots with ribbon number at most four contains 56 polynomials, and we use this set to compute the ribbon numbers for many 12-crossing knots. We also study higher-genus ribbon numbers of knots, presenting some examples that exhibit interesting behavior and establishing that the success of the Alexander polynomial at controlling genus-0 ribbon numbers does not extend to higher genera.

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