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Crosscap numbers of alternating knots via unknotting splices

Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130, USA

https://doi.org/10.1142/S0129167X20500573Cited by:4 (Source: Crossref)

Abstract

Ito–Takimura recently defined a splice-unknotting number $u^{-} (D)$ $u^{-} (D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.

Keywords:

AMSC: 57M25, 57M27

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