Crosscap numbers of alternating knots via unknotting splices

Abstract

Ito-Takimura recently defined a splice-unknotting number $u^{-}(D)$ 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.

Associated data, including crosscap numbers for prime alternating knots through at least 13 crossings, are posted at www.thomaskindred.com

References in corpus (1)

• Crosscap number and knot projections

Cited by in corpus (2)

• Slope norm and an algorithm to compute the crosscap number
• Linear Bounds of the Crosscap Number of Knots