Vol. 318, No. 2, 2022

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The concordance crosscap number and rational Witt span of a knot

Stanislav Jabuka

Vol. 318 (2022), No. 2, 375–400
Abstract

The concordance crosscap number γc(K) of a knot K is the smallest crosscap number γ3(K) of any knot K concordant to K (and with γ3(K) defined as the least first Betti number of any nonorientable surface Σ embedded in S3 with boundary K). This invariant has been introduced and studied by Zhang (2007) using knot determinants and signatures, and has further been studied by Livingston (2008) using the Alexander polynomial. We show in this work that the rational Witt class of a knot can be used to obtain a lower bound on the concordance crosscap number, by means of a new integer-valued invariant we call the Witt span of the knot.

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Keywords
knots, concordance, crosscap number
Mathematical Subject Classification
Primary: 57K10
Secondary: 57K99
Milestones
Received: 24 October 2021
Revised: 9 May 2022
Accepted: 13 May 2022
Published: 20 August 2022
Authors
Stanislav Jabuka
Department of Mathematics and Statistics
University of Nevada, Reno
Reno, NV
United States