The concordance crosscap number and rational Witt span of a knot

Stanislav Jabuka

Vol. 318 (2022), No. 2, 375–400

DOI: 10.2140/pjm.2022.318.375

Abstract

The concordance crosscap number ${\gamma }_c(K)$ of a knot $K$ is the smallest crosscap number ${\gamma }_3(K^{\prime })$ of any knot $K^{\prime }$ concordant to $K$ (and with ${\gamma }_3(K^{\prime })$ defined as the least first Betti number of any nonorientable surface $\mathrm{\Sigma }$ embedded in $S^3$ with boundary $K^{\prime }$). 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.

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