The concordance crosscap number and rational Witt span of a knot

TL;DR

This paper introduces the Witt span, a new invariant derived from the rational Witt class, to provide lower bounds on the concordance crosscap number of knots, advancing understanding of nonorientable surfaces bounded by knots.

Contribution

The paper defines the Witt span invariant from the rational Witt class and demonstrates its effectiveness in bounding the concordance crosscap number.

Findings

Witt span provides new lower bounds for $\, ext{concordance crosscap number}$

The rational Witt class can be used to analyze nonorientable surfaces in knot theory

The approach links algebraic invariants with geometric properties of knots.

Abstract

The concordance crosscap number $\gamma_c(K)$ of a knot $K$ is the smallest crosscap number $\gamma_3(K')$ of any knot $K'$ concordant to $K$ (and with $\gamma_3(K')$ defined as the least first Betti number of any nonorientable surface $\Sigma$ embedded in $S^3$ with boundary $K'$). This invariant has been introduced and studied by Zhang using knot determinants and signatures, and has further been studied by Livingston 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.