Linear Bounds of the Crosscap Number of Knots

TL;DR

This paper investigates bounds on the crosscap number of knots, extending previous results to Conway sums of strongly alternating tangles, and demonstrates that existing linear bounds do not apply universally across all links.

Contribution

It extends bounds on the crosscap number to new classes of links and shows limitations of existing linear bounds for all links.

Findings

Bounds for the crosscap number can be extended to Conway sums of strongly alternating tangles.

There exist families of links where Jones polynomial coefficients and crosscap number grow independently.

Neither of the existing linear bounds applies universally to all links.

Abstract

Kalfagianni and Lee found two-sided bounds for the crosscap number of an alternating link in terms of certain coefficients of the Jones polynomial. We show here that we can find similar two-sided bounds for the crosscap number of Conway sums of strongly alternating tangles. Then we find families of links for which these coefficients of the Jones polynomial and the crosscap number grow independently. These families will enable us to show that neither linear bound generalizes for all links.