Linear bounds of the crosscap number of knots

McConkey, Rob

doi:10.2140/agt.2025.25.4009

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Linear bounds of the crosscap number of knots

Rob McConkey

Algebraic & Geometric Topology 25 (2025) 4009–4035

DOI: 10.2140/agt.2025.25.4009

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.

Keywords

knot theory, crosscap number, Whitehead double, Jones Polynomial, knot invariants

Mathematical Subject Classification

Primary: 57K10, 57K14, 57K16

References

Publication

Received: 16 September 2023

Revised: 15 July 2024

Accepted: 22 August 2024

Published: 29 October 2025

Authors

Rob McConkey
Department of Mathematics and Physics Colorado State University Pueblo Pueblo, CO United States

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Volume 25, issue 7 (2025)