Cosmetic crossings and Seifert matrices

Balm, C., Friedl, S., Kalfagianni, E. and Powell, M. (2012) Cosmetic crossings and Seifert matrices. Communications in Analysis and Geometry, 20(2), pp. 235-253. (doi: 10.4310/CAG.2012.v20.n2.a1)

Abstract

We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for twisted Whitehead doubles of non-cable knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 12 crossings.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Powell, Professor Mark
• Authors: Balm, C., Friedl, S., Kalfagianni, E., and Powell, M.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Communications in Analysis and Geometry
• Publisher: International Press
• ISSN: 1019-8385
• ISSN (Online): 1944-9992
• Published Online: 08 June 2012
• Copyright Holders: Copyright © 2012 The International Press of Boston, Inc.
• First Published: First published in Communications in Analysis and Geometry 20(2): 235-253
• Publisher Policy: Reproduced in accordance with the publisher copyright policy