Communications in Analysis and Geometry

Volume 20 (2012)

Number 2

Cosmetic crossings and Seifert matrices

Pages: 235 – 253



Cheryl Balm (Department of Mathematics, Michigan State University)

Stefan Friedl (Mathematisches Institut, Universität zu Köln, Germany)

Efstratia Kalfagianni (Department of Mathematics, Michigan State University)

Mark Powell (Department of Mathematics, Indiana University)


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.

Full Text (PDF format)