Cosmetic crossings and Seifert matrices

Details Cosmetic crossings and Seifert matrices Cosmetic crossings and Seifert matrices

2011-08-15

arxiv.org/abs/1108.3102v2

Mathematics

Geometric Topology

16 pages, 5 Figures. Minor revisions. This version will appear in Communications in Analysis and Geometry. This paper subsumes

Scientific paper

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.

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