Volume 15, issue 1 (2015)

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The unknotting number and classical invariants, I

Maciej Borodzik and Stefan Friedl

Algebraic & Geometric Topology 15 (2015) 85–135
Abstract

Given a knot K we introduce a new invariant coming from the Blanchfield pairing and we show that it gives a lower bound on the unknotting number of K. This lower bound subsumes the lower bounds given by the Levine–Tristram signatures, by the Nakanishi index and it also subsumes the Lickorish obstruction to the unknotting number being equal to one. Our approach in particular allows us to show for 25 knots with up to 12 crossings that their unknotting number is at least three, most of which are very difficult to treat otherwise.

Keywords
unknotting number, Blanchfield pairing, Alexander module
Mathematical Subject Classification 2010
Primary: 57M27
References
Publication
Received: 30 November 2012
Revised: 18 June 2014
Accepted: 3 July 2014
Published: 23 March 2015
Authors
Maciej Borodzik
Institute of Mathematics
University of Warsaw
ul. Banacha 2
02-097 Warszawa
Poland
http://www.mimuw.edu.pl/~mcboro
Stefan Friedl
Fakultät für Mathematik
Universität Regensburg
D-93053 Regensburg
Germany
http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/