The unknotting number and classical invariants, I

Borodzik, Maciej; Friedl, Stefan

doi:10.2140/agt.2015.15.85

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The unknotting number and classical invariants, I

Maciej Borodzik and Stefan Friedl

Algebraic & Geometric Topology 15 (2015) 85–135

DOI: 10.2140/agt.2015.15.85

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/

Volume 15, issue 1 (2015)