#### 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
Primary: 57M27
##### 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/