HOMFLY-PT homology for general link diagrams and braidlike isotopy

Abel, Michael

doi:10.2140/agt.2017.17.3021

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HOMFLY-PT homology for general link diagrams and braidlike isotopy

Michael Abel

Algebraic & Geometric Topology 17 (2017) 3021–3056

DOI: 10.2140/agt.2017.17.3021

Abstract

Khovanov and Rozansky’s categorification of the homfly-pt polynomial is invariant under braidlike isotopies for any general link diagram and Markov moves for braid closures. To define homfly-pt homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reidemeister moves. In this text we prove that the Reidemeister IIb move fails in homfly-pt homology by using virtual crossing filtrations of the author and Rozansky. The decategorification of homfly-pt homology for general link diagrams gives a deformed version of the homfly-pt polynomial, $P^{b} (D)$ , which can be used to detect nonbraidlike isotopies. Finally, we will use $P^{b} (D)$ to prove that homfly-pt homology is not an invariant of virtual links, even when virtual links are presented as virtual braid closures.

Keywords

braidlike isotopy, Khovanov–Rozansky homology, virtual links

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

References

Publication

Received: 30 October 2016

Revised: 7 March 2017

Accepted: 27 March 2017

Published: 19 September 2017

Authors

Michael Abel
Department of Mathematics Duke University Durham, NC United States
http://services.math.duke.edu/~maabel/

Volume 17, issue 5 (2017)