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2017 HOMFLY-PT homology for general link diagrams and braidlike isotopy
Michael Abel
Algebr. Geom. Topol. 17(5): 3021-3056 (2017). DOI: 10.2140/agt.2017.17.3021


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, Pb(D), which can be used to detect nonbraidlike isotopies. Finally, we will use Pb(D) to prove that homfly-pt homology is not an invariant of virtual links, even when virtual links are presented as virtual braid closures.


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Michael Abel. "HOMFLY-PT homology for general link diagrams and braidlike isotopy." Algebr. Geom. Topol. 17 (5) 3021 - 3056, 2017.


Received: 30 October 2016; Revised: 7 March 2017; Accepted: 27 March 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06791392
MathSciNet: MR3704251
Digital Object Identifier: 10.2140/agt.2017.17.3021

Primary: 57M25, 57M27

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.17 • No. 5 • 2017
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