Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 17, Number 5 (2017), 3021-3056.
HOMFLY-PT homology for general link diagrams and braidlike isotopy
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, , which can be used to detect nonbraidlike isotopies. Finally, we will use to prove that homfly-pt homology is not an invariant of virtual links, even when virtual links are presented as virtual braid closures.
Algebr. Geom. Topol., Volume 17, Number 5 (2017), 3021-3056.
Received: 30 October 2016
Revised: 7 March 2017
Accepted: 27 March 2017
First available in Project Euclid: 16 November 2017
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Abel, Michael. HOMFLY-PT homology for general link diagrams and braidlike isotopy. Algebr. Geom. Topol. 17 (2017), no. 5, 3021--3056. doi:10.2140/agt.2017.17.3021. https://projecteuclid.org/euclid.agt/1510841491