Algebraic & Geometric Topology

HOMFLY-PT homology for general link diagrams and braidlike isotopy

Michael Abel

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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.

Article information

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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

braidlike isotopy Khovanov–Rozansky homology virtual links


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.

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