Geodesic
HOMFLY-PT homology for general link diagrams and braidlike isotopy
Abel, Michael1
1 Department of Mathematics, Duke University, Durham, NC, United States
Algebraic and Geometric Topology, Tome 17 (2017) no. 5, pp. 3021-3056

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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.

DOI : 10.2140/agt.2017.17.3021
Classification : 57M25, 57M27
Keywords: braidlike isotopy, Khovanov–Rozansky homology, virtual links

Abel, Michael 1

1 Department of Mathematics, Duke University, Durham, NC, United States 
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Abel, Michael. HOMFLY-PT homology for general link diagrams and braidlike isotopy. Algebraic and Geometric Topology, Tome 17 (2017) no. 5, pp. 3021-3056. doi: 10.2140/agt.2017.17.3021

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