2013/08/31 by Hao Wu
Mathematics · #math.GT #math.SG #msc:57M25 #msc:57R17
paper · pdf · doi:10.2140/agt.2016.16.41
published as Algebr. Geom. Topol. 16 (2016) 41-127 · 56 pages, 24 figures. Second version: corrected typos and minor mistakes
arxiv created 2014/04/07 · arxiv updated 2016/03/09
We define a homology HN for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential axN+1. Up to a grading shift, H0 is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for N ≥ 1, HN is a ℤ2⊕ℤ⊕ 3-graded ℚ[a]-module that is invariant under transverse Markov moves, but not under negative stabilization/de-stabilization. Thus, for N≥ 1, this homology is an invariant for transverse links in the standard contact S3, but not for smooth links. We also discuss the decategorification of HN and the relation between HN and the \mathfraksl(N) Khovanov-Rozansky homology defined in arXiv:math/0401268.