Braids, transversal links and the Khovanov-Rozansky Theory

Abstract

Abstract. We establish some inequalities for the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal links in standard contact S 3 which are sharper than the well-known bound given by the HOMFLY polynomial. We also introduce a sequence of transversal link invariants and discuss some of their properties.

Some differentials on Khovanov–Rozansky homology

“…To see this, we use an induction scheme introduced by Wu [25]. Suppose D is a braid graph on b strands.…”

“…The KR-complex of a braid graph satisfies decomposition rules analogous to the MOY relations O-III. In the context of Z/2-graded matrix factorizations, such rules were introduced in [11] and later applied to the HOMFLY homology in [12,25]. Similar MOY decompositions also hold in the derived category of Z-graded matrix factorizations.…”

Matrix factorizations and link homology II

“…(see Wu's grading assignment [15] Since each Reidemeister move is local, it is sufficient to prove that for each pair of Reidemeister tangles D 1 and D 2 their complexes C.D 1 / and C.D 2 / are homotopy equivalent. Here we encounter an obstacle: we can not prove the invariance under the Reidemeister move IIb.…”

“…Hao Wu [15] showed that a degree shift related to the first Reidemeister move can be removed if one allows half-integer values for the grading degrees. Here we follow his grading assignment.…”

On transverse invariants from Khovanov homology

“…The first candidate for such an invariant was introduced by Plamenevskaya [Pla06]: to a transverse link of topological type K, this associates a distinguished class in the Khovanov homology of K. Since Plamenevskaya's groundbreaking work, transverse invariants of a similar flavor have been discovered by Wu [Wu08] in sl n Khovanov-Rozansky homology, and by Ozsváth-Szabó-Thurston [OSzT08] and Lisca-Ozsváth-Stipsicz-Szabó [LOSS09] in knot Floer homology. (Knot contact homology produces a transverse invariant of a somewhat different flavor, called transverse homology [EENS,Ng11].…”