Transverse knots and Khovanov homology

By O. Plamenevskaya. 2004

127 citations recorded by the source corpus.

We define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives a bound on the self-linking number in terms of Rasmussen's invariant s(K). We prove that our invariant vanishes for transverse knot stabilizations, and that it is non-zero for quasipositive braids. We also discuss a connection to Heegaard Floer invariants.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/1e4edadb085a1906f1ba2b4717af1e9e0f6d7849

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Citations in the corpus

Legendrian knots and exact Lagrangian cobordisms (2012, background): “f the 5 exact Lagrangian cobordisms L1,...,L5 constructed in Section 8.1, we can assign an element of Kh(Λ). For each Li, the element ΨL i (1) coincides with Plamenevskaya’s transverse knot invariant [Pl], and also lies on Ng’s [Ng] line j −i = C, where C = {max(j −i) | Khi,j(L) 6= 0 }. By Remark 8.”
A Legendrian Thurston-Bennequin bound from Khovanov homology (2005, background): “• tb(K) + |r(K)| ≤ 2g(K) − 1, Bennequin’s original result [2], where g is the genus of K ; • tb(K) + |r(K)| ≤ 2g4(K) − 1 [23], where g4 is the slice genus of K ; • tb(K) + |r(K)| ≤ min-dega HK(a, z) − 1 [8, 16], where HK is the HOMFLYPT polynomial and min-dega is the minimum degree in the variable a”
Transverse knots distinguished by knot Floer homology (2007, methodology): “We remark that the Plamenevskaya invariant of transverse knots from Khovanov homology [29] vanishes for L±1 = L ∓ 2 and (L ′ 1) ± = (L′2) , because in each case it lies in a trivial graded summand of Khovanov homology.”
On the spectral sequence from Khovanov homology to Heegaard Floer homology (2008, methodology): “In [21], Plamenevskaya defines an invariant of transverse links in the tight contact 3-sphere (S3, ξrot) using Khovanov homology.”
On knot Floer homology in double branched covers (2007, background): “For braids in Q′ we can be more precise about Plamenevskaya’s element: Corollary 3.”
Legendrian knots, transverse knots and combinatorial Floer homology (2006, unknown): “53D12, 57R17, 57R58; 57M25”
On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, background): “In [35], Plamenevskaya defines a transverse invariant in reduced Khovanov homology that associates to a transverse braid K .”
The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, background): “HEW HEDDEN AND PHILIP ORDING (4) Quasipositive knots: s(K) = 2τ(K) = 2g4(K), where g4(K) denotes the smooth slice genus of K. This follows from work of Plamenevskaya [27] for τ and from Plamenevskaya [28] and Shumakovitch [34] for s. See also [7]. (5) Knots with up to 10 crossings. [5, 29, 22] (6) “”