On transverse invariants from Khovanov homology

Abstract

Abstract. In [Pla06], O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation (from [BN05]) of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum (from [LSa]). We show that the first of these refinements is invariant under negative flypes and SZ moves; this implies that Plamenevskaya's class is also invariant under these moves. We go on to show that for small-crossing transverse knots K, both refinements are determined by the classical invariants of K.

Positivities of Knots and Links and the Defect of Bennequin Inequality

“…Transverse links T and T ′ are called flype-equivalent [28] if their braid representatives are related to each other by negative flypes, positive stabilizations, positive destabilizations and braid isotopies (the last three operations are just the transverse isotopy). Many known transverse links with the same topological type and the self-linking number are flype equivalent.…”

Topological quantum computation of the Dold-Thom functor

“…The evolution in this case is given by (27) The (27) is interpreted as a topological quantum computer which is able to compute indexes for operators. Connecting (9) and (27) we obtain a possible quantum algorithm for the functor Dold-Thom in algebraic geometry:…”

“…In the context of the algebraic geometry we use the following adaptation (9) where ch is the Chern character-functor and TB(X) is the tangent bundle form the space-manifold X. Some examples of (9) are as follows.…”

“…Recent progress in topological quantum computation [1] have been oriented towards the possible quantum algorithms for Khovanov homology and its generalizations [2,3,4] including Khovanov Homotopy [5,6,7,8,9,10] . In Khovanov homotopy a stable space is associated to each link in such way that the classical topological invariants of the space are given the Khovanov homology of the link.…”

Stability and triviality of the transverse invariant from Khovanov homology

“…It is an open question whether ψ is an effective transverse invariant. Several efforts [LNS15,Wu08,HS16,Col17] have been made to both understand the effectiveness of ψ and to define new invariants related to ψ in the hope that one of these would be effective. Thus far these efforts have not yielded any transverse invariants arising from Khovanov-type constructions that are known to be effective or not.…”