On a Poincaré polynomial from Khovanov homology and Vassiliev invariants

2019/05/14 by Noboru Ιτο, Ito, Noboru, Masaya Kameyama +1
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1905.05664

openalex publication_date 2019/05/14 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We introduce a Poincaré polynomial with two-variable t and x for knots, derived from Khovanov homology, where the specialization (t, x) = (1, -1) is a Vassiliev invariant of order n. Since for every n, there exist non-trivial knots with the same value of the Vassiliev invariant of order n as that of the unknot, there has been no explicit formulation of a perturbative knot invariant which is a coefficient of yⁿ by the replacement q=e^y for the quantum parameter q of a quantum knot invariant, and which distinguishes the above knots together with the unknot. The first formulation is our polynomial.

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