Abstract: We introduce a Poincaré polynomial in two-variables 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 = ey for the quantum parameter q of a quantum knot invariant, which distinguishes the above knots together with the unknot. The first formulation is our polynomial.
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Keywords and phrases: Jones polynomial, Vassiliev invariant, Khovanov polynomial.
Received: April 2, 2022; Accepted: May 12, 2022; Published: June 17, 2022
How to cite this article: Noboru Ito and Masaya Kameyama, On a Poincaré polynomial from Khovanov homology and Vassiliev invariants, JP Journal of Geometry and Topology 27 (2022), 33-48. http://dx.doi.org/10.17654/0972415X22003
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References:
[1] Dror Bar-Natan et al., The Knot Atlas, http://katlas.org. [2] J. S. Birman and X.-S. Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111 (1993), 225-270. [3] T. Kanenobu and Y. Miyazawa, HOMFLY polynomials as Vassiliev invariants, Knot Theory (Warsaw, 1995), Banach Center Publ., 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 165-185. [4] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), 359-426. [5] Y. Ohyama, Vassiliev invariants and similarity of knots, Proc. Amer. Math. Soc. 123 (1995), 287-291. [6] V. A. Vassiliev, Cohomology of knot spaces, Theory of Singularities and its Applications, Adv. Soviet Math., 1, Amer. Math. Soc., Providence, RI, 1990, pp. 23-69. [7] O. Viro, Khovanov homology, its definitions and ramifications, Fund. Math. 184 (2004), 317-342. [8] M. Aganagic and S. Shakirov, Knot homology and refined Chern-Simons index, Comm. Math. Phys. 333(1) (2015), 187-228.
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