Abstract
We study a Khovanov type homology close to the original Khovanov homology theory from Frobenius system. The homology is an invariant for oriented links up to isotopy by applying a tautological functor on the geometric complex. The homology has also geometric descriptions by introducing the genus generating operations. We prove that Jones Polynomial is equal to a suitable Euler characteristic of the homology groups. As an application, we compute the homology groups of (2, k)-torus knots for every k ∈ N.
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References
Bar-Natan, D.: Khovanov’s homology for tangles and cobordisms. Geom. Topol., 9, 1443–1499 (2005)
Kock, J.: Frobenius algebra and 2D topological quantum field theories. London Mathematical Society Student Texts, 59, Cambridge University Press, Cambridge, xiv+240 pp., 2004
Kauffman, L. H.: State models and the Jones polynomial. Topology, 26, 395–407 (1987)
Kadison, L.: New examples of Frobenius extensions. University Lecture Series, 14, American Mathematical Society, Providence, RI,x+84 pp., 1999
Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J., 101, 359–426 (2000)
Khovanov, M.: Link homology and Frobenius extensions. Fund. Math., 190, 179–190 (2006)
Naot, G.: The universal Khovanov link homology theory. Algebr. Geom. Topol., 6, 1863–1892 (2006)
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Supported by NSFC (Grant Nos. 11329101 and 11431009)
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Zhang, M.L., Lei, F.C. A Khovanov type link homology with geometric interpretation. Acta. Math. Sin.-English Ser. 32, 393–405 (2016). https://doi.org/10.1007/s10114-016-4775-1
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DOI: https://doi.org/10.1007/s10114-016-4775-1