A recursive formula for the Khovanov cohomology of Kanenobu knots
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Kanenobu has given infinite families of knots with the same HOMFLY polynomial invariant but distinct Alexander module structure. In this paper, we give a recursive formula for the Khovanov cohomology of all Kanenobu knots $K(p,q)$, where $p$ and $q$ are integers. The result implies that the rank of the Khovanov cohomology of $K(p,q)$ is an invariant of $p+q$. Our computation uses only the basic long exact sequence in knot homology and some results on homologically thin knots.
Abstract
Kanenobu has given infinite families of knots with the same HOMFLY polynomial invariant but distinct Alexander module structure. In this paper, we give a recursive formula for the Khovanov cohomology of all Kanenobu knots $K(p,q)$, where $p$ and $q$ are integers. The result implies that the rank of the Khovanov cohomology of $K(p,q)$ is an invariant of $p+q$. Our computation uses only the basic long exact sequence in knot homology and some results on homologically thin knots.
참고문헌 (26)
- J. A. Baldwin, "Computations of Heegaard-Floer knot homology", ~ (2007)
- D. Bar-Natan, "On Khovanov’s categorification of the Jones polynomial", Algebr. Geom. Topol, 2, pp.337~370 (2002)
- N. Carqueville, "Computing Khovanov-Rozansky homology and defect fusion", Algebr. Geom. Topol, 14(1), pp.489~537 (2014)
- J. Greene, "Homologically thin, non-quasi-alternating links", Math. Res. Lett, 17(1), pp.39~49 (2010)
- J. Greene, "Turaev Torsion, definite 4-manifolds and quasi-alternating knots", Bull. Lond. Math. Soc, 45(5), pp.962~972 (2013)
- T. Kanenobu, "Infinitely many knots with the same polynomial invariant", Proc. Amer. Math. Soc, 97(1), pp.158~162 (1986)
- T. Kanenobu, "Examples on polynomial invariants of knots and links", Math. Ann, 275(4), pp.555~572 (1986)
- M. Khovanov, "A categorification of the Jones polynomial", Duke Math. J, 101(3), pp.359~426 (2000)
- M. Khovanov, "Matrix factorizations and link homology", Fund. Math, 199(1), pp.1~91 (2008)
- M. Khovanov, "Matrix factorizations and link homology II", Geom. Topol, 12(3), pp.1387~1425 (2008)
- E. S. Lee, "On Khovanov invariant for alternating links" (2003)
- E. S. Lee, "An endomorphism of the Khovanov invariant", Adv. Math, 197, pp.554~586 (2005)
- A. Lobb, "The Kanenobu knots and Khovanov-Rozansky homology", Proc. Amer. Math. Soc, 142(4), pp.1447~1455 (2014)
- C. Manolescu, "On the Khovanov and knot Floer homologies of quasi-alternating links", proceedings of the 14th Gökova Geometry-Topology Conference, pp.60~81 (2007), International Press
- P. Ozsv´ath, "Odd Khovanov homology", Algebr. Geom. Topol, 13(3), pp.1465~1488 (2013)
- P. Ozsv´ath, "Holomorphic disks and topological invariants for closed three-manifolds", Ann. of Math, 159(3), pp.1027~1158 (2004)
- P. Ozsv´ath, "On the Heegaard Floer homology of branched double-covers", Adv. Math, 194(1), pp.1~33 (2005)
- K. Qazaqzeh, "A new obstruction of quasi-alternating links", Algebr. Geom. Topol, 15(3), pp.1847~1862 (2015)
- J. A. Rasmussen, "Floer homology and knot complements", Ph.D, pp.2003
- J. A. Rasmussen, "Khovanov homology and the slice genus", Invent. Math, 182(2), pp.419~447 (2010)
- P. Turner, "Calculating Bar-Natan’s characteristic two Khovanov homology", J. Knot Theory Ramifications, 15(10), pp.1335~1356 (2006)
- F. Vafaee, "On the Knot Floer Homology of Twisted Torus Knots", Int. Math. Res. Not, 2015(15), pp.6516~6537 (2015)
- L. Watson, "Any tangle extends to non-mutant knots with the same Jones polynomial", J. Knot Theory Ramifications, 15(9), pp.1153~1162 (2006)
- L. Watson, "Knots with identical Khovanov homology", Algebra. Geom. Topol, 7, pp.1389~1407 (2007)