We show that the generalized Khovanov homology defined by the second author in the framework of chronological cobordisms admits a grading by the group ℤ × ℤ2, in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring ℤπ := ℤ[π]∕(π2 − 1). (Here, setting π to ± 1 results either in even or odd Khovanov homology.) The generalized homology has k := ℤ[X,Y,Z±1]∕(X2=Y 2=1) as coefficients, and the above implies that most automorphisms of k fix the isomorphism class of the generalized homology regarded as a k–module, so that the even and odd Khovanov homology are the only two specializations of the invariant. In particular, switching X with Y induces a derived isomorphism between the generalized Khovanov homology of a link L with its dual version, ie the homology of the mirror image L!, and we compute an explicit formula for this map. When specialized to integers it descends to a duality isomorphism for odd Khovanov homology, which was conjectured by A Shumakovitch.
Keywords: Khovanov homology, odd Khovanov homology, mirror knot
Putyra, Krzysztof 1 ; Lubawski, Wojciech 2
@article{10_2140_agt_2016_16_2021,
author = {Putyra, Krzysztof and Lubawski, Wojciech},
title = {Mirror links have dual odd and generalized {Khovanov} homology},
journal = {Algebraic and Geometric Topology},
pages = {2021--2044},
year = {2016},
volume = {16},
number = {4},
doi = {10.2140/agt.2016.16.2021},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.2021/}
}
TY - JOUR AU - Putyra, Krzysztof AU - Lubawski, Wojciech TI - Mirror links have dual odd and generalized Khovanov homology JO - Algebraic and Geometric Topology PY - 2016 SP - 2021 EP - 2044 VL - 16 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.2021/ DO - 10.2140/agt.2016.16.2021 ID - 10_2140_agt_2016_16_2021 ER -
%0 Journal Article %A Putyra, Krzysztof %A Lubawski, Wojciech %T Mirror links have dual odd and generalized Khovanov homology %J Algebraic and Geometric Topology %D 2016 %P 2021-2044 %V 16 %N 4 %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.2021/ %R 10.2140/agt.2016.16.2021 %F 10_2140_agt_2016_16_2021
Putyra, Krzysztof; Lubawski, Wojciech. Mirror links have dual odd and generalized Khovanov homology. Algebraic and Geometric Topology, Tome 16 (2016) no. 4, pp. 2021-2044. doi: 10.2140/agt.2016.16.2021
[1] , Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443
[2] , A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359
[3] , An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006) 315
[4] , , , Odd Khovanov homology, Algebr. Geom. Topol. 13 (2013) 1465
[5] , Cobordisms with chronologies and a generalization of the Khovanov complex, master’s thesis, Jagiellonian University, Krakow (2008)
[6] , A 2–category of chronological cobordisms and odd Khovanov homology, from: "Knots in Poland III, Part 3" (editors J H Przytycki, P Traczyk), Banach Center Publ. 103, Polish Acad. Sci. Inst. Math. (2014) 291
[7] , Patterns in odd Khovanov homology, J. Knot Theory Ramifications 20 (2011) 203
Cité par Sources :