On a triplygraded generalization of Khovanov homology
 Title:
 On a triplygraded generalization of Khovanov homology
 Author(s):
 Putyra, Krzysztof
 Thesis Advisor(s):
 Khovanov, Mikhail G.
 Date:
 2014
 Type:
 Theses
 Degree:
 Ph.D., Columbia University
 Department(s):
 Mathematics
 Persistent URL:
 https://doi.org/10.7916/D86971RX
 Abstract:
 In this thesis we study a certain generalization of Khovanov homology that unifies both the original theory due to M. Khovanov, referred to as the even Khovanov homology, and the odd Khovanov homology introduced by P. Ozsv´ath, Z. Szab´o, and J. Rasmussen.
The generalized Khovanov complex is a variant of the formal Khovanov bracket introduced by Bar Natan, constructed in a certain 2categorical extension of cobordisms, in which the disjoint union is a cubical 2functor, but not a strict one. This allows us to twist the usual relations between cobordisms with signs or, more generally, other invertible scalars. We prove the homotopy type of the complex is a link invariant, and we show how both even and odd Khovanov homology can be recovered. Then we analyze other link homology theories arising from this construction such as a unified theory over the ring Z_p :=Z[p]/(p²−1), and a variant of the algebra of dotted cobordisms, defined over k := Z[X,Y,Z^±1]/(X² = Y² = 1).
The generalized chain complex is bigraded, but the new grading does not make it a stronger invariant. However, it controls up to some extend signs in the complex, the property we use to prove several properties of the generalized Khovanov complex such as multiplicativity with respect to disjoint unions and connected sums of links, and the duality between complexes for a link and its mirror image. In particular, it follows the odd Khovanov homology of anticheiral links is selfdual. Finally, we explore Bocksteintype homological operations, proving the unified theory is a finer invariant than the even and odd Khovanov homology taken together.
 Subject(s):
 Mathematics
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 Suggested Citation:
 Krzysztof Putyra, 2014, On a triplygraded generalization of Khovanov homology, Columbia University Academic Commons, https://doi.org/10.7916/D86971RX.