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On a triply-graded generalization of...

Putyra, Krzysztof K.

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On a triply-graded generalization of Khovanov homology.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	On a triply-graded generalization of Khovanov homology./
Author:	Putyra, Krzysztof K.
Description:	122 p.
Notes:	Source: Dissertation Abstracts International, Volume: 75-08(E), Section: B.
Contained By:	Dissertation Abstracts International75-08B(E).
Subject:	Theoretical Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3620413
ISBN:	9781303908583

On a triply-graded generalization of Khovanov homology.
Putyra, Krzysztof K.

On a triply-graded generalization of Khovanov homology. - 122 p.

Source: Dissertation Abstracts International, Volume: 75-08(E), Section: B.

Thesis (Ph.D.)--Columbia University, 2014.

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. Ozsvath, Z. Szabo, and J. Rasmussen.

ISBN: 9781303908583Subjects--Topical Terms:

1672766
Theoretical Mathematics.

On a triply-graded generalization of Khovanov homology.
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100 1 $a Putyra, Krzysztof K. $3 2105009
245 1 0 $a On a triply-graded generalization of Khovanov homology.
300 $a 122 p.
500 $a Source: Dissertation Abstracts International, Volume: 75-08(E), Section: B.
500 $a Adviser: Mikhail Khovanov.
502 $a Thesis (Ph.D.)--Columbia University, 2014.
520 $a 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. Ozsvath, Z. Szabo, and J. Rasmussen.
520 $a The generalized Khovanov complex is a variant of the formal Khovanov bracket introduced by Bar Natan, constructed in a certain 2-categorical extension of cobordisms, in which the disjoint union is a cubical 2-functor, 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 pi := Z [pi]/(pi2 - 1), and a variant of the algebra of dotted cobordisms, defined over k := Z [X,Y,Z+/-1]/(X 2 = Y2 = 1).
520 $a 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 self-dual. Finally, we explore Bockstein-type homological operations, proving the unified theory is a finer invariant than the even and odd Khovanov homology taken together.
590 $a School code: 0054.
650 4 $a Theoretical Mathematics. $3 1672766
650 4 $a Mathematics. $3 515831
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710 2 $a Columbia University. $b Mathematics. $3 1684292
773 0 $t Dissertation Abstracts International $g 75-08B(E).
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791 $a Ph.D.
792 $a 2014
793 $a English
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