CALCULATING BAR-NATAN'S CHARACTERISTIC TWO KHOVANOV HOMOLOGY

By P. Turner. 2004

71 citations recorded by the source corpus.

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E2-pages can be described in terms of groups arising from the action of a certain endomorphism on 𝔽2-Khovanov homology. Some simple consequences are discussed.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/245bba19620ef80aa554592903b57d8ff1bba0b6

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A remark on rasmussen's invariant of knots (2005, background): “In characteristic two the condition h2 + 4t = γ2 6= 0 impliesh = γ = 1, and the basis change is α = x andβ = x + 1 which is the change of basis used in [8].”
A spectral sequence for Khovanov homology with an application to (3,q)–torus links (2006, background): “Moreover d1 can be understood explicitly as explained in [8].”
A refinement of Rasmussen's s-invariant (2012, methodology): “r-Natan’s formalcomplex-valued invariant [BN05, Deﬁnition 6.4] and some functor Cob3 /l → KomZ. With coeﬃcients in F2 instead of Zthis is essentially [BN05, Exercise 9.5], and is discussed further in [Tur06]. To obtain the result over Z, by [BN05, Theorem 5] it suﬃces verify that the topological ﬁel”
The universal sl3–link homology (2006, background): “1 Introduction In [8], following his own seminal work in [6] and Lee [11], Bar-Natan [2] and Turner’s [13] subsequent contributions, Khovanov classified all possible Frobenius systems of dimension two which give rise to link homologies via his construction in [6] and showed that there is a universal”
Homological thickness and stability of torus knots (2005, background): “On the other hand, every Eip,p+1 for i = 1, . . . , p− 2 is isotopic (by a sequence of R2 and R1 moves as explained in the proof of Lemma 1) to the diagram Di−1p−2,p−1, while E p−1 p,p+1 is isotopic to the diagram Dp−3p−2,p−1 ∐ U = Dp−2,p−2 ∐ U , where by U we denote the unknot.”
On the geography and botany of knot Floer homology (2014, background): “4 (Turner [94]) Given a knot K in S3 there is a spectral sequence with E1 ∼= K̃ h(K ) and E∞ ∼= F supported in grading u = δ + q = 0.”
Odd Khovanov homology (2007, unknown): “This computation is somewhat disappointing, since it indicates that Kh cannot possess a cancelling differential analogous to the Lee differential [8] and its generalizations [13].”