ON THE QUANTUM FILTRATION OF THE UNIVERSAL sl(2) FOAM COHOMOLOGY

Caprau, Carmen

doi:10.1142/s0218216511009595

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…As noted above we have suppressed the ALD inside of the complex to simplify the diagrams. Proposition 6.3 (Generalization of Rasmussen [39] Prop 4.1 / See also Caprau [6] Prop 3). Let C be an elementary link cobordism from K to K ′ where K and K ′ are related by a birth, death or saddle, then C induces a homomorphism of the canonical generators of Kh(K) and Kh(K ′ ) satisfying:…”

Section: Proof Of Invariancementioning

confidence: 97%

“…Corollary 6.4 (Generalization of Rasmussen [39] Cor 4.2 / See also Caprau [6] Cor 2). Let C be a connected virtual knot cobordism from K to K ′ , then C induces an isomorphism of the canonical generators of Kh(K) and Kh(K ′ ).…”

mentioning

confidence: 93%

“…Generalization of Rasmussen[39] Prop 2.3 / See also Caprau[6] Prop 2). Let C be an elementary link cobordism from D to D ′ where D to D ′ are related by a Reidemeister move.…”

mentioning

confidence: 99%

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Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots

Dye¹,

Kaestner²,

Kauffman³

2014

Preprint

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The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knot and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25] [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial are nonclassical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov-Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen's result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in S g × I × I in the sense of Turaev [42].

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Section: Proof Of Invariancementioning

confidence: 97%

mentioning

confidence: 93%

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Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots

Dye¹,

Kaestner²,

Kauffman³

2014

Preprint

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“…Many others have since modeled the algebraic construction of Rasmussen's invariant to produce more concordance invariants, many of which are generalizations of the s-invariant to sl(n) homology ( [32], [54], [33], [29]) or to the universal sl(2) homology [9].…”

Section: A Survey Of Applications Of Khovanov Homologymentioning

confidence: 99%

On Khovanov Homology and Related Invariants

Caprau¹,

González²,

Lee³

et al. 2020

Preprint

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This paper begins with a survey of some applications of Khovanov homology to lowdimensional topology, with an eye toward extending these results to sl(n) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to sl(n) homology for n ≥ 2, including the universal sl(2) and sl(3) homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.

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