The Categorification of the Kauffman Bracket Skein Module Of

Gabrovšek, Boštjan

doi:10.1017/s0004972713000105

“…(1) 𝛼 APS = (F 2 , F 2 , 0, 0). For this choice of dyad, the unreduced chain complex 𝐶𝐾ℎ 𝛼APS • recovers the chain complex defined in [APS04] and [Gab13] for null homologous knots in RP 3 with coefficients F 2 . Their homology theories work for all knots in RP 3 , not only the null homologous ones.…”

mentioning

confidence: 89%

“…It could appear on the link projection with two crossings, or it could lie on the neglected part. First we quote the following result from [Gab13]: Up to symmetries, there are exactly six singular graph in RP 2 with 2 singular points, as shown in Figure 4. Now we replace each singular point by a positive or negative crossing to get a link with two crossings.…”

Section: Now We Define the Differential Mapmentioning

confidence: 99%

“…For 𝐼-bundles over non-orientable surfaces like RP 2 , their theory was defined with F 2 coefficients. In [Gab13], Gabrovšek extended the definition for links in RP 3 to Z coefficients by fixing a sign convention.…”

Section: Introductionmentioning

confidence: 99%

“…The Euler characteristic of the usual Khovanov homology for links in 𝑆 3 gives the unnormalized Jones polynomial of the link. For (not necessarily null homologous) framed links in RP 3 , [APS04] and [Gab13] construct some Khovanov-type homology whose Euler characteristic gives the Kauffman bracket ⟨𝐿⟩ of the framed link 𝐿, which is an element in the skein module 𝑆(RP 3 ) of RP 3 . 𝑆(RP 3 ) is free Z [︀ 𝐴 ±1 ]︀ -module over two generators.…”

mentioning

confidence: 99%

“…Their homology theories work for all knots in RP 3 , not only the null homologous ones. [APS04] proposes a F 2 version, and [Gab13] fixes some choice of signs so that it works Note that since we have the symmetry 𝛼 APS = 𝛼 * APS , we can pick the point 𝑃 anywhere in the complement of 𝐿 in RP 2 , and it will give the same chain complex, so 𝑃 plays no role in their homology.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Khovanov-type homologies of null homologous links in $\mathbb{RP}^3$

Chen

2021

Preprint

0

View full text Add to dashboard Cite

Let 𝐿 be a null homologous link in RP 3 . We define Khovanov-type homologies of 𝐿 which depend on an extra input 𝛼 = (𝑉 0 , 𝑉 1 , 𝑓, 𝑔) consisting of two graded vectors spaces and two maps between them. With some specific choice of 𝛼 = 𝛼 𝐴𝑃 𝑆 , we recover the categorification of the Kauffman bracket due to Asaeda-Przytycki-Sikora. With another choice of 𝛼 = 𝛼 𝐻𝐹 , we construct a spectral sequence from our theory converging to the Heegaard Floer homology of the even branched double cover of RP 3 .

show abstract

A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds

Manolescu

¹

,

Marengon

²

,

Sarkar

³

et al. 2023

View full text Add to dashboard Cite

We extend the definition of Khovanov-Lee homology to links in connected sums of S 1 × S 2 's, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in S 1 ×S 2 , we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: B 2 × S 2 , S 1 × B 3 , CP 2 , and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from B 4 by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard. Contents1. Introduction 2. Lee's deformation in # r (S 1 × S 2 ) 2.1. Notation and conventions 2.2. The deformed complex and Lee homology 2.3. Behavior under diffeomorphisms 2.4. Lee generators in KC Lee (T ∞ n ) and KC Lee (D) 3. A generalization of Rasmussen's invariant 3.1.

show abstract

A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds

Manolescu¹,

Marengon²,

Sarkar³

et al. 2019

Preprint

View full text Add to dashboard Cite

We extend the definition of Khovanov-Lee homology to links in connected sums of S 1 × S 2 's, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in S 1 ×S 2 , we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications, we prove inequalities relating the Rasmussen-type invariant to the genus of surfaces with boundary in the following four-manifolds: B 2 × S 2 , S 1 × B 3 , CP 2 , and various connected sums and boundary sums of these. We deduce that Rasmussen's invariant also gives genus bounds for surfaces inside homotopy 4-balls obtained from B 4 by Gluck twists. Therefore, it cannot be used to prove that such homotopy 4-balls are non-standard. Contents1. Introduction 2. Lee's deformation in # r (S 1 × S 2 ) 2.1. Notation and conventions 2.2. The deformed complex and Lee homology 2.3. Behavior under diffeomorphisms 2.4. Lee generators in KC Lee (T ∞ n ) and KC Lee (D) 3. A generalization of Rasmussen's invariant 3.1.

show abstract

The Categorification of the Kauffman Bracket Skein Module Of

Cited by 5 publications

References 10 publications

Khovanov-type homologies of null homologous links in $\mathbb{RP}^3$

Khovanov-type homologies of null homologous links in $\mathbb{RP}^3$

A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds

A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds

Contact Info

Product

Resources

About