A type D structure in Khovanov homology

Roberts, Lawrence P.

doi:10.48550/arxiv.1304.0463

Cited by 4 publications

(20 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [14], [15] the author, inspired by bordered Floer homology, describes a similar construction for tangles 1 . The formal algebraic structure of these tangle invariants mimics the formal structure of Ozsváth, Lipshitz, and Thurston's description of bordered Floer homology, [11], which also provided a road map for constructing a gluing theory for Khovanov homology.…”

Section: Background and Motivationmentioning

confidence: 99%

“…More details about this algebra will be given below. For this computation all that is required are the following properties, [14]:…”

Section: The Grothendieck Group Of Bγ Nmentioning

confidence: 99%

“…The vertices of the quiver defining BΓ n correspond to certain planar configurations of circles and decorations, called cleaved links in [14] and [15]. More specifically, let S be an oriented two-dimensional sphere.…”

Section: The Idempotent Sub-algebra Of Bγ Nmentioning

confidence: 99%

“…

In [14], [15] the author showed how to decompose the Khovanov homology of a link L into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for L is decomposed into the union of two tangles. Since Khovanov homology is the categorification of a version of the Jones polynomial, it is natural to ask what the type A and type D structures categorify, and how their pairing is encoded in the decategorifications.

…”

mentioning

confidence: 99%

“…To (D 1 , T 1 ) assign the bigraded differential module T 1 ]] described above. To (D 2 , T 2 ) the construction in[14] assigns a bigraded Abelian group [[ T 2 and a (1, 0) map δ : [[ T 2 −→ BΓ n ⊗ I [[ T , which satisfies the structure relation for a type D structure,[11].…”

mentioning

confidence: 99%

See 4 more Smart Citations

The decategorification of bordered Khovanov homology

Roberts

2014

J. Knot Theory Ramifications

Self Cite

View full text Add to dashboard Cite

Abstract. In [14], [15] the author showed how to decompose the Khovanov homology of a link L into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for L is decomposed into the union of two tangles. Since Khovanov homology is the categorification of a version of the Jones polynomial, it is natural to ask what the type A and type D structures categorify, and how their pairing is encoded in the decategorifications. In this paper, the author constructs the decategorifications of these two structures, in a manner similar to Ina Petkova's decategorification of bordered Floer homology, [13], and shows how they recover the Jones polynomial. We also give a new proof of the mutation invariance of the Jones polynomial which uses these decomposition techniques.

show abstract

Section: Background and Motivationmentioning

confidence: 99%

“…More details about this algebra will be given below. For this computation all that is required are the following properties, [14]:…”

Section: The Grothendieck Group Of Bγ Nmentioning

confidence: 99%

Section: The Idempotent Sub-algebra Of Bγ Nmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

The decategorification of bordered Khovanov homology

Roberts

2014

J. Knot Theory Ramifications

Self Cite

View full text Add to dashboard Cite

show abstract

Planar algebras and the decategorification of bordered Khovanov homology

Roberts

2017

J. Knot Theory Ramifications

Self Cite

View full text Add to dashboard Cite

Abstract. We give a simple, combinatorial construction of a unital, spherical, nondegenerate * -planar algebra over the ring Z[q 1/2 , q −1/2 ]. This planar algebra is similar in spirit to the Temperley-Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author's previous work on bordered Khovanov homology [10]. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology ([9]) and its difference from that in Khovanov's tangle homology, [5], without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of A. Manion, [7]. Furthermore, using Khovanov's conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in Σ × [−1, 1] where Σ is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov's approach to the Jones polynomial.

show abstract

Bordered theory for pillowcase homology

Kotelskiy

2019

Mathematical Research Letters

View full text Add to dashboard Cite

We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A∞ module M (L) over A. Then we prove that Lagrangian Floer homology HF (L, L ) is isomorphic to a suitable algebraic pairing of modules M (L) and M (L ). This extends the pillowcase homology construction given a 2-stranded tangle inside a 3-ball, if one obtains an immersed unobstructed Lagrangian inside the pillowcase, one can further associate an A∞ module to that Lagrangian.

show abstract

A type D structure in Khovanov homology

Cited by 4 publications

References 0 publications

The decategorification of bordered Khovanov homology

The decategorification of bordered Khovanov homology

Planar algebras and the decategorification of bordered Khovanov homology

Bordered theory for pillowcase homology

Contact Info

Product

Resources

About