David E. V. Rose
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View article: A Kirby color for Khovanov homology
A Kirby color for Khovanov homology Open
We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we d…
View article: Link splitting deformation of colored Khovanov–Rozansky homology
Link splitting deformation of colored Khovanov–Rozansky homology Open
We introduce a multiparameter deformation of the triply‐graded Khovanov–Rozansky homology of links colored by one‐column Young diagrams, generalizing the “‐ified” link homology of Gorsky–Hogancamp and work of Cautis–Lauda–Sussan. For each …
View article: Spin Link Homology
Spin Link Homology Open
We put a new spin on Khovanov--Rozansky homology. That is, we equip $Λ^n$-colored $\mathfrak{sl}_{2n}$ Khovanov--Rozansky homology with an involution whose $\pm 1$-eigenspaces are link invariants. When $n=1,2,3$ (and assuming technical con…
View article: Categorification of the internal braid group action for quantum groups, I: 2-functoriality
Categorification of the internal braid group action for quantum groups, I: 2-functoriality Open
We define 2-functors on the categorified quantum group of a simply-laced Kac-Moody algebra that induce Lusztig's internal braid group action at the level of the Grothendieck group.
View article: Bordered invariants from Khovanov homology
Bordered invariants from Khovanov homology Open
To every compact oriented surface that is composed entirely out of 2-dimensional 0- and 1-handles, we construct a dg category using structures arising in Khovanov homology. These dg categories form part of the 2-dimensional layer (a.k.a. m…
View article: A Kirby color for Khovanov homology
A Kirby color for Khovanov homology Open
We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we d…
View article: Categorification of the internal braid group action for quantum groups I: 2-functoriality
Categorification of the internal braid group action for quantum groups I: 2-functoriality Open
We define 2-functors on the categorified quantum group of a simply-laced Kac-Moody algebra that induce Lusztig's internal braid group action at the level of the Grothendieck group.
View article: A note on the Sundaram--Stanley bijection (or, Viennot for up-down tableaux)
A note on the Sundaram--Stanley bijection (or, Viennot for up-down tableaux) Open
We give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a $2k$-fold tensor product of the vector representation of $\mathfrak{sp}_{2n}$ equals the number of $(n+1)$-avoiding match…
View article: Link splitting deformation of colored Khovanov--Rozansky homology
Link splitting deformation of colored Khovanov--Rozansky homology Open
We introduce a multi-parameter deformation of the triply-graded Khovanov--Rozansky homology of links colored by one-column Young diagrams, generalizing the "$y$-ified" link homology of Gorsky--Hogancamp and work of Cautis--Lauda--Sussan. F…
View article: A skein relation for singular Soergel bimodules
A skein relation for singular Soergel bimodules Open
We study the skein relation that governs the HOMFLYPT invariant of links colored by one-column Young diagrams. Our main result is a categorification of this colored skein relation. This takes the form of a homotopy equivalence between two …
View article: Type $C$ Webs
Type $C$ Webs Open
We define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{sp}_{2n})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{sp}_{2n})$ tensor-generated by the funda…
View article: Type $C$ Webs.
Type $C$ Webs. Open
We define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{sp}_{2n})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{sp}_{2n})$ tensor-generated by the funda…
View article: On webs in quantum type $C$
On webs in quantum type $C$ Open
We study webs in quantum type $C$, focusing on the rank three case. We define a linear pivotal category $\mathbf{Web}(\mathfrak{sp}_6)$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\ma…
View article: The sl n foam 2-category: A combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality
The sl n foam 2-category: A combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality Open
We give an elementary construction of colored sln link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between sln webs; applying a (TQFT-like) representable functor recovers (…
View article: Annular Evaluation and Link Homology
Annular Evaluation and Link Homology Open
We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield $\mathfrak{gl}…
View article: Sutured annular Khovanov-Rozansky homology
Sutured annular Khovanov-Rozansky homology Open
We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorificati…
View article: Deformations of colored 𝔰𝔩N link homologies via foams
Deformations of colored 𝔰𝔩N link homologies via foams Open
We prove a conjectured decomposition of deformed [math] link homology, as well as an extension to the case of colored links, generalizing results of Lee, Gornik, and Wu. To this end, we use foam technology to give a completely combinatoria…