Annular Evaluation and Link Homology

Abstract

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 ${\mathfrak{gl}}_{-n}$ link homology, i.e. a link homology theory associated to the Lie superalgebra ${\mathfrak{gl}}_{0|n}$, both for links in $S^3$ and in the thickened annulus. In the $n=2$ case, this produces a categorification of the Jones polynomial that we show is distinct from Khovanov homology, and gives a finite-dimensional categorification of the colored Jones polynomial. This behavior persists for general $n$. Our approach yields simple constructions of spectral sequences relating these theories, and emphasizes the roles of super vector spaces, categorical traces, and current algebras in link homology.

arXiv comment47 pages, many figures

References in corpus (21)

• A diagrammatic approach to categorification of quantum groups I
• 2-Kac-Moody algebras
• Khovanov's homology for tangles and cobordisms
• Knot invariants and higher representation theory
• Categorification of the Kauffman bracket skein module of I-bundles over surfaces
• Sl(N) link homology using foams and the Kapustin-Li formula
• Category O and sl(k) link invariants
• A colored sl(N)-homology for links in S^3
• Super $q$-Howe duality and web categories
• Categorification of the braid groups
• A geometric construction of colored HOMFLYPT homology
• Deformations of colored sl(N) link homologies via foams
• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)
• Categorified quantum sl(2) and equivariant cohomology of iterated flag varieties
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• A categorification of the Jones polynomial
• Clasp technology to knot homology via the affine Grassmannian
• Cyclicity for categorified quantum groups
• A diagrammatic categorification of the q-Schur algebra
• Equivariant colored sl(N)-homology for links
• A Categorification of Quantum sl_3 Projectors and the sl_3 Reshetikhin-Turaev Invariant of Tangles

Cited by in corpus (10)

• A quantum categorification of the Alexander polynomial
• On some $p$-differential graded link homologies
• Anchored foams and annular homology
• Braid group actions from categorical symmetric Howe duality on deformed Webster algebras
• On webs in quantum type $C$
• A categorification of the colored Jones polynomial at a root of unity
• Foams, iterated wreath products, field extensions and Sylvester sums
• Symmetries of ${\mathfrak{gl}}_N$-foams
• A skein relation for singular Soergel bimodules
• Link splitting deformation of colored Khovanov--Rozansky homology