Virtual Khovanov homology using cobordisms

Abstract

We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's $\mathbb{Z}/2$-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a Mathematica based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology.

78 pages, lots of figures, lots of typos, rewritten version, merged with arXiv:1212.0185, added referee's suggestions, to appear in J. Knot Theor. Ramif

References in corpus (9)

• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)
• The sl_3 web algebra
• The Karoubi envelope and Lee's degeneration of Khovanov homology
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n)
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• The sl(N)-web algebras and dual canonical bases
• A large family of indecomposable projective modules for the Khovanov-Kuperberg algebra of $s{l}_3$-webs
• The Khovanov Complex for Virtual Links

Cited by in corpus (7)

• Computations of the slice genus of virtual knots
• Unoriented Virtual Khovanov Homology
• Steenrod square for virtual links toward Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links
• Khovanov-Lipshitz-Sarkar homotopy type for links in thickened surfaces
• Ascent concordance
• Khovanov-Lipshitz-Sarkar homotopy type for links in thickened higher genus surfaces
• On the virtual Rasmussen invariant