A functor-valued invariant of tangles

By Mikhail Khovanov. 2001

315 citations recorded by the source corpus.

We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this invariant descends to the Kauffman bracket of the tangle. When the tangle is a link, the invariant specializes to the bigraded cohomology theory introduced in our earlier work.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cbdacaf0e87e4964e37afd5635f3b57a1fb7040d

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Citations in the corpus

Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology (2006, methodology): “The graded vector space Hn = ⊕ a,b∈Cup(n) bHna〈n〉 with the above multiplication becomes a positively graded algebra [Kho02].”
Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links (2005, background): “We regard FLee([L1]) and FLee([L2]) as V – modules by choosing distinguished points on L1 and L2 near the connected sum point (see [5] for details, where the notation H(1) is used instead of V , and where the topological grading is opposite to the the topological grading used in this paper).”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, background): “This invariant was later recast by Bar-Natan [1] in terms of the category of tangles and cobordisms (see also [22]).”
CROSSINGLESS MATCHINGS AND THE COHOMOLOGY OF (n, n) SPRINGER VARIETIES (2002, unknown): “ar X iv :m at h/ 02 02 11 0v 1 [ m at h.”
Problems on invariants of knots and 3-manifolds (2004, background): “Khovanov [217, 218] defined certain homology groups of a knot whose Euler characteristic is equal to the Jones polynomial, which is called the categorification of the Jones polynomial.”
Man and machine thinking about the smooth 4-dimensional Poincaré conjecture (2009, background): “It should also be possible to use the more algebraic version described in [25], for which the chain complexes live in categories of bimodules over certain rings.”
A link invariant from the symplectic geometry of nilpotent slices (2004, background): “More spectacularly, they can be shown to fit into a topological quantum field theory for two-knots in four-space [18, 13, 16].”
Highest weight categories arising from Khovanov's diagram algebra IV: the general linear supergroup (2009, background): “ralised Khovanov algebras 28 5. Direct limits 33 References 41 1. Introduction This is the culmination of a series of four articles studying various generalisations of Khovanov’s diagram algebra from [K]. The goal is to relate the limiting version H∞ k of this algebra constructed in [BS1] to blocks ”
Frobenius algebras and ambidextrous adjunctions (2005, background): “For example, instances of categorified Frobenius structure have appeared in 3D topological quantum field theory [37], Khovanov homology — the homology theory for tangle cobordisms generalizing the Jones polynomial [20], and the theory of thick tangles [26].”
A Spanning Tree Model for Khovanov Homology (2004, background): “By [Kh-2], there is an isomorphism C(D1#D2) ∼= C(D1) ⊗H1 C(D2){−1}.”