A functor-valued invariant of tangles

Abstract

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.

Gauge theory and knot homologies

“…Moreover, on manifolds with corners, it is expected that F extends to a 2-functor from the 2-category of oriented and decorated 4-manifolds with corners to the 2-category of triangulated categories [19,20,21]. In particular, it should associate:…”

Patterns in Knot Cohomology, I

“…Familiarity with [Kh1,2] or [BN] is assumed. Warning: we use the grading conventions of [Kh2], and the cohomology group that we denote by H i,j is denoted H i,−j in [BN] and [Kh1]. Let h i,j (K) (or simply h i,j ) be the rank of H i,j (K).…”

Categorification of level two representations of quantum $sl_n$ via generalized arc rings

“…Arc ring A n k;k . We first recall the definition of H n from [4]. Let A be a free abelian group of rank 2 spanned by 1 and X with 1 in degree 1 and X in degree 1.…”

“…Khovanov constructed in [4] a family of rings H n , for n 0, which is a categorification of Inv(n), the space of U q .sl 2 /-invariants in V˝2 n : These rings lead to an invariant of (even) tangles which to a tangle assigns a complex of .H n ; H m /-bimodules, up to chain homotopy equivalence. Khovanov and the author [1] built subquotients of H n and used them to categorify the action of tangles on V˝n: The same rings were also introduced by Stroppel [6].…”