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A functor-valued invariant of tangles

Mikhail Khovanov

Algebraic & Geometric Topology 2 (2002) 665–741

arXiv: math.QA/0103190

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.

Keywords
tangles, Jones polynomial, Kauffman bracket, TQFT, complexes, bimodules
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27, 16D20, 18G60
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Publication
Received: 21 February 2002
Accepted: 25 April 2002
Published: 6 September 2002
Authors
Mikhail Khovanov
Department of Mathematics
University of California
Davis, CA 95616
USA