Geometry & Topology

Khovanov's homology for tangles and cobordisms

Dror Bar-Natan

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We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2–knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a “TQFT”) to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.

Article information

Geom. Topol., Volume 9, Number 3 (2005), 1443-1499.

Received: 3 November 2004
Accepted: 4 July 2005
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

2–knots canopoly categorification cobordism Euler characteristic Jones polynomial Kauffman bracket Khovanov knot invariants movie moves planar algebra skein modules tangles trace groups


Bar-Natan, Dror. Khovanov's homology for tangles and cobordisms. Geom. Topol. 9 (2005), no. 3, 1443--1499. doi:10.2140/gt.2005.9.1443.

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