Algebraic & Geometric Topology

A functor-valued invariant of tangles

Mikhail Khovanov

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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.

Article information

Source
Algebr. Geom. Topol., Volume 2, Number 2 (2002), 665-741.

Dates
Received: 21 February 2002
Accepted: 25 April 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882739

Digital Object Identifier
doi:10.2140/agt.2002.2.665

Mathematical Reviews number (MathSciNet)
MR1928174

Zentralblatt MATH identifier
1002.57006

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds 16D20: Bimodules 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]

Keywords
tangles Jones polynomial Kauffman bracket TQFT complexes bimodules

Citation

Khovanov, Mikhail. A functor-valued invariant of tangles. Algebr. Geom. Topol. 2 (2002), no. 2, 665--741. doi:10.2140/agt.2002.2.665. https://projecteuclid.org/euclid.agt/1513882739


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