## Geometry & Topology

### Khovanov's homology for tangles and cobordisms

Dror Bar-Natan

#### Abstract

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

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

Dates
Accepted: 4 July 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799641

Digital Object Identifier
doi:10.2140/gt.2005.9.1443

Mathematical Reviews number (MathSciNet)
MR2174270

Zentralblatt MATH identifier
1084.57011

#### Citation

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. https://projecteuclid.org/euclid.gt/1513799641

#### References

• M M Asaeda, J H Przytycki, A S Sikora, Khovanov homology of links in $I$–bundles over surfaces.
• M M Asaeda, J H Przytycki, A S Sikora, Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces, \agtref420045211771210
• D Bar-Natan, On Khovanov's categorification of the Jones polynomial, \agtref2200216337370
• J S Carter, M Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs 55, American Mathematical Society, Providence, RI (1998)
• J S Carter, M Saito, S Satoh, Ribbon-moves for 2–knots with 1–handles attached and Khovanov-Jacobsson numbers.
• M Jacobsson, An invariant of link cobordisms from Khovanov homology, \agtref420045312111251
• M Jacobsson, Khovanov's conjecture over $\bbZ[c]$.
• V Jones, Planar algebras, I, New Zealand Journal of Mathematics (to appear) \arxivmath.QA/9909027
• L H Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395–407
• M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
• M Khovanov, A functor-valued invariant of tangles, \agtref2200230665741
• M Khovanov, Patterns in knot cohomology. I, Experiment. Math. 12 (2003) 365–374
• M Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. (to appear) \arxivmath.QA/0207264
• E S Lee, On Khovanov invariant for alternating links.
• J H Przytycki, Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999) 45–66
• J A Rasmussen, Khovanov homology and the slice genus, e-print (2004) \arxivmath.GT/0402131
• J A Rasmussen, Khovanov's invariant for closed surfaces, e-print (2005) \arxivmath.GT/0502527
• R Scharein, KnotPlot, http://www.pims.math.ca/knotplot/
• A N Shumakovitch, Torsion of the Khovanov homology, e-print (2004) \arxivmath.GT/0405474
• J Stallings, Centerless groups–-an algebraic formulation of Gottlieb's theorem, Topology 4 (1965) 129–134
• O Viro, Remarks on definition of Khovanov homology, e-print (2002) \arxivmath.GT/0202199
• S Wolfram, The Mathematica$\sp \circledR$ book, Wolfram Media, Inc. Champaign, IL (1999)