sl(2) tangle homology with a parameter and singular cobordisms

Caprau, Carmen Livia

doi:10.2140/agt.2008.8.729

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sl(2) tangle homology with a parameter and singular cobordisms

Carmen Livia Caprau

Algebraic & Geometric Topology 8 (2008) 729–756

DOI: 10.2140/agt.2008.8.729

Abstract

We construct a bigraded cohomology theory which depends on one parameter $a$ , and whose graded Euler characteristic is the quantum $s l (2)$ link invariant. We follow Bar-Natan’s approach to tangles on one side, and Khovanov’s $s l (3)$ theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov $s l (2)$ invariant (or Lee’s modification of it) corresponds to $a = 0$ (or $a = 1$ ). In particular, the construction naturally resolves the sign ambiguity in the functoriality of Khovanov’s $s l (2)$ theory.

Keywords

categorification, cobordisms, Euler characteristic, Jones polynomial, functoriality, Khovanov homology, knots and links, movie moves, webs and foams

Mathematical Subject Classification 2000

Primary: 57M27, 57M25

Secondary: 18G60

References

Publication

Received: 9 January 2008

Accepted: 28 January 2008

Published: 25 May 2008

Authors

Carmen Livia Caprau
Department of Mathematics California State University 5245 North Backer Avenue M/S PB 108 Fresno CA 93740-8001 USA
http://zimmer.csufresno.edu/~ccaprau

Volume 8, issue 2 (2008)