ON THE sl(2) FOAM COHOMOLOGY COMPUTATIONS

Abstract

Abstract. We show how to use Bar-Natan's 'divide and conquer' approach to computation to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.

A cohomology theory for colored tangles

“…These techniques provide computational efficiency of the sl(2) foam cohomology groups (and, implicitly, of the original Khovanov homology groups). For more details about efficient computations we refer the reader to [6]. In particular, it follows that the category Kof /h has a natural structure of an oriented planar algebra.…”

“…Specifically, we cut a colored oriented framed knot (K, n) (using horizontal lines) into subtangles (T i , n), compute the invariants C n (T i ) and assemble them into C n (K), as prescribed in this subsection. Before the assembling operation, we simplify each C n (T i ) as much as possible by simplifying the cochain objects of C n (T i ) (thus we simplify the formal complexes [D i,s ] ∈ Kof by making use of the "delooping" and "Gaussian elimination" procedures, as described in [6]). Once that is taken care of, we apply the functor F to arrive at the complex FC n (D), and take its cohomology.…”

Twin TQFTs and Frobenius Algebras

“…There are many similarities between the algebraic structure of the category of openclosed cobordisms and certain relations satisfied by the singular cobordisms, although the two types of cobordisms are topologically different. The original Khovanov homology relies on a 2D TQFT, and it would be quite desirable and refreshing to have some kind of TQFT defined on foams/singular cobordisms, and use it to obtain another method for defining the universal sl(2) foam cohomology theory (specifically, the purely topological one-with no dots on cobordisms-discussed in [6], Section 4), and a generalization of the Khovanov homology. In particular, this would provide us with knowledge of the algebraic structure that governs this cohomology theory that provides a properly functorial Khovanov homology theory.…”

“…In this paper we make the first step in achieving this goal. The singular cobordisms considered here are a particular case of those used in [4,5,6], in the sense that the 1-manifolds are disjoint unions of oriented circles and bi-webs (webs with exactly two bivalent vertices). The second step in reaching our goal will be treated in a subsequent paper, where we also show that it suffices to work with bi-webs, as opposed to arbitrary webs (webs with an even number of bivalent vertices).…”

“…In [6], the author described a method that computes fast and efficient the sl(2) foam cohomology groups, and also provided a purely topological version of the sl(2) foam theory in which no dots are required on cobordisms.…”