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

Abstract

729sl(2) tangle homology with a parameter and singular cobordisms CARMEN LIVIA CAPRAUWe construct a bigraded cohomology theory which depends on one parameter a, and whose graded Euler characteristic is the quantum sl.2/ link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl.3/ theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl.2/ invariant (or Lee's modification of it) corresponds to a D 0 (or a D 1/: In particular, the construction naturally resolves the sign ambiguity in the functoriality of Khovanov's sl.2/ theory.

Fixing the functoriality of Khovanov homology

“…All of these variations are displayed in Figure 12. Note that the interleaved variations were not treated at all in version 1 of this paper on the arXiv, or in Caprau's paper [6] on the disoriented version of Khovanov homology.…”

The Rasmussen invariant and the Milnor conjecture

“…A similar result is most likely to hold in the filtred Lee case. Moreover, the sign issue can be fixed at the cost of a more involved construction; see [8,5,3]. But anyway, this (up to sign) invariance of the induced maps is not necessary to prove the Milnor conjecture.…”

The universal sl(2) cohomology via webs and foams

“…Using Bar-Natan's [1] approach to local Khovanov homology and Khovanov's work in [12], the author showed in [3] how to construct a bigraded tangle cohomology theory depending on one parameter, via a setup with webs and foams-singular cobordisms-modulo a finite set of relations; see also [2] for a longer, more detailed version of [3].…”

“…In the first half of this paper, we generalize the construction described in [3] to obtain the universal sl(2)-link cohomology-universal in the sense of [ a bigraded tangle cohomology theory. The good news is that the generalized theory still satisfies the functoriality property with no sign indeterminacy.…”

“…Besides generalizing the construction in [3] and thus obtaining a better invariant, we go further in the second half of the paper to show more insights about the new theory. Since any surface-link can be regarded as a cobordism between empty links, our construction yields an invariant of such surfaces.…”