The (colored) sln link homologies and the categorical branching rules: Two methods combined
Final Report Abstract
Knot homologies have attracted a lot of attention in the past few years. In particular, these are interesting for topologist, representation theorists, physicist etc. alike. These homologies are categorifications of the knot polynomials who were mostly found, in groundbreaking work, in the 1980ties. These homologies “lift” the information and connection given by the knot polynomials to a “higher, deeper” level. Nowadays a lot of variations of knot homologies are known. For example, Khovanov- Rozansky homologies for slN /glN - for all N and all “colorings” with (sufficiently nice) representations of slN /glN . Sadly, the combinatorial complexity of these homologies grows rapidly with growing N (or “growing colors”). The question if one can relate these homologies for “big N ” with those for “small N” naturally arises. It turns out that the original question about “branching of Khovanov-Rozansky homologies” (i.e. reducing N) can not be answered in a satisfactory way - too much information is lost. The homologies can only be “deformed”. Still, two variations of the original question have a nice answer. Instead of “branching Khovanov-Rozansky slN /glN homology reducing N ” one can: • Connect Khovanov-Rozansky slN /glN homologies for various “colors” (still conjecturally for the homologies, but now, by our work in connection to the DFG project, known in a conceptual way for the associated polynomials.) • The existing connections between Khovanov-Rozansky slN /glN homologies and their deformations can be used to prove open problems in the field. In particular, as a result of the DFG funded research, we are able to prove functoriality of (colored) Khovanov-Rozansky homologies.
Publications
- Super q-Howe duality and web categories
D. Tubbenhauer, P. Vaz, and P. Wedrich
(See online at https://doi.org/10.2140/agt.2017.17.3703)
