Combinatorial link homologies and their applications

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This project deals with structures and applications of combinatorial link homologies. The first theme of the proposed research is link homologies and embeddings of Lie algebras. It is known that certain embeddings of Lie algebras induce state sum formulas for the corresponding polynomial link invariants. The Principal Investigator (PI) will study how to "lift" such formulas to the categorifications of these link polynomials. The main motivation for this is a potential explicit categorification of the Kauffman polynomial using certain additional structures on the Khovanov-Rozansky homology. The second theme is twistings in link diagrams and the Khovanov-Rozansky homology. Krasner simplified the Khovanov-Rozansky chain complex of a two-strand twisting, which has led to new topological and structural results about the Khovanov-Rozansky homology. The PI will further study applications of this simplified chain complex. The final theme of this project is the spanning tree model. The PI will study potential applications of the explicit spanning tree model for the odd Khovanov homology of knots recently discovered by Roberts, Jaeger and Manion.In the late 1990s, Khovanov pointed to a new direction in the quantum invariant approach to low-dimensional topology. His work showed that, instead of working inside a linear space or a module, one should work inside a category. This way, the construction will lead to homological invariants which retain more topological information than the polynomial invariants. Such homological invariants are called categorifications of the corresponding polynomial invariants. Khovanov categorified the Jones polynomial and, with Rozansky, the HOMFLY-PT polynomial. Experts have used these categorifications to prove topological theorems that were only accessible by geometric analysis before. Among these, the best known is Rasmussen's combinatorial proof Milnor's Conjecture. The goal of this project is to further expand the scope of the categorification approach to knot theory by better understanding existing categorifications, constructing new categorifications and finding new applications of these categorifications.