A categorification of the quantum sl(N)-link polynomials using foams
arXiv:0807.2658
Abstract
In this thesis we define and study a categorification of the sl(N)-link polynomial using foams, for N\geq 3. For N=3 we define the universal sl(3)-link homology, using foams, which depends on three parameters and show that it is functorial, up to scalars, with respect to link cobordisms. Our theory is integral. We show that tensoring it with Q yields a theory which is equivalent to the rational universal Khovanov-Rozansky sl(3)-link homology. For N\geq 4 we construct a rational theory categorifying the sl(N)-link polynomial using foams. Our theory is functorial, up to scalars, with respect to link cobordisms. To evaluate closed foams we use the Kapustin-Li formula. We show that for any link our homology is isomorphic to the Khovanov-Rozansky homology. We conjecture that the theory is integral and we compute the conjectured integral sl(N)-link homology for the (2,m)-torus links and show that it has torsion of order N.
PhD thesis, 141 pages, lots of figures
References in corpus (21)
- Khovanov's homology for tangles and cobordisms
- Spiders for rank 2 Lie algebras
- Khovanov-Rozansky Homology and Topological Strings
- An invariant of link cobordisms from Khovanov homology
- Note on Khovanov link cohomology
- Surface Operators and Knot Homologies
- Category O and sl(k) link invariants
- Some differentials on Khovanov-Rozansky homology
- Khovanov homology and the slice genus
- A slice genus lower bound from sl(n) Khovanov-Rozansky homology
- Homological thickness and stability of torus knots
- Matrix Factorizations and Kauffman Homology
- On Khovanov's cobordism theory for su(3) knot homology
- Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial
- A geometric model for Hochschild homology of Soergel bimodules
- An sl(2) tangle homology and seamed cobordisms
- Homology of torus links
- Properties of Khovanov homology for positive braid knots
- The Khovanov-Rozansky Cohomology and Bennequin Inequalities
- A combinatorial approach to functorial quantum sl(k) knot invariants
- The Universal sl_2 Link Homology Theory