Khovanov-Rozansky Homologies and Cabling

Abstract: In the cabling procedure for HOMFLY polynomials colored HOMFLY polynomials of a knot are obtained from ordinary HOMFLY of the cabled knot with extra twists added. Thus colored polynomials can be seen as relation between HOMFLYs of cabled knot with different twists. In present work we search for relations of such type in Khovanov-Rozansky homologies and investigate, why no generalizations of Khovanov-Rozansky homologies to non-skew-symmetric representations have been constructed. We consider the simplest possible case, i.e. the unknot colored in (11)- and (2)-representations. Naive t-deformation of HOMFLY relation failed to exist in both cases. In case of (11)-representation we have succeeded after a switch to a homological description and that led us to a conjecture about Khovanov-Rozansky homologies of torus knots $T(2,k)$. The (2)-case provides only framing-dependent answers and no simple rule of transformation, such as q,t-shift is seen. We have shown that in our procedure the number of whose nontrivial components with different t-gradings depends on the choice of unknot framing.