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I read through parts of Khovanov/Rozansky's paper on the categorification of the HOMFLY polynomial using Matrix Factorizations. Technically, I can follow (though it seems to me that quite a lot of details are missing and tedious to fill in) - intuitively, however, I have no idea why one is lead to consider matrix factorizations when studying knot theory, in particular the RT invariants obtained from interpreting colored tangles as morphisms between modules over the quantum group. Until now, it feels quite mysterious to me why Khovanov and Rozansky choose particular potentials like $x_1^n+x_2^n-x_3^n-x_4^n$ in their construction, and why one should expect that in the end we get something invariant under the Reidemeister moves.

Can somebody explain to me the motivation behind this construction? What is the relation between the morphism of modules over the quantum group a wide edge represents and the matrix factorization associated to it?

Thank you!

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up vote 10 down vote accepted

Many knot homologies are expected to have Floer-theoretic interpretations. However, in Floer theory often the chain "complex" $CF(L_0,L_1)$ is not a complex but rather an a-infty bimodule over a pair of curved a-infty algebras; matrix factorizations are more or less a special case of these, where the curvature of the ainfty algebras is a multiple of the identity. Under some nice (monotone or exact) assumptions $CF(L,L)$ has differential that squares to zero, since the curvatures of the a-infty algebras on both sides occur with opposite sign and so cancel if the Lagrangians are the same (or related by a symplectomorphism). But even in the monotone or exact situations $CF(L_0,L_1)$ can have a differential which gives a matrix factorization. Now Manolescu has proposed a symplectic interpretation of Khovanov-Rozansky for links, which looks like $CF(L,\beta(L))$ where \beta is a braid presentation of the link, but so far there doesn't seem to be a proposal for graphs. Probably for graphs the invariant would be the homotopy type of $CF(L_0,L_1)$ (or some more general quilted Floer chain group) which under suitable monotonicity assumptions will be a matrix factorization. Note that Kamnitzer has a proposal for a Floer-theoretic version for arbitrary G; it would be interesting to see if one could extend this proposal to the graph case, which one would probably need for an exact triangle.

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