We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.

For each braid β ∈ Brn we construct a 2-periodic complex S β of quasi-coherent C * × C *equivariant sheaves on the non-commutative nested Hilbert scheme Hilb f ree 1,n . We show that the triply graded vector space of the hypercohomology H(S β ⊗ ∧ • (B)) with B being tautological vector bundle, is an isotopy invariant of the knot obtained by the closure of β. We also show that the support of cohomology of the complex S β is supported on the ordinary nested Hilbert scheme Hilb 1,n ⊂ Hilb f ree 1,n , that allows us to relate the triply graded knot homology to the sheaves on Hilb 1,n List of Symbols 81References 82

ABSTRACT. We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n − 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q, t-Catalan numbers, and of the last three authors relating knot homology to Hilbert schemes on singular curves.

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The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities y k = x n , whose links are the (k, n) torus knots, and for the singularity y 4 = x 7 − x 6 + 4x 5 y + 2x 3 y 2 , whose link is the (2,13) cable of the trefoil. SMOOTH POINTS, NODES, AND CUSPSWe illustrate the conjecture with some elementary examples. Denote by C 2,n the formal germ at the origin of the curve cut out by y 2 = x n , and by O 2,n its ring of functions. The link of this singularity is the right-handed (2, n) torus link T 2,n . The first few of these:Computing P(T 2,n ) is an elementary exercise in the skein relation: smoothing a crossing yields T 2,n−1 and switching a crossing gives T 2,n−2 . This yields the recurrence P(T 2,n ) = −a(q − q −1 )P(T 2,n−1 ) + a 2 P(T 2,n−2 ) T 2,1 is the unknot, and T 2,0 is two unlinked circles. It is immediate from the skein relation that the HOMFLY polynomial of n unlinked circles is ((a − a −1 )/(q − q −1 )) n−1 . Thus: