This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact R 3 and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian filling is quasi-positive. On the other hand, we show that if a knot type is positive, then it has a Legendrian representative with an exact embedded Lagrangian filling. Further, we produce examples that show that strong quasi-positivity and fillability are independent conditions. arXiv:1307.7683v1 [math.SG]

We construct infinitely many Legendrian links in the standard contact R 3 with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in S 3 that bound topologically distinct pieces of algebraic curves in B 4 ⊂ C 2 , is applied to find contact 3-manifolds with topologically distinct symplectic fillings, and is generalized to higher dimensions.

We show that C 2 contains pairs of properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms of C 2 . The construction is based on realizing corks as branched covers of holomorphic disks in the 4-ball. These disks can also be described using exotic factorizations of quasipositive braids.

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From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant ν is an invariant of the smooth 4-manifold associated to a knot K ⊂ S 3 by attaching an n-framed 2-handle to B 4 along K . We also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants τ and are not invariants of such 4-manifolds. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct S 1 × S 2 surgeries, resolving a question from Problem 1.16 in Kirby's list [28].