In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invariant is arbitrarily far from maximal. We also exhibit Legendrian knots requiring arbitrarily many stabilizations before they become Legendrian isotopic. Similar new phenomena are observed for transverse knots. To achieve these results we define and study "partially thickenable" tori, which allow us to completely classify solid tori representing positive torus knots.Comment: 34 pages, 6 figure

For a nullhomologous Legendrian knot in a closed contact 3manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery, and we characterize the spin c structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery on a Legendrian knot in the standard 3sphere, generalizing previous results of Lisca-Stipsicz and Golla. In fact our methods allow direct calculation of the contact invariant in terms of the rational surgery mapping cone of Ozsváth and Szabó. The proof involves a construction called reducible open book surgery, which reduces in special cases to the capping-off construction studied by Baldwin.

This paper completely answers the question of when contact (r)-surgery on a Legendrian knot in the standard contact structure on S 3 yields a symplectically fillable contact manifold for r ∈ (0, 1]. We also give obstructions for other positive r and investigate Lagrangian fillings of Legendrian knots.

A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in C 2 , with either orientation. A related question asks whether every compact contractible 4-manifold admits the structure of a Stein domain. We verify Gompf's conjecture, with one orientation, for a family of Brieskorn spheres of which some are known to admit a smooth embedding in C 2 . With the other orientation our methods do not resolve the question, but do give rise to an example of a contractible, boundaryirreducible 4-manifold that admits no Stein structure with either orientation, though its boundary has Stein fillings with both orientations.

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