We classify Legendrian rational unknots with tight complements in the lens spaces L(p, 1) up to coarse equivalence. As an example of the general case, this classification is also worked out for L(5, 2). The knots are described explicitly in a contact surgery diagram of the corresponding lens space.2010 Mathematics Subject Classification. 53D10, 57M25, 57M27.and likewise for K 2 . For p = 2 this reduces in fact to one possibility, and for p > 2, q ∈ {1, p − 1} the knot ±K 1 is isotopic to ±K 2 , both being fibres in an S 1 -bundle 1 When we speak of a 'rational unknot' in L(p, q) we always mean a rational unknot that is not an honest unknot, i.e. the homological order of the knot is supposed to be greater than 1.

A Legendrian or transverse knot in an overtwisted contact 3-manifold is nonloose if its complement is tight and loose if its complement is overtwisted. We define three measures of the extent of nonlooseness of a nonloose knot and show they are distinct.

In this note, we define a new invariant of a Legendrian knot in a contact 3manifold using an open book decomposition supporting the contact structure. We define the support genus sg(L) of a Legendrian knot L in a contact 3-manifold (M, ξ) as the minimal genus of a page of an open book of M supporting the contact structure ξ such that L sits on a page and the framings given by the contact structure and the page agree. We show any null-homologous loose knot in an overtwisted contact structure has support genus zero. To prove this, we show that any topological knot or link in any 3-manifold M sits on a page of a planar open book decomposition of M .1991 Mathematics Subject Classification. 57R17.

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It is known that any contact 3-manifold can be obtained by rationally contact Dehn surgery along a Legendrian link L in the standard tight contact 3-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link L describing a given contact 3-manifold under consideration.In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three.In the second part, we compute contact surgery numbers of all contact structures on the 3-sphere. Moreover, we completely classify the contact structures with contact surgery number one on S 1 × S 2 , the Poincaré homology sphere and the Brieskorn sphere Σ(2, 3, 7). We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere. We further obtain results for the 3-torus and lens spaces.As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.