On overtwisted contact surgeries

In this note, we obtain a new result concluding when contact (+1/n)-surgery is overtwisted. We give a counterexample to a conjecture by James Conway on overtwistedness of manifolds obtained by contact surgery. We list some problems related to the contact surgery.Comment: 6 pages, 1 figur

Legendrian rational unknots in lens spaces

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.Comment: 25 pages, 12 figure

Legendrian lens space surgeries

We show that every tight contact structure on any of the lens spaces $L(ns^2-s+1,s^2)$ with $n\geq 2$, $s\geq 1$, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot $T(s,-(sn-1))$ in the tight or an overtwisted contact structure on the 3-sphere.Comment: 16 pages, 8 figure

Legendrian torus knots in lens spaces

In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.Comment: 16 pages, 2 figures. Title is changed. Definitions are cleared, focused only on positive torus knot

Invariants of Legendrian knots from open book decompositions

In this note, we define a new invariant of a Legendrian knot in a contact manifold 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, \xi) as the minimal genus of a page of an open book of M supporting the contact structure \xi such that L sits on a page and the framing given by the contact structure and by the page agree. We show any null-homologous loose knot in an overtwisted contact structure has support genus zero. To prove this, we observe any topological knot or link in any 3-manifold M sits on a page of a planar open book decomposition of M.Comment: 20 pages, 20 figures, proof of Lemma 2.1 added, minor corrections and clarification

Legendrian knots in Lens spaces

In this note, we first classify all topological torus knots lying on the Heegaard torus in Lens spaces, and then we classify Legendrian representatives of torus knots. We show that all Legendrian torus knots in universally tight contact structures on Lens spaces are determined up to contactomorphism by their knot type, rational Thurston-Bennequin invariant and rational rotation number

Legendrian Hopf links

We completely classify Legendrian realisations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.Comment: 36 pages, 18 figure