Contact \((+1)\)-surgeries on rational homology \(3\)-spheres (Q6042866)

As indicated by its title, this paper studies contact manifolds \((Y,\xi)\) which result from contact \((+1)\)-surgery along a Legendrian knot \(L\) in a contact rational homology 3-sphere. In particular, a variety of sufficient conditions are given for the contact invariant of \((Y,\xi)\) to vanish, and two conditions are given under which \((Y,\xi)\) must be overtwisted. The conditions are mostly stated in terms of the (rational) Thurston-Bennequin invariant and the (rational) rotation number of \(L\), with one condition appealing to the \(\tau\) invariant, and the proofs largely rely on computations in (the chain complex of) knot Floer homology. Among the most fundamental questions one can ask about a contact manifold \((Y,\xi)\) is whether it is tight or overtwisted, and the contact invariants \[ c(\xi) \in \widehat{HF}(-Y) \quad\text{and}\quad c^+(\xi) \in HF^+(-Y), \] defined by Ozsváth-Szabó as elements of the Heegaard Floer homologies of \(-Y\), are powerful tools for attempting to answer this question. Namely, \(c(\xi)\) is trivial whenever \(\xi\) is overtwisted and, according to Ghiggini, neither \(c(\xi)\) nor \(c^+(\xi)\) is trivial when \((Y,\xi)\) admits a strong symplectic filling. Establishing the triviality of the contact invariants for a fixed \((Y,\xi)\) thus shows that \((Y,\xi)\) is not strongly fillable, and perhaps overtwisted. By a result of Ding-Geiges, contact \((\pm 1)\)-surgery allows us to access all closed, connected contact 3-manifolds. That is, any such \((Y,\xi)\) may be expressed as the result of contact surgery along a Legendrian link in \((S^3,\xi_{\mathrm{std}})\), with \((+1)\)-surgery performed along some components of the link and \((-1)\)-surgery performed along others. It is thus important to understand the effect of contact \((\pm 1)\)-surgery on the contact invariants \(c(\xi)\) and \(c^+(\xi)\). The case of contact \((-1)\)-surgery -- also known as Legendrian surgery -- is well-studied and reasonably well-behaved, while positive contact surgeries are more mysterious. This paper concerns itself with the latter. The criteria given in this paper for the vanishing of the contact invariants all follow from a direct investigation of the knot Floer homology, but not from a single central theorem. A representative criterion is the following: Proposition 1.3. Suppose \((Y,\xi)\) is a contact \(L\)-space, and \(L\) is a Legendrian knot in \((Y,\xi)\). If \(tb_{\mathbb{Q}}(L)<-1\), then the contact invariant \(c^+(\xi_{+1}(L))\) vanishes. Here \(\xi_{+1}(L)\) is the contact structure obtained by contact \((+1)\) surgery along \(L\).