Overtwisted positive contact surgeries (Q2417917)

A contact structure on a smooth 3-manifold \(M\) is a completely non-integrable 2-plane field \(\xi\subset TM\), i.e., there exists a globally defined 1-form \(\alpha\) on \(M\) such that \(\alpha\wedge d\alpha\) is a volume form of \(M\). The standard contact structure \(\xi_{\text{std}}\) on \(S^3\subset\mathbb{R}^4\) is given in Cartesian coordinates as \(\xi_{\text{std}}=\ker(x_1 dy_1-y_1 dx_1+x_2 dy_2-y_2 dx_2)\). An embedded disc \(D\to M\) in a contact manifold \((M,\xi)\) is called overtwisted if \(T_pD=\xi(p)\) for all points \(p\in\partial D\). A contact structure is tight if there is no overtwisted disc, otherwise it is overtwisted. A Legendrian knot in a contact manifold \((M,\xi)\) is an embedding \(S^1\to M\) so that the tangent space of the image is contained in \(\xi\). If \(L\) is a null homologous oriented Legendrian knot in a closed contact 3-manifold \((M,\xi)\) with \(M\) being a rational homology sphere, \(\Sigma\) is a rational Seifert surface for \(L\) with connected binding, that is, \(\partial\Sigma\) is connected and is homologous to \(r\cdot[L]\), where \(r\) is the smallest positive integer such that \(r\cdot[L]=0\in H_1(M;\mathbb Z)\), and \(L'\) is another Legendrian knot, then the rational linking \(\mathsf{lk}\) is defined as \(\mathsf{lk}(L.L')=\frac1r[\Sigma]\cdot[L']\). If the Legendrian knot \(L'\) is a push-off of \(L\) in the direction of the framing of the normal bundle of \(L\) induced by \(\xi\vert_{L}\), then the rational Thurston-Bennequin invariant \(\mathsf{tb}\) of \(L\) is defined as \(\mathsf{tb}(L)=\mathsf{lk}(L.L')\). If \(\imath:\Sigma\to M\) is an embedding on the interior of \(\Sigma\), \(\tau\) is a trivialisation of the pull-back bundle \(\imath^*(\xi)\) over \(\Sigma\), then the rational rotation number \(\mathsf{rot}\) of \(L\) is defined as \(\mathsf{rot}(L)=\frac1r\text{wind}_\tau(v)\), where \(\text{wind}_\tau(v)\) measures the winding number of \(v\) in \(\mathbb R^2\) with respect to the trivialisation \(\tau\). For a null-homologous Legendrian knot \(L\subset(M,\xi)\) contact \((+r)\)-surgery on \(L\) is defined for some rational \(r>0\). The surgery coefficient \(r\) is understood to be given with respect to the contact framing on \(L\), and so contact \((+r)\)-surgery on \(L\) results in the manifold given by smooth \(\mathsf{tb}(L) +r\)-surgery on the underlying smooth knot type \(K\) of \(L\). For a Legendrian knot \(L\) in \((S^3,\xi_{\text{std}})\) with \(\mathsf{tb}(L)\le-2\), \textit{P. Lisca} and \textit{A. I. Stipsicz}, in [Pac. J. Math. 228, No. 2, 277--295 (2006; Zbl 1172.57012)], proved that contact \((+1)\)-surgery on \(L\) has vanishing Heegaard Floer contact invariant. It is natural to ask whether these are indeed overtwisted. \par In this paper, the authors answer this question for a large class of such knots, as well as more generally in an arbitrary contact 3-manifold. They prove that if \(L\subset(M,\xi)\) is a null-homologous Legendrian knot, such that \(c_1(\xi)\) is torsion and \(\mathsf{tb}(L)\le-2\), then if \(|n\cdot\mathsf{rot}(L)-(n-1)\cdot\mathsf{tb}(L)|>n(2g(L)-1)+\mathsf{tb}(L)\) for a positive integer \(n<|\mathsf{tb}(L)|\), where \(g(L)\) is the genus of \(L\), then contact \((+n)\)-surgery on \(L\) is overtwisted. In particular, if \(\mathsf{tb}(L)\le-2\) and \(|\mathsf{rot}(L)|>2g-1+\mathsf{tb}(L)\), then contact \((+1)\)-surgery is overtwisted. Also, the authors prove that if \(L\subset(M,\xi)\) is a null-homologous Legendrian knot, such that \(c_1(\xi)\) is torsion and \(g(L)\) is the genus of \(L\), then (i)\,if \(\mathsf{tb}(L)-\mathsf{rot}(L)<-2g(L)-1\), then all natural positive contact surgeries on \(L\) are overtwisted, and (ii)\, if \(\mathsf{tb}(L)+\mathsf{rot}(L)<-2g(L)-1\), then all, including non-natural, positive contact surgeries on \(L\) are overtwisted.