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Contact (+1)-surgeries along Legendrian two-component links. (English) Zbl 1446.57003

If there exists a globally defined 1-form \(\alpha\) on a smooth 3-manifold \(Y\) such that \(\alpha\wedge d\alpha> 0\), then a contact structure on \(Y\) is a completely non-integrable 2-plane field \(\xi\subset TM\), locally satisfying \(\xi=\ker\alpha\). An embedded disc \(D\to Y\) in a contact manifold \((Y,\xi)\) 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. Any closed oriented 3-manifold admits an overtwisted contact structure, but it is much harder to find tight contact structures on a closed oriented 3-manifold. One way of obtaining new contact manifolds from the existing one is through contact surgery. If \(L\) is tangent to the given contact structure \(\xi\) on \(Y\), or \(L\) is a Legendrian knot in \((Y,\xi)\), then a contact surgery is a version of Dehn surgery that is adapted to the contact category. A tubular neighborhood of \(L\) is deleted, re-glued, and then a contact structure on the surgered manifold is obtained by extending \(\xi\) from the complement of the tubular neighborhood of \(L\) to a tight contact structure on the re-glued solid torus. In [Math. Proc. Camb. Philos. Soc. 136, No. 3, 583–598 (2004; Zbl 1069.57015)], F. Ding and H. Geiges proved that every closed contact 3-manifold \((Y,\xi)\) can be obtained by contact \((\pm1)\)-surgery along a Legendrian link in \((S^3,\xi_{\mathrm{std}})\), where \(\xi_{\mathrm{std}}\) denotes the standard contact structure on \(S^3\). For a Heegaard Floer homology with coefficients in \(\mathbb{F}=\mathbb{Z}/2\mathbb{Z}\), Heegaard Floer theory associates an abelian group \(\widehat{HF}(Y,t)\) to a closed, oriented \(\mathrm{Spin}^c\) 3-manifold \((Y,t)\), and a homomorphism \(F_{W,\mathfrak{s}}:\widehat{HF}(Y_1,t_1)\to\widehat{HF}(Y_2,t_2)\) to a \(\mathrm{Spin}^c\) cobordism \((W,\mathfrak{s})\) between two \(\mathrm{Spin}^c\) 3-manifolds \(\widehat{HF}(Y_1,t_1)\) and \(\widehat{HF}(Y_2,t_2)\). In [Duke Math. J. 129, No. 1, 39–61 (2005; Zbl 1083.57042)], P. Ozsváth and Z. Szabó introduced an invariant \(c(Y,\xi)\in\widehat{HF}(-Y)\) for the closed contact 3-manifold \((Y,\xi_{\mathrm{std}})\), where \(\widehat{HF}(Y)\) is the direct sum \(\oplus_t\widehat{HF}(Y,t)\) over all \(\mathrm{Spin}^c\) structures \(\mathfrak{t}\) on \(Y\) and \(F_W\) is the sum \(\sum_{\mathfrak{s}}F_{W,\mathfrak{s}}\) over all \(\mathrm{Spin}^c\) structures \(\mathfrak{s}\) on \(W\). The authors showed that \(c(Y,\xi)=0\) if \((Y,\xi)\) is overtwisted. If the contact manifold \((Y_K,\xi_K)\) is obtained from \((Y,\xi)\) by contact \((+1)\)-surgery along a Legendrian knot \(K\), then \(F_{-W}(c(Y,\xi))=c(Y_K,\xi_K)\), where \(-W\) stands for the cobordism induced by the surgery with reversed orientation. It is natural to ask whether the contact invariant of a contact 3-manifold obtained by contact surgery along a Legendrian link is trivial or not. All known results concern contact surgeries along Legendrian knots. For example, contact \(\frac1n\)-surgeries along certain Legendrian knots in \((S^3,\xi_{\mathrm{std}})\) yield contact 3-manifolds with nonvanishing contact invariants for any positive integer \(n\), and contact 3-manifolds obtained from \((S^3,\xi_{\mathrm{std}})\) by contact \(n\)-surgeries along Legendrian knots.
In this paper, the authors study contact surgeries along Legendrian links in the standard contact 3-sphere. The first result states that if \(L=L_1\cup L_2\) is an oriented Legendrian two-component link in \((S^3,\xi_{\mathrm{std}})\) whose two components have nonzero linking number, \(L\) satisfies \(\nu^+(L_2)=\nu^+(\overline{L_2})=0\), where \(\overline{L_2}\) denotes the mirror of \(L_2\), then contact \((+1)\)-surgery on \((S^3,\xi_{\mathrm{std}})\) along \(L\) yields a contact 3-manifold with vanishing contact invariant. In the special case, where \(L\) is a Legendrian unknot with \(\mathsf{tb}(L_2)=-1\), contact \((+1)\)-surgery on \((S^3,\xi_{\mathrm{std}})\) along \(L_2\) yields the unique, up to isotopy, tight contact structure \(\xi_t\) on \(S^1\times S^2\). Hence, the theorem may be interpreted as a result of contact \((+1)\)-surgery along a Legendrian knot in \((S^1\times S^2,\xi_t)\). As a corollary, the authors show that if \(L\) is a Legendrian knot in \(\#^k(S^1\times S^2,\xi_t)\), the contact connected sum of \(k\) copies of \((S^1\times S^2,\xi_t)\), \(L\) is not null-homologous, then contact \(\frac1n\)-surgery on \(\#^k(S^1\times S^2,\xi_t)\) along \(L\) yields a contact 3-manifold with vanishing contact invariant for any positive integer \(n\). Furthermore, the authors prove that contact \((+1)\)-surgery on \((S^3,\xi_{\mathrm{std}})\) along a Legendrian Whitehead link yields a contact 3-manifold with vanishing contact invariant. Finally, they show that if there exists a front projection of a Legendrian two-component link \(L=L_1\cup L_2\) in the standard contact 3-sphere \((S^3,\xi_{\mathrm{std}})\) that contains certain configurations, then contact \((+1)\)-surgery on \((S^3,\xi_{\mathrm{std}})\) along \(L\) yields an overtwisted contact 3-manifold.

MSC:

57K10 Knot theory
57K33 Contact structures in 3 dimensions
57R17 Symplectic and contact topology in high or arbitrary dimension
57R58 Floer homology
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References:

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