An Ozsváth-Szabó Floer homology invariant of knots in a contact manifold (Q936615)

The paper under review is to extend a fundamental result of Bennequin-Eliashberg in terms of the knot Floer homology defined by Ozsváth-Szabó. Section 2 begins with the background on the knot Floer homology filtration. The knot Floer chain complex \(CFK^{\infty}(Y, [S], K, {\mathbf s})\) is generated by triples \([x, i, j]\) with \(x\in T_{\alpha}\cap T_{\beta}\), intersections of Lagrangian tori from Heegaard diagrams of the closed oriented 3-manifold \(Y\), and \({\mathbf s} (x) + (i-j)PD (\mu ) = {\mathbf s}_0\) of the meridian \(\mu\) of the knot \(K\) and the \(\text{Spin}^c\) structure \({\mathbf s}_0\) in \(Y_0(K)\). The \({\mathbb Z}\oplus {\mathbb Z}\) filtered chain complex is given by \({\mathcal F}: C_* \to {\mathbb Z}\oplus {\mathbb Z}\) such that \({\mathcal F}(\partial x) \leq {\mathcal F}(x)\) for every \(x\in C_*\), where \({\mathcal F}([x, i, j]) = (i, j)\) and \({\mathcal F}(\sum_k [x_k, i_k, j_k]) = (\max_k i_k, \max_k j_k)\). For instance, \(\hat{CF}(Y, {\mathbf s}) = C_{\mathbf s}\{i=0\}\). For two different Seifert surfaces \(S_+, S_-\) of the knot \(K\) and \(x\in \hat{CF}(Y, {\mathbf s})\), \[ {\mathcal F}_{\mathbf s}^+(x) - {\mathcal F}_{\mathbf s}^- (x) = \tfrac{1}{2}\langle c_1({\mathbf s}), [S_+ - S_-]\rangle, \] for any \(\text{Spin}^c\) structure \({\mathbf s}\). The knot Floer homology groups are given by \[ H_*\left({\mathcal F}_{\mathbf s}(Y, [S], K, m)/ {\mathcal F}_{\mathbf s}(Y, [S], K, m-1)\right) \] and denoted by \(\widehat{HFK}_{\mathbf s}(Y, [S], K, m)\). Note that \(\sum_m \chi \left( \widehat{HFK}_{\mathbf s}(Y, [S], K, m)\right)\cdot T^m = \Delta_K (T)\) the classical Alexander-Conway polynomial. Relative to the subcomplex, \([x]\neq 0 \in \hat{HF}(Y, {\mathbf s})\), one can measure the minimal level in the filtration as \(\tau_{[x]}(Y, [S], K)\) given by the image \(I_m\) induced from the inclusion \(l_m: {\mathcal F}_{\mathbf s} (Y, [S], K, m) \hookrightarrow \hat{CF}(Y, {\mathbf s})\) (see definition 11). Let \({\mathbf s}_{\xi}\) be the \(\text{Spin}^c\) structure on the contact 3-manifold \((Y, \xi)\) associated to the contact structure \(\xi\). The Ozsváth-Szabó contact invariant \(c(\xi)\) is the class of a generator of \(H_*({\mathcal F}_{{\mathbf s}_{\xi}}(-Y, [S], K - g(S))\) under the homology induced from the inclusion \(i_{- g(S)}: {\mathcal F}_{{\mathbf s}_{\xi}}(-Y, [S], K - g(S) \to \hat{CF}(-Y, {\mathbf s}_{\xi})\). Then the author defines \(\tau_{\xi}([S], K) = \tau^*_{c(\xi)}(Y, [S], K)\) in definition 23. Some basic properties of \(\tau_{[x]}(Y, K)\) and its dual version \(\tau^*_{[y]}(Y, K)\) on the knot Floer homology and knot Floer cohomology are discussed in section 3. The main result of the paper is given in theorem 2: for a null-homologous knot \(K\hookrightarrow Y\) and a Seifert surface \(S\) with \(c(\xi) \neq 0\), the Thurston-Bennequin number \(tb(\tilde{K})\) and the rotation number \(rot_{\mathbf s}(\tilde{K})\) satisfy \[ tb(\tilde{K}) + |rot_{\mathbf s}(\tilde{K})| \leq 2 \tau_{\xi}(K) - 1, \] for any Legendrian representative \(\tilde{K}\) of the knot \(K\). The proof is given in section 3. One can first prove that \(tb(\tilde{K}) + rot_{\mathbf s}(\tilde{K}) \leq 2 \tau_{\xi}(K) - 1\), since the orientation changing switches the sign of its rotation number. Note that \(tb(\tilde{K}) = [\hat{S}]\cdot [\hat{S}] + 1\) and \(rot_{\mathbf s}(\tilde{K}) = \langle c_1(t), [\hat{S}]\rangle\) by Gompf's work on the canonical \(\text{Spin}^c\) structure \(t\) associated to a symplectic cobordism \((W_k, \omega)\) between \((Y, \xi)\) and \((Y_k, \xi_K)\). One thus stabilizes \(\tilde{K}\) to keep \(tb(\tilde{K}) + rot_{\mathbf s}(\tilde{K})\) constant. Therefore \(tb(\tilde{K}) + rot_{\mathbf s}(\tilde{K})\leq 2\tau_{\xi}(K) +1\) by the nontrivial filtration pairing. By applying the double technique, one gets \[ 2(tb(\tilde{K}) + rot_{\mathbf s}(\tilde{K}))+1 \leq \max[tb(\tilde{K\# K}) + rot_{\mathbf s}(\tilde{K\# K})]\leq 4\tau_{\xi} (K) + 1. \] Theorem 2 follows from the oddness of the sum \(tb(\tilde{K}) + rot_{\mathbf s}(\tilde{K})\). Note that \(c(\xi)\neq 0\) implies that the contact structure \(\xi\) is tight and \(\tau_{\xi}(K) \leq g(S)\) by an adjunction inequality. Hence the Bennequin-Eliashberg inequality \(tb(\tilde{K}) + |rot_{\mathbf s}(\tilde{K})| \leq 2g(S) -1\) follows from theorem 2. In other preprints of the author, examples of Legendrian knots with arbitrarily negative \(\tau_{\xi}\) can be constrcuted. Thus, theorem 4 shows that for any \(N > 0\) there is a knot with \(tb(\tilde{K}) + |rot_{\mathbf s}(\tilde{K})| \leq - N\). The author proposes an interesting question in the introduction for a criterion on tight contact structure.