Formality of Floer complex of the ideal boundary of hyperbolic knot complement (Q2236439)

A knot \(K\) in the three-sphere \(S^{3}\) is an embedding of the circle \(S^{1}\) into \(S^{3}\). Since topology of the knot complement \(S^{3}\setminus K\) determines that of the knot \(K\) (or mirror of \(K\)), understanding topology of the knot complement is important in knot theory. Given a knot \(K\subset S^{3}\), the boundary \(T:=\partial (U(K))\) of a tubular neighborhood \(U(K)\) of \(K\) in \(S^{3}\) is a torus. The authors take the conormal \(\nu^{*}T\) of the torus \(T\) in the cotangent bundle \(T^{*}(S^{3}\setminus K)\) of the knot complement, which is a Lagrangian submanifold of \(T^{*}(S^{3}\setminus K)\) and defined the wrapped Floer complex which is generated by the set of Hamiltonian chords of kinetic energy Hamiltonian associated to a Riemannian metric on the complement with boundaries on the Lagrangian [\textit{Y. Bae} et al., ``A wrapped Fukaya category of knot complement'', Preprint, \url{arXiv:1901.02239}]. The \(A_{\infty}\)-structure on the complex is defined by counting pseudo-holomorphic disks bounding the Hamiltonain chords and the Langrangian submanifolds (under the Hamiltonian diffeomorphism). A knot is called hyperbolic if the knot complement admits a hyperbolic metric. For example, the figure-eight knot is hyperbolic. Using special properties of hyperbolic geometry, the authors prove that the \(A_{\infty}\) structure maps of the wrapped Floer complex of hyperbolic-knot complement vanishes except degree \(2\) and the resulting reduced wrapped Floer cohomology(equivalently, knot Floer cohomology) is concentrated to degree \(0\) [\textit{Y. Bae} et al., Asian J. Math. 25, No. 1, 117--176 (2021; Zbl 07413486)]. They also showed that, for torus knots, the reduced wrapped Floer cohomology is non-zero for degree \(0,1\) and that the wrapped Floer cohomology distinguishes hyperbolic knots and torus knots. A very rough outline of the proof of the main theorem is following: The hyperbolic-knot complement has the hyperbolic \(3\)-space \(\mathbb{H}^{3}:=\{(x,y,z)\in \mathbb{R}^{2}\times \mathbb{R}^{+}\}\) with isometric \(\mathrm{PSL}(2,\mathbb{C})\)-action as its universal cover. By the Margulis' thick-thin decomposition, one can take the torus \(T:=\partial (U(K))\) as a level set of the Busemann function which is lifted to \(log z\), under which \(T\) is lifted to a horosphere centered at \(\{z=+\infty\}\). Using property of horo-sphere and \(\mathrm{PSL}(2,\mathbb{C})\)-action, the set of geodesic cord in \(S^{3}\setminus K\) is bijective to the set of images of all infinite tame geodesics in \(S^{3}\setminus K\) (Proposition 3.6). Moreover, the set of geodesic cord in \(S^{3}\setminus K\) is bijective to the set of Hamiltonian chord of \(\nu^{*}T\subset S^{3}\setminus K\) (Lemma 5.2). From the non-negativity of the second variation of the energy functional, both nullity and Morse index of any geodesic cord vanish (Theorem 1.1). Therefore, all the degrees of generators are \(0\), which implies, by the dimension formula, that the structure maps vanish except degree 2 (Theorem 1.3). By the maximum principle, for any Floer disk bounding triple of Hamiltonian chords, there exists a totally geodesic immersed ideal triangle \(\delta\subset S^{3}\setminus K\) such that the ideal edges contain geodesic triples corresponding to the chords and that it contains the projected image of the Floer disk (Theorem 1.5). Technical results include vertical \(C^{0}\) estimates, uniform horizontal \(C^{0}\) estimates and the maximum principle which lead to the well-definedness of the wrapped Floer cohomology. There are also other contact/symplectic approaches to knot invariants, for example, [\textit{T. Ekholm} et al., Geom. Topol. 17, No. 2, 975--1112 (2013; Zbl 1267.53095); \textit{T. Ekholm} et al., Invent. Math. 211, No. 3, 1149--1200 (2018; Zbl 1385.57015); \textit{V. Shende}, Forum Math. Pi 7, Paper No. e6, 16 p. (2019; Zbl 1426.57021)].