arXiv 1901.022582019-01math.SGmath.GT

双曲纽结补理想边界的Floer复形的形式

Formality of Floer complex of the ideal boundary of hyperbolic knot complement

Youngjin Bae, Seonhwa Kim, Yong-Geun Oh

备注：: 52 pages, 1 figure; v2 57 pages, 2 figures, calculations for torus knots added, abstract and introduction rewritten, mistakes in the statements and proofs of Proposition 2.1 and Lemma 9.4 corrected, old Section 11 moved to Appendix B, typos corrected

摘要：

这是作者文章[BKO]（arXiv：1901.02239）的续篇。本文考虑了闭三维流形$M$中的双曲纽结$K$及其补流形$M \setminus K$的余切丛.我们用双曲度量$h$赋予$M \setminus K$，用诱导动能哈密顿量$H_h = \frac{1}{2} |p|_h^2$和Sasakian几乎复结构$J_h$赋予它的余切丛$T^*(M \setminus K)$，并把一个包裹的范畴关联到$T^*(M\setminus K)$，其包裹由$H_h$给出.然后，我们考虑其对象的horo-torus $T$的余法$ν^*T$。我们证明了所有的非常数哈密顿弦是横向的和Morse指数0相对于horo环面$T$，使结构映射满足$\widetilde{\mathfrak m}^k = 0$，除非$k \neq 2$和一个$A_\infty$ -代数与$ν^*T$减少到一个非交换代数集中到度0。我们证明了包装Floer上同调$HW(ν^*T; H_h)$关于$H_h$是定义良好的，同构的结Floer上同调$HW(\partial_\infty(M \setminus K))$，在[BKO]中介绍了任意结$K \subset M$。我们还定义了一个约化的上同调，表示为$\widetilde{HW}^d(\partial_\infty(M \setminus K))$，通过修改出常数弦，并证明了，如果$\widetilde{HW}^d(\partial_\infty(M \setminus K))\neq 0$的一些$d \geq 1$，那么$K$不能是双曲的。另一方面，我们证明了所有的环面纽结有$\widetilde{HW}^1(\partial_\infty(M \setminus K)) \neq 0$。

英文摘要：

This is a sequel to the authors' article [BKO](arXiv:1901.02239). We consider a hyperbolic knot $K$ in a closed 3-manifold $M$ and the cotangent bundle of its complement $M \setminus K$. We equip $M \setminus K$ with a hyperbolic metric $h$ and its cotangent bundle $T^*(M \setminus K)$ with the induced kinetic energy Hamiltonian $H_h = \frac{1}{2} |p|_h^2$ and Sasakian almost complex structure $J_h$, and associate a wrapped Fukaya category to $T^*(M\setminus K)$ whose wrapping is given by $H_h$. We then consider the conormal $ν^*T$ of a horo-torus $T$ as its object. We prove that all non-constant Hamiltonian chords are transversal and of Morse index 0 relative to the horo-torus $T$, and so that the structure maps satisfy $\widetilde{\mathfrak m}^k = 0$ unless $k \neq 2$ and an $A_\infty$-algebra associated to $ν^*T$ is reduced to a noncommutative algebra concentrated to degree 0. We prove that the wrapped Floer cohomology $HW(ν^*T; H_h)$ with respect to $H_h$ is well-defined and isomorphic to the Knot Floer cohomology $HW(\partial_\infty(M \setminus K))$ that was introduced in [BKO] for arbitrary knot $K \subset M$. We also define a reduced cohomology, denoted by $\widetilde{HW}^d(\partial_\infty(M \setminus K))$, by modding out constant chords and prove that if $\widetilde{HW}^d(\partial_\infty(M \setminus K))\neq 0$ for some $d \geq 1$, then $K$ cannot be hyperbolic. On the other hand, we prove that all torus knots have $\widetilde{HW}^1(\partial_\infty(M \setminus K)) \neq 0$.

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