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

Yong-Geun Oh (Center for Geometry and Physics, Institute for Basic Sciences (IBS), Pohang, South Korea; and Department of Mathematics, POSTECH, Pohang, South Korea)

Abstract

This is a sequel to the authors’ article [BKO]. 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 $T^\ast (M \setminus K)$ with the induced kinetic energy Hamiltonian $H_h = \frac{1}{2} {\lvert p \rvert}^2_h$ and Sasakian almost complex structure $J_h$, and associate a wrapped Fukaya category to $T^\ast (M \setminus K)$ whose wrapping is given by $H_h$. We then consider the conormal $\nu^\ast T$ 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 $\tilde{\mathfrak{m}}^k = 0$ unless $k \neq 2$ and an $A_\infty$-algebra associated to $\nu^\ast T$ is reduced to a noncommutative algebra concentrated to degree $0$. We prove that the wrapped Floer cohomology $HW (\nu^\ast T ; H_h)$ 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 $\widetilde{HW}^1 (\partial_\infty (M \setminus K)) \neq 0$.

Keywords

hyperbolic knots, Knot Floer algebra, horo-torus, formality, totally geodesic triangle

2010 Mathematics Subject Classification

Primary 53D35. Secondary 57M27.

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Y.B. was partially supported by IBS-R003-D1 and JSPS International Research Fellowship Program.

S.K. is supported by the IBS project IBS-R003-D1. He is also partially supported by the National Research Foundation(NRF) no. 2019R1C1C1003383.

Y.O. is supported by the IBS project IBS-R003-D1. He was also partially supported by the National Science Foundation under Grant No. DMS-1440140 during his residence at the Mathematical Sciences Research Institute in Berkeley, California in the fall of 2018.

Received 13 May 2019

Accepted 20 May 2020

Published 30 September 2021