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

TL;DR

This paper studies the algebraic structures associated with hyperbolic knots in 3-manifolds, showing that wrapped Floer cohomology relates to knot Floer cohomology and providing obstructions for hyperbolic knots.

Contribution

It establishes the formality of the Floer complex for the ideal boundary of hyperbolic knot complements and relates wrapped Floer cohomology to knot Floer cohomology, revealing new invariants.

Findings

Wrapped Floer cohomology is well-defined and isomorphic to knot Floer cohomology.

The $A_$-algebra reduces to a noncommutative algebra concentrated in degree 0.

Non-vanishing reduced Floer cohomology obstructs hyperbolic knots.

Abstract

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 $\nu^*T$ 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 $\nu^*T$ is reduced to a noncommutative algebra concentrated to degree 0. We prove that the wrapped Floer cohomology $HW(\nu^*T; H_h)$ 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 $\widetilde{HW}^1(\partial_\infty(M \setminus K)) \neq 0$.