2019/01/08 by Youngjin Bae, Seonhwa Kim, Bae, Youngjin +3
Mathematics · Medicine · #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1901.02258
openalex publication_date 2019/01/08 · openalex created_date 2019/01/25 · openalex updated_date 2026/07/28
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 ∖ K. We equip M ∖ K with a hyperbolic metric h and its cotangent bundle T^*(M ∖ K) with the induced kinetic energy Hamiltonian Hh = (1)/(2) |p|h2 and Sasakian almost complex structure Jh, and associate a wrapped Fukaya category to T^*(M∖ K) whose wrapping is given by Hh. 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 mk = 0 unless k ≠ 2 and an A_∞-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; Hh) with respect to Hh is well-defined and isomorphic to the Knot Floer cohomology HW(∂_∞(M ∖ K)) that was introduced in [BKO] for arbitrary knot K ⊂ M. We also define a reduced cohomology, denoted by \widetildeHWd(∂_∞(M ∖ K)), by modding out constant chords and prove that if \widetildeHWd(∂_∞(M ∖ K))≠ 0 for some d ≥ 1, then K cannot be hyperbolic. On the other hand, we prove that all torus knots have \widetildeHW1(∂_∞(M ∖ K)) ≠ 0.