A wrapped Fukaya category of knot complement

Abstract

This is the first of a series of two articles where we construct a version of wrapped Fukaya category WF(M\K; H-g0) of the cotangent bundle T *(M\K) of the knot complement M\K of a compact 3-manifold M , and do some calculation for the case of hyperbolic knots K ? M. For the construction, we use the wrapping induced by the kinetic energy Hamiltonian H-g0 associated to the cylindrical adjustment (g0) on M\K of a smooth metric g defined on M. We then consider the torus T = ?N(K) as an object in this category and its wrapped Floer complex CW*(?*T; H-g0) where N(K) is a tubular neighborhood of K ? M. We prove that the quasi-equivalence class of the category and the quasi-isomorphism class of the A00 algebra CW*(?*T; H-g0) are independent of the choice of cylindrical adjustments of such metrics depending only on the isotopy class of the knot K in M. In a sequel (Bae et al. in Asian J Math 25(1):117-176, 2019), we give constructions of a wrapped Fukaya category WF(M\K; H-h) for hyperbolic knot K and of A00 algebra CW*(?*T; H-h) directly using the hyperbolic metric h on M\K , and prove a formality result for the asymptotic boundary of (M\K , h).