Fiber Floer cohomology and conormal stops

Let S be a closed orientable spin manifold. Let K ⊂ S be a sub-manifold and denote its complement by M_K . In this paper we prove that there exists an isomorphism between partially wrapped Floer cochains of a cotangent fiber stopped by the unit conormal Λ_K and chains of a Morse theoretic model of the based loop space of M_K, which intertwines the A_∞-structure with the Pontryagin product. As an application, we restrict to codimension 2 spheres K ⊂ Sⁿ where n = 5 or n ≥ 7. Then we show that there is a family of knots K so that the partially wrapped Floer cohomology of a cotangent fiber is related to the Alexander invariant of K. A con-sequence of this relation is that the link Λ_K ∪ Λ_x is not Legendrian isotopic to Λ_unknot ∪ Λ_x where x ∈ M_K .