Sutured contact homology, conormal stops and hyperbolic knots

We apply the conormal construction to a hyperbolic knot $K\subset S^3$, and study the sutured contact manifold $(V,\xi )$ obtained by taking the complement of a standard neighbourhood of the unit conormal ${\text{La}}_K\subset (S{T}^{\ast }{S}^3,{\xi }_{\text{st}})$. We show that the sutured Legendrian contact homology of a unit fiber ${\text{La}}_0$, with its product structure, is a complete invariant of the knot (up to mirror). This can also be seen as the computation of the homology of the fiber in $S{T}^{\ast }{S}^3$, stopped at ${\text{La}}_K$. Our main tool is, for any submanifold $N\subset M$, an explicit relationship between the complement of a unit conormal ${\text{La}}_N$, and the unit bundle of $M\setminus N$.