The characterizing contact slope conjecture for Legendrian figure-eight and torus knots

A characterizing contact slope for a Legendrian knot is a contact surgery slope that distinguishes the knot from every Legendrian knot with contactomorphic surgery. The claim concerns any Legendrian figure-eight knot or torus knot, including the unknot. Characterizing contact slope conjecture. The slope $\pm1$ is a characterizing contact slope for any Legendrian figure-eight knot or torus knot, including the unknot. The conjecture is motivated by the fact that the knots considered in the paper's main characterization results are smoothly characterized by surgeries; its resolution status is not specified in the supplied text.