Non-loose negative torus knots

We study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. Additionally, we study the Legendrian invariants of these knots in the minus version of the knot Floer homology, obtaining that $U\cdot\mathfrak L(L)$ vanishes for any Legendrian negative torus knot $L$ in any overtwisted structure, and that the strongly non-loose transverse realizations $T$ are characterized by having non-zero invariant $\mathfrak T(T)$. Along the way, we relate our Legendrian realizations to the tight contact structures on the Legendrian surgeries along them. Specifically, we realize all tight structures on the lens spaces $L(pq+1,p^2)$ as a single Legendrian surgery on a Legendrian $T_{(p,-q)}$, and we relate transverse realizations in overtwisted structures to the non-fillable tight structures on the large negative surgeries along the underlying knots.