Non-loose negative torus knots

Abstract

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 show that the strongly non-loose transverse realizations $T$ are classified by their non-zero invariants $\mathfrak{T}(T)$ in the minus version of the knot Floer homology. However, not all the elements of $HF{K}^{-}(T_{(p,q)})$ can be realized. 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.

arXiv comment17 pages, 3 figures. To appear in Quantum Topol

References in corpus (12)

• Floer homology and knot complements
• On the classification of tight contact structures I
• On the existence of tight contact structures on Seifert fibered 3-manifolds
• Knots and Contact Geometry
• Heegaard Floer invariants of Legendrian knots in contact three--manifolds
• Sutured Floer homology and invariants of Legendrian and transverse knots
• Non-looseness of non-loose knots
• The Legendrian knot complement problem
• Fillability of small Seifert fibered spaces
• On loose Legendrian knots in rational homology spheres
• On invariants for Legendrian knots
• On contact surgery and knot Floer invariants

Cited by in corpus (2)

• On contact surgery and knot Floer invariants
• Strongly exceptional Legendrian connected sum of two Hopf links