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On small Seifert fibered spaces M(e0;r1,r2,r3) with e₀≠−1,−2, all tight contact structures are Stein fillable. This is not the case for e₀=−1 or −2. However, for negative twisting structures it is expected that they are all symplectically fillable. Here, we characterise fillable structures among zero-twisting contact structures on small Seifert fibered spaces of the form M(−1;r1,r2,r3). The result is obtained by analysing monodromy factorizations of associated planar open books.

We establish some general relations between Heegaard-Floer-based contact invariants. In particular, we observe that if the contact invariant of large negative, respectively positive, contact surgeries along a Legendrian knot does not vanish, then the Legendrian invariant, respectively the Legendrian inverse limit invariant, of that knot is non-zero. We use sutured Floer homology and the limit constructions due to Golla and Etnyre, Vela-Vick and Zarev.

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 T(T ) in the minus version of the knot Floer homology. However, not all the elements of HFK_(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 space L(pq + 1; p2) 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.