Non-loose negative torus knots


Matkovic, I


Quantum Topology

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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 look at the Legendrian invariants of these knots in the
minus version of the knot Floer homology, obtaining that U · L(L) vanishes for
any Legendrian negative torus knot L in any overtwisted structure, and that
the strongly non-loose transverse realizations T are classified by their non-zero
invariant 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 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.

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Journal Article