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Register a new userLegendrian torus knots in S^1times S^2
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We classify the Legendrian torus knots in S^1times S^2 with its standard tight contact structure up to Legendrian isotopy.
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We establish a number of results about smooth and topological concordance of knots in $S^1times S^2$. The winding number of a knot in $S^1times S^2$ is defined to be its class in $H_1(S^1times S^2;mathbb{Z})cong mathbb{Z}$. We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell in the topological category. We say a knot in $S^1times S^2$ is slice (resp. topologically slice) if it bounds a smooth (resp. locally flat) disk in $D^2times S^2$. We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than $pm 1$, there are infinitely many topological concordance classes even within the collection of slice knots. Additionally we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and topologically concordant, but not smoothly concordant.
In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.
All knots in $R^3$ possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $R^3$ can be defined. The definitions extend easily to null-homologous knots in any $3$-manifold $M$ endowed with a contact structure $xi$. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the $3$-torus $T^3$. We show how to compute the Thurston-Bennequin and rotation invariants in a tight oriented contact structure on $T^3$ using projections.
The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e. Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. This part was essentially written more than 10 years ago, but only a short version, without the detailed proofs, was published (in CRM Proc. Lecture Notes, Vol. 15, 1998). That paper also briefly discussed the overtwisted case. The final part of the present paper contains a more systematic discussion of Legendrian knots in overtwisted contact manifolds, and in particular, gives the coarse classification (i.e. classification up to a global contactomorphism) of topologically trivial Legendrian knots in overtwisted contact S^3.
An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. In contrast to Legendrian knots, virtual Legendrian knots enjoy the property that there is a bijective correspondence between the virtual Legendrian knots and the equivalence classes of Gauss diagrams. We study virtual Legendrian isotopy classes of Legendrian links and show that every such class contains a unique irreducible representative. In particular we get a solution to the following conjecture of Cahn, Levi and the first author: two Legendrian knots in $ST^*S^2$ that are isotopic as virtual Legendrian knots must be Legendrian isotopic in $ST^*S^2.$
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