Related papers: Legendrian rational unknots in lens spaces
Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of…
Each ruling of a Legendrian link can be naturally treated as a surface. For knots, the ruling is 2-graded if and only if the surface is orientable. For 2-graded rulings of homogeneous (in particular, alternating) knots, we prove that the…
The quantum lens spaces form a natural and well-studied class of noncommutative spaces which can be subjected to classification using algebraic invariants by drawing on the fully developed classification theory of unital graph…
We study the space of real rational curves of low degree in the quadric of signature $(3,2)$ and provides a classificaton of real rational knots and nodal curves. Apart from the classification, we also study the relationship between the…
We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove…
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…
We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology…
Berge in [1] defined doubly primitive knots, which yield lens spaces by Dehn surgery. At the same paper he listed the knots into several types. In this paper we will prove the list is complete when $\tau>1$. The invariant $\tau$ is a…
We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize…
We classify all the lattices realized as the intersection form of a positive definite four manifold with boundary $S_n^3(K)$ for a knot $K$ in the three sphere and a positive integer $n$ greater than $4g_4(K)+3$. We then use this result to…
In this short note we observe that a result of Eliashberg and Polterovitch allows to use the doubly slice genus as an obstruction for a Legendrian knot to be a slice of a concordance from the trivial Legendrian knot with maximal…
We give an $h$--principle type result for a class of Legendrian embeddings in contact manifolds of dimension at least $5$. These Legendrians, referred to as loose, have trivial pseudo-holomorphic invariants. We demonstrate they are…
In the first part of the paper we present a classification of fake lens spaces of dimension >= 5 whose fundamental group is the cyclic group of order N >= 2. The classification uses and extends the results of Wall and others in the case N =…
The curves of zero intensity of a complex optical field can form knots and links: optical vortex knots. Both theoretical constructions and experiments have so far been restricted to the very small families of torus knots or lemniscate…
We study the properties of glued knots, a sub-class of real rational knots, that can be constructed by gluing ellipses. We define an invariant called the gluing degree and relate it to various classical properties of knots and classify all…
The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of…
For a knot in the 3-sphere, the Upsilon invariant is a piecewise linear function defined on the interval [0,2]. For an L-space knot, the Upsilon invariant is determined only by the Alexander polynomial of the knot. We exhibit infinitely…
We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite…
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can…
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the…