Legendrian Hopf links in L(p,1)
We classify Legendrian realisations, up to coarse equivalence, of the Hopf link in the lens spaces L(p,1) with any contact structure.
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arXiv subjects
Sinem Onaran
Publications and source records attributed to Sinem Onaran.
Non-looseness of boundaries of Legendrian ribbons
Every null-homologous link in an oriented 3-manifold is isotopic to the boundary of a ribbon of a Legendrian graph for any overtwisted contact structure. However this is not the case if the boundary is required to be non-loose. Here, we define the `Tight Reattachment Property' for a Legendrian graph and show that it implies the boundary of its ribbon is non-loose. We also discuss the applicability of this property and examine examples and constructions of Legendrian graphs with this property.
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Strongly exceptional Legendrian connected sum of two Hopf links
In this paper, we give a complete coarse classification of strongly exceptional Legendrian realizations of connected sum of two Hopf links in contact 3-spheres. This is the first classification result about exceptional Legendrian representatives for connected sums of link families.
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Contact surgery graphs
We define a graph encoding the structure of contact surgery on contact 3-manifolds and analyze its basic properties and some of its interesting subgraphs.
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Contact surgery numbers
It is known that any contact 3-manifold can be obtained by rational contact Dehn surgery along a Legendrian link L in the standard tight contact 3-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link L describing a given contact 3-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three. In the second part, we compute contact surgery numbers of all contact structures on the 3-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $S^1\times S^2$, the Poincar\'e homology sphere, and the Brieskorn sphere $\Sigma(2,3,7)$. We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $3$-sphere. We further obtain results for the 3-torus and lens spaces. As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.
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Minimal number of singular fibers in a nonorientable Lefschetz fibration
We show that there exists an admissible nonorientable genus $g$ Lefschetz fibration with only one singular fiber over a closed orientable surface of genus $h$ if and only if $g \geq 4$ and $h \geq 1$.
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Real tight contact structures on lens spaces and surface singularities
We classify the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on $S^3$ and real lens spaces $L(p,\pm 1)$. We prove that there is a unique real tight $S^3$ and $\mathbb{R}P^3$. We show there is at most one real tight $L(p,\pm 1)$ with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bounds we explicitly construct real tight manifolds through equivariant contact surgery, real open book decompositions and isolated real algebraic surface singularities. As a by-product we observe the existence of an invariant torus in an $L(p,p-1)$ which cannot be made convex equivariantly.
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On the Support Genus of Legendrian Knots
In this paper, we show that any topological knot or link in $S^1 \times S^2$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in $S^1 \times S^2$ has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces $L(p,1)$.
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L-space knots with tunnel number >1 by experiment
In Dunfield's catalog of the hyperbolic manifolds in the SnapPy census which are complements of L-space knots in $S^3$, we determine that $22$ have tunnel number $2$ while the remaining all have tunnel number $1$. Notably, these $22$ manifolds contain $9$ asymmetric L-space knot complements. Furthermore, using SnapPy and KLO we find presentations of these $22$ knots as closures of positive braids that realize the Morton-Franks-Williams bound on braid index. The smallest of these has genus $12$ and braid index $4$.
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Legendrian Hopf links
We completely classify Legendrian realisations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.
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Exceptional Legendrian torus knots
We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bounds on the number of tight contact structures on the complements of torus knots; the classification of exceptional realisations of these torus knots is then achieved by exhibiting sufficiently many realisations in terms of contact surgery diagrams. We also discuss a couple of general theorems about the existence of exceptional Legendrian knots.
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On overtwisted contact surgeries
In this note, we obtain a new result concluding when contact (+1/n)-surgery is overtwisted. We give a counterexample to a conjecture by James Conway on overtwistedness of manifolds obtained by contact surgery. We list some problems related to the contact surgery.
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Legendrian torus knots in lens spaces
In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.
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Legendrian lens space surgeries
We show that every tight contact structure on any of the lens spaces $L(ns^2-s+1,s^2)$ with $n\geq 2$, $s\geq 1$, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot $T(s,-(sn-1))$ in the tight or an overtwisted contact structure on the 3-sphere.
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Non-looseness of non-loose knots
A Legendrian or transverse knot in an overtwisted contact 3-manifold is non-loose if its complement is tight and loose if its complement is overtwisted. We define three measures of the extent of non-looseness of a non-loose knot and show they are distinct.
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Legendrian rational unknots in lens spaces
We classify Legendrian rational unknots with tight complements in the lens spaces L(p,1) up to coarse equivalence. As an example of the general case, this classification is also worked out for L(5,2). The knots are described explicitly in a contact surgery diagram of the corresponding lens space.
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