More Like this

1. Computing a Link Diagram from its Exterior

   Dunfield, Nathan M.; Obeidin, Malik; Rudd, Cameron Gates (January 2022, Leibniz international proceedings in informatics)

   Goaoc, Xavier; Kerber, Michael (Ed.)

   A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture. 

   more » « less
2. Contact surgery numbers

   https://doi.org/10.4310/JSG.2023.v21.n6.a4  

   Etnyre, John; Kegel, Marc; Onaran, Sinem (January 2023, Journal of Symplectic Geometry)

   It is known that any contact $$3$$-manifold can be obtained by rationally 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. 

   more » « less
3. Fillable contact structures from positive surgery

   https://doi.org/10.1090/btran/200  

   Mark, Thomas; Tosun, Bülent (August 2024, Transactions of the American Mathematical Society, Series B)

   We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot $K$in a contact 3-manifold $Y$, gives rise to a weakly fillable contact structure. We show that this happens if and only if $Y$itself is weakly fillable, and $K$is isotopic to the boundary of a properly embedded symplectic disk inside a filling of $Y$. Moreover, if $Y^{\prime }$is a contact manifold arising from positive contact surgery along $K$, then any filling of $Y^{\prime }$is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of $Y$. Using this result we deduce several necessary conditions for a knot in the standard 3-sphere to admit a fillable positive surgery, such as quasipositivity and equality between the slice genus and the 4-dimensional clasp number, and we give a characterization of such knots in terms of a quasipositive braid expression. We show that knots arising as the closure of a positive braid always admit a fillable positive surgery, as do knots that have lens space surgeries, and suitable satellites of such knots. In fact the majority of quasipositive knots with up to 10 crossings admit a fillable positive surgery. On the other hand, in general, (strong) quasipositivity, positivity, or Lagrangian fillability need not imply a knot admits a fillable positive contact surgery. 

   more » « less
4. A remark on the geography problem in Heegaard Floer homology

   https://doi.org/10.1090/pspum/102/08  

   Hanselman, Jonathan; Kutluhan, Çagatay; Lidman, Tye (July 2019, Proceedings of symposia in pure mathematics)

   Gay, David; Wu, Weiwei (Ed.)

   We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this corollary to show that the chain complex depicted by Ozsváth, Stipsicz, and Szabó to argue that there is no algebraic obstruction to the existence of knots with trivial epsilon invariant and non-trivial upsilon invariant cannot be realized as the knot Floer complex of a knot. 

   more » « less
5. Bordered Floer homology and contact structures

   https://doi.org/10.1017/fms.2023.19  

   Alishahi, Akram; Földvári, Viktória; Hendricks, Kristen; Licata, Joan; Petkova, Ina; Vértesi, Vera (January 2023, Forum of Mathematics, Sigma)

   Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $$(M, \xi , \mathcal {F})$$ whose convex boundary is equipped with a signed singular foliation $$\mathcal {F}$$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $$c_D$$ and $$c_A$$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $$\mathcal {F}$$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić. 

   more » « less