Search

Casals-Gorsky-Gorsky-Simental realized all positroid strata of the complex Grassmannian as augmentation varieties of Legendrians called positroid links. We prove that the partial order on strata induced by Zariski closure also has a symplectic interpretation, given by exact Lagrangian cobordism., 16 pages, 15 figures

This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher-dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many non-orientable exact Lagrangian fillings., Comment: 20 pages

Given an immersed, Maslov-$0$, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-$0$, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-$0$, exact Lagrangian filling with genus $g \geq 1$ and $p$ double points can be obtained from such a Lagrangian surgery on a filling of genus $g-1$ with $p+1$ double points. To show this, we establish the connection between the existence of an immersed, Maslov-$0$, exact Lagrangian filling of a Legendrian $\Lambda$ that has $p$ double points with action $0$ and the existence of an embedded, Maslov-$0$, exact Lagrangian cobordism from $p$ copies of a Hopf link to $\Lambda$. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-$0$, exact Lagrangian cobordisms between Legendrian links., Comment: 51 pages, 19 figures. Comments welcome!

We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of Legendrian links with infinitely many distinct planar exact Lagrangian fillings, which can be viewed as the smallest Legendrian links currently known to have infinitely many distinct exact Lagrangian fillings. As an application we find new examples of infinitely many exact Lagrangian spheres and tori $4$-dimensional Milnor fibers of isolated hypersurface singularities with positive modality. We also provide a generalization of a criterion for when the symplectic homology of a Weinstein $4$-manifold is non-vanishing given an explicit Weinstein handlebody diagram.

We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centered condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the toric divisor supports an explicit Weinstein structure. Many examples which fail this condition also fail to have Weinstein (or even exact) complement to the partially smoothed divisor. We investigate the combinatorial possibilities of Delzant polytopes that realize such Weinstein domain complements. We also develop an algorithm to construct a Weinstein handlebody diagram in Gompf standard form for the complement of such a partially smoothed toric divisor. The algorithm we develop more generally outputs a Weinstein handlebody diagram for any Weinstein 4-manifold constructed by attaching 2-handles to the disk cotangent bundle of any surface $F$, where the 2-handles are attached along the co-oriented conormal lifts of curves on $F$. We discuss how to use these diagrams to calculate invariants and provide numerous examples applying this procedure. For example, we provide Weinstein handlebody diagrams for the complements of the smooth and nodal cubics in $\mathbb{CP}^2$., Comment: 87 pages, 93 figures, comments welcome! Many expositional improvements, expansions, and clarifications in v2

In this article, we provide an introduction to an algorithm for constructing Weinstein handlebodies for complements of certain smoothed toric divisors using explicit coordinates and a simple example. This article also serves to welcome newcomers to Weinstein handlebody diagrams and Weinstein Kirby calculus. Finally, we include one complicated example at the end of the article to showcase the algorithm and the types of Weinstein Kirby diagrams it produces., Comment: 21 pages, 28 figures. Improvements and clarifications thanks to referee suggestions, as well as added references to the new article

An $n$-crossing is a point in the projection of a knot where $n$ strands cross so that each strand bisects the crossing. An \"ubercrossing projection has a single $n$-crossing and a petal projection has a single $n$-crossing such that there are no loops nested within others. The \"ubercrossing number, $\text{\"u}(K)$, is the smallest $n$ for which we can represent a knot $K$ with a single $n$-crossing. The petal number is the number of loops in the minimal petal projection. In this paper, we relate the \"{u}bercrossing number and petal number to well-known invariants such as crossing number, bridge number, and unknotting number. We find that the bounds we have constructed are tight for $(r, r+1)$-torus knots. We also explore the behavior of \"{u}bercrossing number under composition., Comment: 13 pages, 8 figures

An $n$-crossing is a point in the projection of a knot where $n$ strands cross so that each strand bisects the crossing. An ��bercrossing projection has a single $n$-crossing and a petal projection has a single $n$-crossing such that there are no loops nested within others. The ��bercrossing number, $\text{��}(K)$, is the smallest $n$ for which we can represent a knot $K$ with a single $n$-crossing. The petal number is the number of loops in the minimal petal projection. In this paper, we relate the ��bercrossing number and petal number to well-known invariants such as crossing number, bridge number, and unknotting number. We find that the bounds we have constructed are tight for $(r, r+1)$-torus knots. We also explore the behavior of ��bercrossing number under composition., 13 pages, 8 figures