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We produce the first examples relating non-orientable exact Lagrangian fillings of Legendrian links to cluster theory, showing that the ungraded augmentation variety of certain max-tb representatives of Legendrian $2$-bridge links is isomorphic to a product of $A_n$-type cluster varieties. As part of this construction, we describe a surjective map from the set of (possibly non-orientable) exact Lagrangian fillings to cluster seeds, producing a product of Catalan numbers of distinct fillings. We also relate the ruling stratification of the ungraded augmentation variety to Lam and Speyer's anticlique stratification of acyclic cluster varieties, showing that the two coincide in this context. As a corollary, we apply a result of Rutherford to show that the cluster-theoretic stratification encodes the information of the lowest $a$-degree term of the Kauffman polynomial of the smooth isotopy class of the $2$-bridge links we study., Comment: 46 pages, 18 figures; v2: corrected statement of Prop. 5.15 and added clarifications following referee comments

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: 21 pages, 7 figures. A published version of this manuscript will appear in the Bulletin of the London Mathematical Society

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., Comment: 18 pages, 14 figures. Final version to appear in Pacific J. Math

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 use exact Lagrangian fillings and Weinstein handlebody diagrams to construct infinitely many distinct exact Lagrangian tori in $4$-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 diagrams., Comment: Major revisions. Theorem 1.1 from version 1 was incorrect and has been removed. New examples added in section 5

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

In the symplectization of standard contact $3$-space, $\mathbb R \times \mathbb R^3$, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus $0$. We show that any Legendrian knot has a non-orientable Lagrangian endocobordism, and that the crosscap genus of such a non-orientable Lagrangian endocobordism must be a positive multiple of $4$. The more restrictive exact, non-orientable Lagrangian endocobordisms do not exist for any exactly fillable Legendrian knot but do exist for any stabilized Legendrian knot. Moreover, the relation defined by exact, non-orientable Lagrangian cobordism on the set of stabilized Legendrian knots is symmetric and defines an equivalence relation, a contrast to the non-symmetric relation defined by orientable Lagrangian cobordisms., Comment: 23 pages, 18 figures

A multi-crossing (or n-crossing) is a singular point in a projection at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite, which gives the upper bound on the span of the bracket polynomial of K as 4c_2(K), to the n-crossing number: span is bounded above by ([n^2/2] + 4n-8) c_n(K) for all integers n at least 3. We also explore n-crossing additivity under composition, and find that for n at least 4, there are examples of knots such that the n-crossing number is sub-additive. Further, we present the first extensive list of calculations of n-crossing numbers for knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum., Comment: 28 pages, 19 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 \"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