Abstract

I'll discuss a number of (arguably) new holomorphic-curve invariants of Legendrian knots and links. These come from an L-infinity structure on commutative Legendrian contact homology, derived from rational symplectic field theory. The new invariants include a symplectic structure on the augmentation variety of a Legendrian link, as well as a Poisson bracket (on a polynomial ring) associated to any positive braid. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.

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Three-manifolds (and closed braids inside them) admit descriptions called open book decompositions; in this setting a surface with boundary and a mapping class describe the 3-manifold (and braid). One invariant of an open book is the fractional Dehn twist coefficient (FDTC). The FDTC is a real number invariant of a mapping class of a surface with boundary, which has connections to contact topology and foliation theory. I'll show that the FDTC of a given mapping class can be computed using a multitude of geometrically defined left-orders on the mapping class group. This is joint work with Diana Hubbard.

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The band rank of a braid is the length of its shortest decomposition into a product of conjugates of Artin generators. Using braided surfaces the band rank of a braid can be used to express the ribbon genus of its closure, and thus it is very difficult to compute in general. In this talk I will outline computational approaches to producing upper and lower bounds on the band rank of a braid. This is joint work with Justin Meiners.

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Braided embeddings are a natural generalization of closed braids in three dimensions, which gives a way to construct many higher dimensional embeddings. We will discuss the lifting and isotopy problem for braided embeddings. Time permitting, we will also mention some applications to the existence of nice branched coverings, contact geometry and homomorphisms between braid groups.

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Kodaira classified all singular fibers that can arise in an algebraic fibration with genus 1 fiber. Sakali and I have been looking into a similar classification for genus 2 fibrations by Ogg, Iitaka and later Namikawa and Ueno and we determine how these singular fibrations deform into Lefschetz fibrations. The gamut of tools runs from deformation theory, to Lefschetz fibrations and open books, to braids and braided surfaces, and the branched covers that relate all of them. This is joint work with S. Sakalli.

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The Nielsen-Thurston classification theorem states that mapping classes fall into three types: periodic, reducible, and pseudo-Anosov. One of the famous open problems in the study of mapping class groups is to show that pseudo-Anosov mapping classes are generic. This problem remains open, but it has been proven in one sense. In this talk, I will introduce this open problem and share the progress which has been made in solving the problem.

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In 3-dimensional contact geometry, tight contact structures constitute a geometrically interesting class of contact structures because they form natural boundary conditions for symplectic and certain complex 4-manifolds, and moreover they are deeply related to the topology of 3-manifolds, and Heegaard and Monopole Floer theories. An outstanding open problem in 3-dimensional contact geometry concerns the classification of tight contact structures: When a closed oriented 3-manifold admits a tight contact structure, can one classify all tight contact structures on the manifold? A great deal of important work in the last 25 years has been put towards the resolution of this fundamental question. But at the moment it is fair to say a thorough understanding is far from complete. For example, if one considers the classification question on 3-manifolds obtained by Dehn filling of knots in three sphere, then currently the only such complete classification result for all surgeries available is for the unknot. In this talk, I will report on ongoing joint work with J. Etnyre and H. Min that gives the first complete classification result for all surgeries on an infinite family of torus knots. I will provide further context and motivations for the result, and give some details of the proof.

Abstract

We investigate braid invariants, such as the writhe and the fractional Dehn twist coefficient (FDTC), that arise as the homogenization of a concordance homomorphism, such as tau and Upsilon from the Heegaard Floer tool box. After providing a new characterization of the FDTC, we turn to connections with low-dimensional topology via the concept of homogenization of knot invariants. Concretely, we view the slice Bennequin inequality---a celebrated  inequality due to Kronheimer-Mrowka and Rudolph that relates a knot invariant (the smooth 4-genus) and a braid invariant (the writhe)---as a special case of relating knot concordance homomorphisms and their homogenizations. As an application we find that the slice-Bennequin inequality holds with the FDTC in place of the writhe. Teaser: As a motivation for the concept of homogenization, this talk features a neat construction of the field of real numbers you probably dont know about.

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In this talk, I will discuss the pure braided plat presentation for knots and links in lens spaces. Using such presentations, I will present an algorithm to put the knots on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists and discuss its applications.

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We construct 3-dimensional genus-g handlebodies H and H' in the 4-sphere so that H and H' have the same boundary and are homeomorphic rel boundary, but are not smoothly isotopic rel boundary (for all g ≥ 2). In fact, H and H' are not even topologically isotopic rel boundary, even when their interiors are pushed into the 5-ball. This proves a conjecture of Budney and Gabai for g ≥ 2 in a very strong sense, and is a surprising answer to a 1-dimension up version of an open question about Seifert surfaces in S^3. In this talk, I'll give 3- and 4-dimensional motivation and discuss some interesting theorems about knotted surfaces that go into the construction. (This is joint with Mark Hughes and Seungwon Kim.)

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In recent joint works with Maxim Prasolov and Vladimir Shastin we developed a method to decide algorithmically whether or not two given Legendrian knots are equivalent. The method is based on the formalism of rectangular diagrams, which we have extended to Giroux' convex surfaces and Legendrian graphs. In many cases, the approach allows to distinguish Legendrian knots that are not distinguished by other known means or have so large complexity that other methods are not feasible to apply to them. We provide an example of two non-equivalent Legendrian knots that cobound an annulus embedded in the three-sphere and tangent to the standard contact structure along the entire boundary. Such examples have not been known earlier.

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Since McDuff proved that embedded contact homology can be used to characterize 4D ellipsoid embeddings in 2011 and McDuff and Schlenk discovered the first "infinite staircase" of ellipsoid embeddings in 2012, a growing body of work has analyzed which toric domains in R^4 (regions symmetric under the natural torus action from C^2) admit infinite staircases of ellipsoid embeddings. From the ECH (embedded contact homology) perspective, symplectic embeddings into a toric domain are determined by a certain set of torus knots on its boundary. We will discuss an algorithm used to identify these torus knots and find a fundamentally new type of infinite staircase in recent work with Magill and McDuff, as well as its possible generalizations and limitations.

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Khovanov homology provides a powerful tool for studying knots and links in 3-space and surfaces in 4-space. I will discuss recent developments that use Khovanov homology to distinguish non-isotopic surfaces in the 4-ball. We will see how braids relate two seemingly disparate strengths of these tools from Khovanov homology: their amenability to calculation (including recent software), and their sensitivity to complex curves.
Based on joint work with Isaac Sundberg and with Alan Du.

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Birman-Menasco proved the remarkable finiteness theorem: modulo exchange move, the number of the closed n-braid representatives of genus g knots/links is finite. In this talk, I would like to explain various topics related or inspired by Birman-Menasco finiteness theorem. Among them, I will explain its quantitative refinement and applications.

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The braid group is a particular example of the mapping class group of a surface with boundary. An invariant that roughly measures how much a mapping class “twists” about the boundary of the surface is the fractional Dehn twist coefficient. In this talk I will discuss some reasons why we might care about this invariant, several results about the fractional Dehn twist coefficient for braids, and explore to what extent these results can be extended to more general mapping class groups. This talk will include joint work with Peter Feller and Hannah Turner.

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This talk can be seen as the second part of the one given by Luis Paris last February. To make the exposition self-contained, I will start with the definition (and the motivation) of this new family of groups and therefore I will present some results (and questions) on pure subgroups and crystallographic quotients of virtual Artin groups. At the end, if time permits, I will present other families of groups that admit similar notions of virtual extension. Joint work with Luis Paris and Anne Laure Thiel.

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I will review some classic results about transverse knots in contact manifolds and then introduce the definition of the transverse unknotting number. For smooth knots, an “ancestor-descendant” relation has been defined by Cantarella-Henrich-Magness-O’Keefe-Perez-Rawdon-Zimmer; I will describe a transverse analog of this relation and then define and calculate some “transverse family trees.” This is joint work with Blossom Jeong.

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We will discuss motivation for and approaches to the question of when the monoid in the mapping class group of a surface with boundary corresponding to monodromies of open book decompositions of Stein fillable contact 3-manifolds differs from the monoid of mapping classes which admit factorizations into positive Dehn twists. In particlar, combining new results with previous work of several people, we give a complete solution to this problem for all but the case of the genus 1 surface with 1 boundary component. This is joint work with Vitalijs Brejevs.

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The L-space conjecture has been in the news a lot lately: this conjecture predicts that three seemingly different ways to measure the "size" of a 3-manifold are equivalent. In particular, it predicts that a manifold with the "extra" geometric structure of a taut foliation also has "extra" Heegaard Floer homology. In this talk, I'll discuss the motivation for this conjecture, and describe some new results which produce taut foliations by leveraging special properties of positive braid knots. Along the way, we will produce some novel obstructions to braid positivity. I will not assume any background knowledge in Floer or foliation theories; all are welcome!

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I will discuss some open problems regarding symmetric knots, and group actions on 3- and 4-manifolds. In the second part of the talk I will discuss how techniques from Floer theory can be employed to approach some of these problems.

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In this talk I’ll outline a proof that Khovanov homology detects the T(2,5) torus knot. In particular, I’ll explain why a knot with the same Khovanov homology as T(2,5) must be fibered, and then use some deep results in Floer homology to see what we can deduce about the monodromy of that fibration. This is based on joint work with John Baldwin and Ying Hu.

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In this talk I'll explain how knot Floer homology detects whether the monodromy of a fibered knot is right-veering. This gives a purely Floer-theoretic characterization of tight contact structures, and has applications to Dehn surgery and taut foliations. Our proof uses a relationship between Heegaard Floer homology and the dynamics of surface diffeomorphisms. This is based on joint work with Yi Ni and Steven Sivek.

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