NCNGT 2020

Exotic slice disks and symplectic surfaces- Kyle Hayden (Columbia University)

Abstract: One approach to understanding the smooth topology of a 4-dimensional manifold is to study the embedded surfaces it contains. I'll construct new examples of "exotically knotted" surfaces in 4-manifolds, i.e. surfaces that are isotopic through ambient homeomorphisms but not through diffeomorphisms. We'll begin with simple examples of exotic disks in the 4-ball. Then we'll turn to the symplectic setting and exhibit new types of exotic phenomena among symplectic, holomorphic, and Lagrangian surfaces.

Doubly slice pretzels- Clayton McDonald (Boston College)

Abstract: A knot K in S^3 is slice if it is the cross section of an embedded sphere in S^4, and it is doubly slice if the sphere is unknotted. Although slice knots are very well studied, doubly slice knots have been given comparatively less attention. We prove that an odd pretzel knot is doubly slice if it has 2n+1 twist parameters consisting of n+1 copies of a and n copies of -a for some odd integer a. Combined with the work of Issa and McCoy, it follows that these are the only doubly slice odd pretzel knots. Time permitting, we might go over some preliminary results involving links as well.

Topologically embedding spheres in knot traces- Patrick Orson (Boston College)

Abstract: Knot traces are smooth 4-manifolds with boundary, that are homotopic to the 2-sphere, and obtained by attaching a 2-handle to the 4-ball along a framed knot in the 3-sphere. I will give a complete characterisation for when the generator of the second homotopy group of a knot trace can be represented by a locally flat embedded 2-sphere with abelian exterior fundamental group. The answer is in terms of classical and computable invariants of the knot. This result is directly analogous to the the result of Freedman and Quinn that says a knot with Alexander polynomial 1 is topologically slice, and can be used to exhibit new exotic 4-dimensional phenomena. This is a joint project with Feller, Miller, Nagel, Powell, and Ray.

Regular homotopy and Gluck twists- Hannah Schwartz (MPIM Bonn)

Abstract: Any 2-sphere K smoothly embedded in the 4-sphere is related to the unknotted sphere through a finite sequence of locally supported homotopies called finger moves and Whitney moves. We call the minimum number of finger moves (or equivalently Whitney moves) in any such a homotopy the "Casson-Whitney number" of the sphere K. In this talk, I will discuss joint work with Joseph, Klug, and Ruppik showing that if the Casson-Whitney number of K is equal to one, then the unknotted torus can be obtained by attaching a single 1-handle to K. I will also present an application of this result, from joint work with Naylor, that the Gluck twist of any sphere with Casson-Whitney number equal to one is diffeomorphic to the standard 4-sphere, and give some well-known families of 2-spheres for which this is the case.

Unknotting with a single twist- Samantha Allen (Dartmouth College)

Abstract: Ohyama showed that any knot can be unknotted by performing two full twists, each on a set of parallel strands. We consider the question of whether or not a given knot can be unknotted with a single full twist, and if so, what are the possible linking numbers associated to such a twist. It is observed that if a knot can be unknotted with a single twist, then some surgery on the knot bounds a rational homology ball. With this, a wider range of tools become available, including classical invariants and invariants arising from Heegaard Floer theory. Using these tools, if a knot $K$ can be unknotted with a single twist of linking number $l$, we give restrictions on the genus, signature function, Upsilon function, and $V$ and $\nu^+$ invariants of $K$ in terms of $l$ and the sign of the twist. In this talk, I will discuss some of these restrictions, their implications, and some unanswered questions. This talk is based on joint work with Charles Livingston.

Characterising homotopy ribbon discs- Anthony Conway (MPIM Bonn)

Abstract: After reviewing some notions from knot concordance, we explore the following question: how many slice discs does a slice knot admit? This is joint work with Mark Powell.

Fox-Milnor Conditions for 1-solvable Knots and Links- Shawn Williams (Rice University)

Abstract: A well known result of Fox and Milnor states that the Alexander polynomial of slice knots factors as $f(t)f(t^{-1})$, providing us with a useful obstruction to a knot being slice. In this talk, I will present a generalization of this result to certain localized Alexander polynomials of 1-solvable boundary links, and first order Alexander polynomials of 1-solvable knots.

Geodesics on hyperbolic surfaces and volumes of link complements in Seifert-fibered spaces- José Andrés Rodriguez Migueles (University of Helsinki)

Abstract: Let Γ be a link in a Seifert-fibered space M over a hyperbolic surface Σ that projects injectively to a collection of closed geodesics γ in Σ. When γ is filling, the complement of Γ in M admits a hyperbolic structure of finite volume. We give bounds of its volume in terms of topological invariants of (γ,Σ).

Uniform models for random 3-manifolds- Gabriele Viaggi (University of Heidelberg)

Abstract: As discovered by Thurston, hyperbolic 3-manifolds are abundant among all 3-manifolds. In many examples, the generic element in a family of 3-manifolds sharing a common combinatorial description admits such a hyperbolic structure. The family of random 3-manifolds (Dunfield and Thurston model) is one of these examples. The existence of a hyperbolic metric on such random objects has been established by Maher, exploiting the solution of the Geometrization conjecture by Perelman. In this talk, I will describe a more constructive approach to this result and give an explicit construction for the metric that only uses tools from the deformation theory of Kleinian groups. The metric obtained is explicit enough to allow the computation of geometric invariants such as volume and diameter. Joint with Peter Feller and Alessandro Sisto.

Hyperbolic Limits of Cantor sets complements in the Sphere- Franco Vargas Pallete (Yale University)

Abstract: In this talk we will show that if M is a hyperbolic manifold that embeds in $S^3$ with no $Z^2$ in $pi_1$, then M can be approximated (in the geometric sense) by hyperbolic metrics on Cantor set complements in $S^3$. This is joint work with Tommaso Cremaschi.

Stable commutator length in graphs of groups- Lvzhou (Joe) Chen (University of Chicago)

Abstract: The stable commutator length (scl) is a relative version of the Gromov-Thurston norm. For a given null-homologous loop L in a space X, its scl is the infimal complexity of surfaces bounding L, measured in terms of Euler characteristics. Surfaces realizing the minimal complexity are called extremal. They are pi_1-injective, and can only exist when scl is rational. We show that scl takes rational values for all loops in a space X if pi_1(X) is certain graphs of groups, inclusing Baumslag-Solitar groups. Moreover, there is a linear programming algorithm to compute scl.

Flows, Thurston norm, and surfaces: homology to isotopy- Michael Landry (Washington University)

Abstract: Let M be a closed hyperbolic 3-manifold. We begin by reviewing a classical picture due to Thurston, Fried, and Mosher in which a single pseudo-Anosov flow organizes the data of a fibered face F of the Thurston norm ball of M as well as certain nice surface representatives of homology classes lying in the cone over F. We then announce a new theorem which strengthens the above by collating all isotopy classes of incompressible surfaces representing classes in the cone over F. We explain that the result follows from a more general theorem, involving veering triangulations, which also applies to other faces of the Thurston norm ball (possibly nonfibered, possibly lower-dimensional).

Dehn filling and knot complements that irregularly cover- William Worden (Rice University)

Abstract: It is a longstanding conjecture of Neumann and Reid that exactly three knot complements can irregularly cover a hyperbolic orbifold---the figure-8 knot and the two Aitchison--Rubinstein dodecahedral knots. This conjecture, when combined with work of Boileau--Boyer--Walsh, implies a more recent conjecture of Reid and Walsh, which states that there are at most 3 knot complements in the commensurability class of any hyperbolic knot. We give a Dehn filling criterion that is useful for producing large families of knot complements that satisfy both conjectures. The work we will discuss is partially joint with Hoffman and Millichap, and partially joint with Chesebro, Deblois, Hoffman, Millichap, and Mondal.

Embedded totally geodesic submanifolds in small volume hyperbolic manifolds- Michelle Chu (University of Illinois, Chicago)

Abstract: The smallest volume cusped hyperbolic 3-manifolds are arithmetic and contain many immersed but not embedded closed totally geodesic surfaces. In this talk we discuss nonexistence of codimension-1 closed embedded totally geodesic submanifolds in small volume hyperbolic manifolds of higher dimensions. This is joint work with Long and Reid.

Arithmetic manifolds and their geodesic submanifolds- Nicholas Miller (University of California Berkeley)

Abstract: It is a well known consequence of the Margulis dichotomy that when an arithmetic hyperbolic manifold contains one totally geodesic hypersurface, it contains infinitely many. Both Reid and McMullen have asked conversely whether the existence of infinitely many geodesic hypersurfaces implies arithmeticity of the corresponding hyperbolic manifold. In this talk, I will discuss recent results answering this question in the affirmative. In particular, I will describe how this follows from a general superrigidity style theorem for certain natural representations of fundamental groups of hyperbolic manifolds. I will also discuss a recent extension of these techniques into the complex hyperbolic setting, which requires the aforementioned superrigidity theorems as well as some theorems in incidence geometry. This is joint work with Bader, Fisher, and Stover.

Deligne-Mostow lattices and line arrangements in complex projective 2-space- Irene Pasquinelli (Institut de Mathématiques de Jussieu, Paris)

Abstract: In 1983, Hirzebruch considers some arrangements of complex lines in complex projective 2-space. Then he shows that a suitable branched cover ramified along the line arrangement is a complex hyperbolic manifold. This manifold turns out to be one of the well known Deligne-Mostow lattices. In the first part of this talk I will introduce you to the complex hyperbolic space, to its group of isometries and to the Deligne-Mostow lattices. Then I will tell you about Hirebruch's construction. In the second part, I will first explain how Hirzebruch's work has been generalised by Bartel, himself and Hoefer to all of the Deligne-Mostow lattices. Then I will tell you how, in a joint work with Elisha Falbel, we created an explicit dictionary between line arrangements and fundamental domains for the lattices. I will also explain one of the applications of this. In fact, we use this result to contribute to the problem of creating a complex analogue to the hybridisation construction.

Counting simple closed multi-geodesics on hyperbolic surfaces with respect to the lengths of individual components- Francisco Arana-Herrera (Stanford)

Abstract: In her thesis, Mirzakhani showed that on any closed hyperbolic surface of genus g, the number of simple closed geodesics of length at most L is asymptotic to a polynomial in L of degree 6g-6. Wolpert conjectured that analogous results should hold for more general countings of multi-geodesics that keep track of the lengths of individual components. In this talk we will present a proof of this conjecture which combines techniques and results of Mirzakhani as well as ideas introduced by Margulis in his thesis.

Cutting and pasting along measured laminations- Aaron Calderon (Yale)

Abstract: One of the fundamental techniques of low-dimensional topology is cutting and pasting along embedded codimension 1 submanifolds. In surface theory these submanifolds are just simple closed curves, and cutting and pasting gives rise to Fenchel-Nielsen coordinates for Teichmüller space, normal forms for simple closed curves, and many other foundational results. The set of simple closed curves completes to the space of “measured laminations,” and in my first talk, I will summarize how to build coordinates for Teichmüller space by cutting and pasting along any lamination, generalizing the shear coordinates of Bonahon and Thurston. In my second talk, I will explain how these coordinates lead to an extension of Mirzakhani’s conjugacy between the earthquake and horocycle flows, two notions of unipotent flow coming from hyperbolic, respectively flat, geometry. This represents joint work with James Farre.

Volumes and filling collections of multicurves- Andrew Yarmola (Princeton)

Abstract: Consider a link L in a Seifert-fibered space N over a surface S of negative Euler characteristic. If the fiber-wise projection of L to S is a collection C of closed curves in minimal position, then N \ L is hyperbolic if and only if C is filling and N \ L is acylindrical. The behavior of vol(N \ L) in terms of the topology and geometry of C have been studied in recent years, but effective lower bounds have been elusive. In this talk we will focus on the case where C is a collection of simple closed curves. In the special case where N = PT(S) is the projectivized tangent bundle and L is the canonical lift of a pair of filling multicurves, we show that vol(N \ L) is quasi-isometric to the Weil-Petersson distance between the corresponding strata in Teichmuller space. In the more general setting, we show that vol(N \ L) is quasi-isometric to expressions involving distances in the pants graph whenever L is a stratified link. This is joint work with T. Cremaschi and J. A. Rodriguez-Migueles.

PSL(2,R) representations and left-orderablility of Dehn filling- Xinghua Gao (KIAS)

Constructing non-trival $\widetilde{PSL}(2, R)$ representations has been proven to be a useful tool for showing the left-orderablity of a 3-manifold group. In this talk, I will show how to use $\widetilde{PSL}(2, R)$ representations of the fundamental group of a knot complement to determine which Dehn filling of it has left-orderable fundamental group. In particular, I will compute slopes of left-orderable Dehn filling of a class of two-bridge knots as an example.

Promoting circular-orderability to left-orderability- Ty Ghaswala (UQAM)

I will present new necessary and sufficient conditions for a circularly-orderable group to be left-orderable, and introduce the obstruction spectrum of a circularly-orderable group. I will then show how newly developed machinery can help us compute the obstruction spectrum in a variety of examples, including 3-manifold groups relevant to the L-space conjecture, and mapping class groups. I will finish the talk with new progress in answering the question of when the direct product of two circularly-orderable groups is circularly-orderable, a fundamental question which is frustratingly difficult to say anything about.

Left-orderability of 3-manifold groups and foliations of $3$-manifolds- Ying Hu (Nebraska)

In this talk, we will discuss how the existence of certain nice dimension 1 and dimension 2 foliations of $3$-manifolds can lead to the left-orderability of their fundamental groups. We will give some applications of these observations to cyclic branched covers of a knot. Limitations of the techniques will also be discussed. This is joint work with Steve Boyer and Cameron Gordon.

L-space knots do not have essential Conway spheres- Tye Lidman (NC State)

Abstract: The properties of a knot are heavily governed by the essential surfaces that sit in the exterior. We will study a relation between essential planar surfaces in a knot exterior and knot Floer homology. This is joint work with Allison Moore (VCU) and Claudius Zibrowius (UBC).

Introduction to L-space links- Beibei Liu (Bonn)

Abstract: L-spaces are simplest 3-manifolds in terms of Heegaard Floer homology and L-space links are links such that all large surgeries are L-spaces. In this talk, we will concentrate 2-component L-space links which is a family of ``simple” links in the sense that their Alexander polynomials contain full information of the link Floer complex, and give explicit answers to questions relating to the link itself and its surgeries such as some detection results, sharp slice genus bounds and Thurston polytope.

Fibred knots, positivity and L-spaces- Filep Misev (Regensburg)

Abstract: Torus knots are lens space knots: they admit surgeries to lens spaces. This classical theorem has a modern analogue in terms of Floer homology: algebraic knots are L-space knots. I will present knots which do not admit L-space surgeries despite strikingly resembling algebraic knots and L-space knots in general. More precisely, we will see a method which allows to construct infinite families of knots of arbitrary fixed genus g > 1 which are all algebraically concordant to the torus knot T(2,2g+1) of the same genus and which are fibred and strongly quasipositive. Besides the study of L-spaces, these knots are of interest in the context of knot concordance, in particular Fox's slice-ribbon question, as well as Boileau-Rudolph's question, or Baker's conjecture, on the independence of strongly quasipositive fibred knots in the concordance group. Joint work with Gilberto Spano.

Non-left-orderable surgeries on iterated 1-bridge braids- Zipei Nie (Princeton)

Abstract: We prove that the L-space conjecture holds for those L-spaces obtained from Dehn surgery on knots which are closures of iterated 1-bridge braids, i.e., the braids obtained from satellite operations on 1-bridge braids. In the proof, we emphasize the power of fixed points in the Homeo_+(R) representation, and introduce property (D) to handle the satellite operation.

From Floer homology to spectral theory, and hyperbolic geometry- Francesco Lin (Columbia)

Abstract: In the first part of the talk, I will review some spectral theory of three manifolds, and discuss its relation with Floer homology, and in particular L-spaces. In the second part of the talk, I will discuss how spectral theory on a hyperbolic three manifold can be understood in terms of natural geometric quantities. This is joint work with M. Lipnowski.

On the monopole Lefschetz number of finite order diffeomorphisms- Jianfeng Lin (UC San Diego)

Abstract: Let K be a knot in an integral homology 3-sphere Y, and Σ the corresponding n-fold cyclic branched cover. Assuming that Σ is a rational homology sphere (which is always the case when n is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of Σ. Our formula is motivated by a Witten-style conjecture relating the two gauge theoretic invariants of homology S1 cross S3s (the Furuta-Ohta invariant and the Casson-Seiberg-Witten invariant). As applications, we give a new obstruction (in terms of the Jones polynomial) for the branched cover of a knot in S3 being an L-space and we define a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers. This is a joint work with Danny Ruberman and Nikolai Saveliev.

L-spaces in instanton Floer homology- Steven Sivek (Imperial)

Abstract: Framed instanton homology $I^\#(Y)$ is a gauge-theoretic invariant which appears to coincide with the hat version of Heegaard Floer homology. Inspired by the notion of a Heegaard Floer L-space, we say that a rational homology sphere Y is an “instanton L-space" if the rank of $I^\#(Y)$ is as small as possible, namely $|H_1(Y)|$. In this talk I’ll summarize what is known about instanton L-spaces, especially those which arise as Dehn surgeries on knots in $S^3$, and what this tells us about fundamental groups, A-polynomials of knots, and the L-space conjecture. Various parts of this are joint with Antonio Alfieri, John Baldwin, Irving Dai, and Raphael Zentner.