Berkeley topology seminar

Marc Kegel: The Conway knot has infinite concordance order

We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite concordance order. This is based on joint work with Chiara Donatone, Lukas Lewark, and Paula Truöl.

Ian Biringer: Ranks of cusped mapping tori 

The rank of a manifold is the minimal number of generators for its fundamental group. If S is a surface and f : S -> S is a homeomorphism, the mapping torus M_f is obtained from S x [0,1] by gluing S x 1 to S x 0 via the map (x,1) -> (f(x),0). It’s easy to show that rank M_f <= rank S + 1.

In 2015, Souto and I showed that if S is a closed surface and f is complicated enough, then we have rank M_f = rank S + 1. It turns out that the analogous result isn’t true when S has cusps, due to a possible subtle interaction between generating sets for pi_1 M_f and the cusps of M_f. However, if f is complicated enough, one can prove that rank M_f >= rank S. This is a joint project-in-progress with Dave Futer and Matt Zevenbergen.

Lvzhou Chen: The Wiegold problem and the weight of free products

The normal rank (or weight) of a group G is the smallest number of elements that normally generate G. This plays an important role in 3-manifold topology, but it is poorly understood. It is extremely difficult to give lower bounds apart from looking at the abelianization. The Wiegold problem asks if there is a finitely generated perfect group that has normal rank greater than one, namely the group cannot be normally generated by any single element. In a joint work with Yash Lodha, we show free products of nontrivial left-orderable groups all have normal rank greater than one, which solves the Wiegold problem. Our result also has nice implications about Dehn surgeries of 3-manifolds. I will explain the topological and dynamical ingredients in the proof of our theorem.

Sam Taylor: Pseudo-Anosov flows, hyperbolic geometry, and the curve graph

Starting with a pseudo-Anosov flow on a hyperbolic manifold M, it’s natural to wonder how the dynamics of the flow interact with the hyperbolic geometry of M. In joint work with Junzhi Huang, we study this question through the lens of the curve graph of a transverse surface S. Specifically, the flow determines a canonical pair of multicurves on S (when S is not a fiber surface); we show that if the curve graph distance between these multicurves is large, then the geometry of M near S is complicated. This builds on foundational work concerning fibered manifolds and connects to the more recent study of endperiodic mapping tori.

Sara Maloni: The space of d-pleated surfaces and its topology

In this talk, we will discuss joint work with Giuseppe Martone, Filippo Mazzoli, and Tengren Zhang about d-pleated surfaces, which are surface group representations into 𝖯𝖲𝖫_d(C) obtained by "bending" a Hitchin representation along a geodesic lamination \lambda and can be considered as a higher rank generalization of the classical notion of pleated surfaces in three-dimensional hyperbolic space. We will describe how to define these representations, how to describe the global topology of the space of d-pleated surfaces and how to show that every connected component of the character variety contains exactly one connected component of such space.

Eli Grigsby (in dept colloquium): The geometry, topology, and combinatorics of neural networks

Deep neural networks are a class of parameterized functions that have proven remarkably successful at making predictions about unseen data from finite labeled data sets. They do so even in settings when classical intuition suggests that they ought to be overfitting (memorizing) the data. 

I will begin by describing the structure of neural networks and how they learn. I will then advertise one of the theoretical questions animating the field: how does the relationship between the number of parameters and the size of the data set impact learning dynamics? Along the way I will emphasize the many ways in which geometry, topology, and combinatorics play a role in the field.

Caglar Uyanik: Singularity of Cannon-Thurston maps

Cannon and Thurston showed that a hyperbolic 3-manifold that fibers over the circle gives rise to a sphere-filling curve. The universal cover of the fiber surface is quasi-isometric to the hyperbolic plane, whose boundary is a circle, and the universal cover of the 3-manifold is 3-dimensional hyperbolic space, whose boundary is the 2-sphere. Cannon and Thurston showed that the inclusion map between the universal covers extends to a continuous map between their boundaries, whose image is onto. In particular, any measure on the circle pushes forward to a measure on the 2-sphere using this map. We compare several natural measures coming from this construction.

Lily Li: Finiteness of Singular Orbits of pseudo-Anosov Flows 

Fixing a closed 3-manifold M, it's a long-standing program to understand the pseudo-Anosov flows it supports. In this talk, we discuss a finiteness result constraining the set of singular orbits of such flows. Using Gabai's Kneser normal form, we give a pictorial proof that given a closed atoroidal 3-manifold, there are at most finitely many possible links up to isotopy that arise as singular orbits of pseudo-Anosov flows on it. 

Jason Deblois: Notes from a search for knots

I will give some details of a successful search, joint with Arshia Gharagozlou and Neil Hoffman, for knot complements in S^3 with two properties — (1) they admit complete hyperbolic structures containing closed, embedded totally geodesic surfaces, and (2) they have hidden symmetries — among covering spaces of the rich class of “prism orbifolds”. Key search phases were pre-filtering: ruling out the prism orbifolds that wouldn’t yield knot complement covers; enumeration; filtering out covers lacking necessary conditions to be knot complements; and recognizing knot complements. I’ll describe the computational tools we used and the theoretical results undergirding them.

Susan Hermiller (in dept colloquium): A tale of three unknotting conjectures

Unknotting number is a fundamental measure of how complicated a knot is, measuring how far it is from the unknot via crossing changes. Unknotting number is a challenging invariant to compute; a vast array of tools have been applied to its calculation, and many conjectures have grown up around it. In this talk I will discuss three conjectures, each aimed at simplifying the task of computing unknotting numbers. I will describe how our resolution of one of these conjectures several years ago led us recently to resolve another - the (non)additivity of unknotting number under connected sum. This is joint work with Mark Brittenham.

Mark Brittenham: The unknotting number of 10_6 is...

The knot 10_6 currently holds the distinction of being the `simplest' knot whose unknotting number we do not know. It also holds a different distinction in that it is the key to identifying the `simplest' counterexample to one of two conjectures on unknotting number (we just don't know which...). We will discuss this knot, some of its useful properties, the search for its unknotting number, and the larger question of how to `efficiently' compute unknotting numbers. This is joint work with Susan Hermiller.

Nathan Dunfield: Ribbon concordances and slice obstructions: experiments and examples

There are 352.2 million prime knots in the 3-sphere with at most 19 crossings.  In joint work with Sherry Gong, I studied which of these knots are slice, in both the smooth and topological categories. While no algorithm is known for deciding whether a given knot is slice in either setting, we were able to determine it smoothly for all but about 11,400 knots (0.003% or 1 in 30,000) and topologically for all but about 1,400 knots (0.0004% or about 1 in 250,000). In particular, we showed that some 1.6 million of these knots (0.46%) are smoothly slice (in fact ribbon) and that 350.5 million are not even topologically slice (99.54%).  I’ll discuss the varied tools and techniques we used for this, and explain how our data is consistent with several important conjectures and suggests new ones.

Matt Hedden: Floer homology and (no) Anosov flows

I'll discuss an obstruction to a 3-manifold admitting an Anosov flow coming from (Heegaard) Floer homology and use it to produce infinite families of closed hyperbolic 3-manifolds which cannot admit such a flow. This is joint work with Katherine Raoux and Jeremy Van Horn-Morris.

Gheehyun Nahm: Barbells and knotted things

In recent work, Budney and Gabai defined barbell diffeomorphisms and used them to construct knotted 3-balls in S^4. Barbell diffeomorphisms provide a concise and hands-on method for producing interesting diffeomorphisms of 4-manifolds. In this talk, I will first introduce barbell diffeomorphisms, and then explore further applications and construct the following (time permitting):

1. Knotted genus-2 handlebodies in S^4. This provides an alternative proof of a theorem of Hughes, Kim, and Miller.

2. Knotted S^3’s in S^5 with exactly four critical points with respect to the standard height function. This answers a question of Kuiper and in particular produces knotted solid tori in S^4, proving the remaining case of a conjecture of Budney and Gabai.

3. Brunnian links of 3-balls in S^4.

This is joint work with Seungwon Kim and Alison Tatsuoka.

Danny Calegari: A tale of two Cannons

What do Cannon-Thurston maps have to do with the Cannon conjecture? Come to the talk and find out!

Qianhe Qin: How to Pick Out the Slicing Degree of Knots Using a Spork

The slicing degree of a knot K is the smallest integer k such that K is k-slice (i.e., bounds a disk with self-intersection number –k) in #n(-CP)^2 for some n. In this talk, we establish bounds on the slicing degrees of knots using Rasmussen’s s-invariant, knot Floer homology, and singular instanton homology. We also introduce sporks, defined as pairs (W, f) consisting of a contractible 4-manifold W and a boundary diffeomorphism f that extends smoothly inside. Sporks appear naturally in certain k-RBG links and produce knots with the same k-trace; although a bit blunt to produce exotic smooth structures, they turn out to be unexpectedly effective for detecting slicing degree in the examples we consider.

Colby Kelln: Aspherical cone offs of hyperbolic manifolds with boundary

We give geometric conditions which imply that the space obtained by coning off the boundary components of a hyperbolic manifold M is negatively curved.  Moreover, we give explicit geometric conditions under which a locally convex subset of M gives rise to a locally convex subset of the cone-off. Group-theoretically, we conclude that the fundamental group of the cone-off is hyperbolic of cohomological dimension n and the pi_1--image of the coned-off locally convex subset is a quasi-convex subgroup. This is joint work with Jason Manning. 

Judson Kuhrman: Spinning Constructions and Exotic RP^2-knots

Recent developments demonstrate that invariants coming from real Seiberg-Witten gauge theory are fruitful for the detection of exotically knotted surfaces in the 4-sphere. Notably, Miyazawa introduces an invariant for small-genus knotted surfaces and proves the existence of an infinite family of (smoothly mutually distinct) exotic unknotted RP^2-knots. I will show how, by a shift in perspective, we can construct a larger (bi-)infinite family of exotic unknotted RP^2s distinguished by the same invariant, in a way that is relatively straightforward and procedural.

Jo Nelson: Contact toric structures and concave boundaries of plumbings

Little is known about tight contact structures which are not fillable.  Algebraic torsion measurements in embedded contact homology are useful for obstructing symplectic fillability and overtwistedness of the contact 3-manifold, but has been left unexplored. We discuss the methods we developed, focusing on concave linear plumbings of symplectic disk bundles over spheres admitting a concave contact boundary. This talk is based on joint work with Aleksandra Marinkovic, Ana Rechtman, Laura Starkston, Shira Tanny, and Luya Wang.  Time permitting, we will discuss our work in progress to determine nonfillable tight contact 3-manifolds obtained from more general plumbings.

Patrick Orson: Exotic pseudo mapping classes

A pseudo-isotopy is a weakening of the concept of isotopy, removing the insistence that it is level-preserving. Pseudo-isotopies play a key role in the study of mapping class groups in dimensions 4 and above. We investigate the question: if a self-diffeomorphism of a 4-manifold is topologically pseudo-isotopic to the identity, must it always be smoothly so? We produce the first examples where the answer is “no” -- the first exotic pseudo mapping classes. On the other hand, we derive a set of conditions on the fundamental group of a 4-manifold that identify a large class of examples where this topological to smooth upgrade can always be made. I will explain how these results make progress on our original motivating question of whether topological isotopy implies smooth stable isotopy for diffeomorphisms of 4-manifolds. This is joint work with Mark Powell and Oscar Randal-Williams.

Danny Ruberman: Closed exotic aspherical 4-manifolds

An aspherical space is one with vanishing higher homotopy groups; under mild assumptions this means that it is determined up to homotopy type by its fundamental group. A famous conjecture of Borel states that closed aspherical manifolds are in fact determined up to homeomorphism by their fundamental group. One could also ask (although Borel famously didn’t) whether smooth aspherical manifolds are determined up to diffeomorphism by their fundamental group; this is known to hold in dimensions at most 3 and to be false in dimension at least 5. We resolve the remaining case by exhibiting closed smooth aspherical 4-manifolds that are homeomorphic but not diffeomorphic. This is joint with Davis, Hayden, Huang, and Sunukjian.