Berkeley topology seminar

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!

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.

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.

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.

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.

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.

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.

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.

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

The weight (or normal rank) 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 group that has weight 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 weight greater than one, which solves the Wiegold problem. I will explain the topological and dynamical ingredients in the proof of our theorem.