TRG Fall 2021

September 16th: Fraser Binns

• Title: Torus knots and Nonorientable Surfaces

• Abstract: It is a fact that every knot bounds a nonorientable surface in the 4-ball. Given such, we can define a measure of the complexity of a knot called ``the nonorientable 4 ball genus" as the minimal first Betti number of a nonorientable surface bounding that knot. In this talk I will discuss some (old) upper and (new!) lower bounds on the nonorientable 4 ball genus, with a particular focus on torus knots. This talk is based on joint work in progress with Sunkyung Kang, Jonathan Simone and Paula Truöl.

September 23rd: Gage Martin

• Title: Braids, fibered links, and annular Khovanov homology

• Abstract: Birman-Hilden give a construction that relates braid closures with certain fibered links via taking a branched double cover. In this talk we will see how the construction can be used to give topological applications of annular Khovanov homology. As an example we will use the Birman-Hilden construction to show that annular Khovanov homology detects a specific 4-braid representative of the unknot. This is joint work with Fraser Binns.

September 30th: Jacob Caudell

• Title: Surface intersections and a genus bound on reducible Dehn fillings

• Abstract: The topology of a 3-manifold both informs and is informed by the essential surfaces it contains. The intersection pattern of a pair of properly embedded surfaces in a 3-manifold with torus boundary can be recorded by the data of a pair of graphs with special decorations, and the combinatorics of these kinds of objects often dictate topological properties of the surfaces and 3-manifold from which they originate. This idea was first used by Litherland and was later leveraged by Gordon and Luecke to tremendous effect in the pre-Floer/Khovanov world of low-dimensional topology. In this talk we will introduce pairs of surface intersection graphs and their combinatorics, deduce a genus bound on reducible Dehn fillings, and discuss an application of this genus bound in my paper Three lens space summands from the Poincaré homology sphere ([2101.01256] Three lens space summands from the Poincaré homology sphere (arxiv.org)).

October 5th* (Tuesday at 3pm): Siddharth Mahendraker

• Title: Geometric Lie theory and a symplectic interpretation of Khovanov homology

• Abstract: Khovanov homology is a useful tool used to study knots. In principal, it can be computed entirely combinatorially, and its (graded) Euler characteristic yields the Jones polynomial. In their 2006 paper, Seidel-Smith suggest a geometric interpretation of Khovanov homology in terms of the geometry of a certain symplectic fibration which arises naturally in Lie theory. In this talk we introduce the basic ingredients from Lie theory required to understand this fibration. Expect lots of concrete computations!

October 14th: Braeden Reinoso

• Title: Fixed Points of Pseudo-Anosov Maps and Knot Floer Homology

• Abstract: Recently, Baldwin-Hu-Sivek showed that Khovanov homology detects the torus knot T(2,5). Their proof veered into interactions of knot homology theories with pseudo-Anosov maps on surfaces and contact geometry (among several other areas of low-dimensional topology), and their idea was motivated in part by a still-open question: does knot Floer homology detect T(2,5)? One strategy for an affirmative result in this direction would be to show that no pseudo-Anosov automorphism on a genus 2 surface has a unique fixed point. In a joint two-week talk with Ethan Faber, we will discuss an infinite family of counterexamples to this strategy, originating from joint work in progress between the two of us with Luya Wang. In this first half, I'll introduce the detection problem, and the knot Floer homology, contact geometry and fibered knot background needed to discuss the strategy and our counterexamples.

October 21st: Ethan Farber

• Title: A working mathematician’s guide to pseudo-Anosovs

• Abstract: Last week Braeden Reinoso explained how proving the non-existence of pseudo-Anosovs (pA’s) on the genus 2 surface with 1 boundary component and satisfying certain conditions, is sufficient to show that knot Floer homology detects the torus knot T(2,5). In this companion talk I will present an infinite family of pA’s that satisfy nearly all of the necessary conditions. These examples were previously thought not to exist by experts, and they problematize this proof strategy. This is work in progress joint with Braeden Reinoso (BC) and Luya Wang (Berkeley). In preparation for these examples, I will introduce the notion of a train track and show, through several examples, how they provide a combinatorial way of understanding pA’s. The bulk of the talk will be focused on manipulating concrete examples, and there will be many pictures.

October 28th: Laura Seaberg

• Title: Crash course: Morse theory

• Abstract: It is not uncommon for a topology student to hear the advice that it might be useful to learn a little Morse theory. The now-classical subject seeks to answer the question: what is the relationship between a smooth manifold and the real-valued functions defined on it? It turns out that topological properties of manifolds are recoverable from the analytical information in these functions--specifically, through their non-degenerate critical points. We will present an introduction to Morse theory and some generalizations such as Morse-Bott theory. If there's time, we will indicate how Morse theory has informed more contemporary methods. Our goal is to make this talk accessible and visual, with concrete examples on several of your favorite manifolds.

November 4th: Mac Krumpak

• Title: A gentle introduction to gauge theory

• Abstract: I'll discuss some of the key features of gauge theory and try to outline some of the ways in which gauge theory has been used to answer topological questions. I don't want to be too heavy on the differential topology, so I won't expect much background. You should probably know what a bundle is. I will be discussing principal bundles, vector bundles, connections on bundles, etc., though I can also try to explain some of those ideas depending on audience interest.

November 11th* (at 12pm): Nicolas Petit

• Title: What is... combinatorial knot theory?

• Abstract: This introductory talk is meant to go over the basics of knot theory, specifically from a combinatorial perspective, and discuss some invariants arising from this approach. After a brief historical note, we will discuss some important knot polynomials, then leave participants some time to work on simple examples in small groups. Finally, we will broadly discuss some recent (~20 years) directions combinatorial knot theory has taken (e.g. virtual knots, categorifications). The talk should be accessible to any graduate student, with little to no background required.

• Worksheet (with solutions): link

November 18th: Marius Huber

• Title: The Classification of Spherical 3-Manifolds

• Abstract: In this talk, I will outline the classification of spherical 3-manifolds, i.e. 3-manifolds that can be endowed a metric of constant curvature +1. The talk will be mostly a survey of results, into which I will go at different levels of depths.

December 2nd: Ian Montague

• Title: Seiberg-Witten Floer K-Theory and Cyclic Group Actions on Spin 4-Manifolds with Boundary

• Abstract: Over two decades ago, Furuta proved his celebrated "10/8ths" theorem gives a bound on the magnitude of the signature of a (smooth) spin 4-manifold in terms of its second Betti number. Soon afterwards, Bryan showed that if a spin 4-manifold admits a cyclic group action (e.g., a cyclic branched cover over a surface), then Furuta's bound can be strengthened. In another direction, Manolescu showed in 2013 that given a spin 4-manifold X with boundary a fixed rational homology sphere Y, the 4-manifold X satisfies a "10/8ths-type" inequality involving a correction term dubbed the kappa invariant of Y, defined by analyzing the Pin(2)-equivariant complex K-theory of the Seiberg-Witten Floer spectrum associated to Y. As a common generalization of Bryan and Manolescu's work, I will describe work-in-progress which establishes a 10/8ths-type inequality for spin 4-manifolds with rational homology sphere boundary equipped with a Z/m action. In particular, the corresponding correction term in the inequality involves a family of "equivariant kappa invariants" associated to the bounding rational homology sphere, which take the form of local minima of a semi-infinite sub-lattice of Q^{m} or Q^{m-1} depending on the parity of m.