I organised a learning seminar on "exotic" surfaces in the 2022 Fall semester.
September 20th: Mira Wattal
Khovanov homology - it's as easy as 1, 2, 3!
In this talk, we'll construct Khovanov homology in three easy steps: 1) buy magic beans from a questionable street seller, 2) plant beans and grow giant beanstalk, and 3) steal a golden egg from a giant. Kidding! Khovanov homology isn't quite this fantastical, but it does involve a complex recipe that can be broken down into three steps: defining the "cube of resolutions", realizing this cube as a commutative diagram in the (1+1)-cobordism category, and defining a functor that takes this diagram to a commutative diagram in the category of abelian groups. By the end of this talk, you'll be saying: "Khovanov homology - it's as easy as 1, 2, 3!
September 27th: Kevin Yeh
Khovanov homology and exotic surfaces in the 4-ball
Given a link cobordism, there is an induced map on the Khovanov homology between the links. I will begin by introducing this induced map, and then use it to work towards the contents of the Hayden-Sundberg paper (https://arxiv.org/abs/2108.04810). The main result of which is that for each g, there exists a knot bounding a pair of genus-g surfaces in B^4, such that the two surfaces are isotopic topologically, but not isotopic smoothly.
October 4th: Jacob Caudell
Everything You Ever Wanted To Know About Kirby Calculus But Were Afraid To Ask
In this collaborative session, we will review Morse functions and handle decompositions---the analog of CW decompositions in the smooth(able) category, introduce the basics of Kirby calculus (the practice of describing and manipulating handle decompositions in dimension 4), and produce recipes for obtaining Kirby diagrams of (1) exteriors of properly embedded surfaces in the 4-ball and (2) two-fold branched covers of the 4-ball (which necessarily have branch locus a properly embedded surface in the 4-ball). Time permitting, we will then proceed to group work on a collection of exercises---the only way to truly grok Kirby calculus is to Do The Work.
October 18th: Ali Naseri Sadr
A Pair of Exotic Surfaces
We will see how one can use the link T(4,-6) to construct a pair of Seifert surfaces with minimal genus for a knot K which are not isotopic to each other when pushed into the 4-ball. Then we modify these surfaces by Whitehead doubling process and we show they become topologically isotopic to each other; however, they can still be distinguished in the smooth category using Khovanov homology.
October 25th: Ali Naseri Sadr
Exotic Surfaces, part two
I will continue the talk by proving the relation between Seifert pairing and the intersection form of branched double cover. After that, I will introduce the Whitehead doubling operation and symmetry group of a Knot.
November 1st: Ali Naseri Sadr
Exotic Surfaces in the Smooth Category
We will start by showing that the Whitehead double of the two surfaces we have constructed are isotopic in the topological category. After that, we distinguish these surfaces in the smooth category using their induced maps in Khovanov homology.
November 8th: Fraser Binns
Bing Doubles of slice disks
I will discuss some work of Hayden-Kjuchukova-Krishna-Miller-Powell-Sunukjian. Specifically I will introduce the Bing doubling operation for disk links and give an example of a pair of Brunnian links that are topologically but not smoothly isotopic. Our obstruction to the smooth isotopy will combine symplectic and hyperbolic geometry. Separately I may discuss how to produce Kirby diagrams for double branched covers of Seifert surfaces pushed into the 4-ball.
November 29th: Fraser Binns
Rim Surgery and Cobordism Maps in Knot Floer homology
Rim surgery is an operation on smoothly embedded surfaces in 4-manifolds. Knot Floer homology is a theory which assigns smoothly embedded surfaces (with some extra decorations) in 4-manifolds to linear maps. In this talk I will discuss a theorem of Juhasz-Zemke on the effect of rim-surgery on these linear maps. No background in Heegaard Floer homology will be assumed.
December 6th: Mira Wattal
Lee's TQFT, the s-invariant, and overcoming my fear of spectral sequences
Freedman and Quinn showed that from a topologically slice knot with a positive smooth slice genus, one can construct a smooth structure on $\mathbb{R}^4$ that is not diffeomorphic to the standard smooth structure. Freedman in particular showed that if the Alexander polynomial of a knot is 1, then it is topologically slice. The next natural question to ask is whether there exists a sufficient condition for a knot to have positive smooth slice genus. This question is positively answered by Rasmussen with the s-invariant.
Today, I won't explicate the details behind this sufficient condition, but I will
1) describe an algorithm for computing the s-invariant, by way of a
2) quick primer on spectral sequences, and (time-willing)
3) compute the s-invariant for the knot that Lisa Piccarillo considered in her proof that the Conway knot is not smoothly slice.
December 13th: Kevin Yeh
Khovanov-Jacobsson Number is NOT a Good Invariant
Viewing an embedded closed surface in four-space as a cobordism between empty links, the functoriality of Khovanov homology induces a group homomorphism from Z to Z. It was shown by Rasmussen and Khovanov that this map is an invariant of the embedding; so naturally one can ask whether this invariant is any good at detecting different smooth embeddings of the surface. The answer is: no, not really. Not only does this invariant vanish for all surfaces of genus not 1, Rasmussen shows that in the genus 1 case, the invariant is trivially determined, irregardless of the embedding. This reaffirms the observation that it is generally harder to detect exotica of closed manifolds than it is to do so when one has non-empty boundaries, cf. the recent works of Hayden we looked at previously. If time permits I will mention recent works of Sundberg and Swann to adapt the Khovanov-Jacobsson number to the relative (i.e. with boundary) setting.