Schedule for: 24w5291 - What's your trick? A Non-Traditional Conference in Low-Dimensional Topology

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I would like to present the following trick, which is the base of the theory of finite type invariants: It may be hard to determine the variation of a $3$-manifold invariant $\lambda$ under a surgery. However, computing the variation of such a variation under a disjoint surgery is sometimes possible, and such a computation may characterize $\lambda$. Andrew Casson applied this trick to exhibit a Dehn surgery formula for his invariant that counts conjugacy classes of fundamental group $SU(2)$-representations in 1984. Greg Kuperberg and Dylan Thurston applied this trick to identify a count of configurations of the Theta-graph in a $3$-manifold with this Casson invariant in 1999. In 2018, Tadayuki Watanabe adapted this trick to produce a topologically trivial $S^4$-bundle over $S^2$ that is not smoothly trivial. Doing so, he proved that the group of orientation-preserving diffeomorphisms of $S^4$ is not homotopy equivalent to $SO(5)$, which contradicted the so-called four-dimensional Smale conjecture. I propose to show how this trick works in the last two examples above.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Our "trick" will be a basic lemma which gives a sufficient condition for a chain complex of R-modules to be formal. This lemma is rather broad, and encompasses many constructions from classical homological algebra. This trick was key in the proof of myself, M. Borodzik and B. Liu that the link Floer complexes of plumbed L-space links are formal, and is also appears in my proof of the isomorphism of lattice homology and Heegaard Floer homology. The goal of the talk is to be rather accessible.

You have a menu of lattices which come attached to familiar objects in topology -- star-shaped lattices, graph lattices -- you have another menu coming from gauge theoretic obstructions -- changemaker lattices, cubiquitous lattices -- and you want to know which lattices from Menu A are also on Menu B -- say to know which lens spaces are integer surgery along a knot, or which alternating knots have unknotting number one. The trick is to compare the indecomposable elements in the lattices on the two menus. I'll explain the trick and some new potential targets for it.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

My first project in graduate school was to understand the 3-manifolds and contact structures supported by genus 1 open books with connected binding. Since then, open books have been unifying thread in a lot of my research. They provide a very useful way of thinking about problems related to Dehn surgery and Floer homology. I'll flesh this out with several examples.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I'll describe a very simple trick for simplifying Khovanov complexes related to full twists and cables which appears throughout many of my results; as I'll show, this is "enough" to prove that the Khovanov skein lasagna module vanishes for knot traces with framing coefficient less than the Thurston-Bennequin number of the knot (no contact geometry necessary).

Any time you have a framing, take it to the limit.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A tangle decomposition along a Conway sphere breaks a knot or link into simpler pieces, each of which is a two-string tangle. Whether the goal is direct (investigate knot invariants under mutation), indirect (investigate unknotting number), or about something else entirely (Dehn surgery), tangle decompositions can be used to address classic problems in knot theory and three-manifold topology. The famous ‘Montesinos trick’ establishes the connection between rational tangle replacements and surgery. I plan to discuss several results that rely on tangles and Montesinos’s correspondence in joint works with several sets of coauthors.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

My go-to strategy for thinking about knots and manifolds often involves braids. In this talk, I'll demonstrate how and why factoring braids in creative ways is surprisingly helpful. The goal of this talk is to advertise the utility of this perspective, while also introducing some general strategies that you, too, might find helpful (and could implement if you needed to!).

Although Donaldson's diagonalization theorem prima facie concerns closed 4-manifolds, it can be used to obtain a large number of results about knots and 3-manifolds. I will explain some of the lesser-known applications.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.