5/3/22: Sarah Blackwell (University of Georgia)

Title: Triple Knot Grid Diagrams

Abstract: The complex projective plane satisfies two special properties that set it apart from its peers: it admits (1) a symplectic structure and (2) a genus one trisection. Together these properties inspire the notion of a ``triple knot grid diagram,’’ the primary object of study in this talk. Triple knot grid diagrams are specific shadow diagrams of surfaces in the complex projective plane that naturally arise as grid diagrams on the central surface of its standard (genus one) trisection. Interestingly, diagrams satisfying one extra condition represent Lagrangian surfaces, thus capturing this geometric information combinatorially.

Link to recording

4/5/22: Sashka Kjuchukova (Notre Dame University)

Title: H-sliceness in definite 4-manifolds

Abstract: Let $K\subset S^3$ be a knot and let $X$ be a closed smooth four-manifold. Does $K$ bound a smooth/locally flat null-homologous disk properly embedded in $X$ minus an open ball? (If so, we say $K$ is smoothly/topologically H-slice in $X$.) The classification of H-slice knots in a 4-manifold $X$ can help detect exotic smooth structures on $X$. I will describe new tools to compute the (smooth or topological) $\mathbb{CP}^2$ slicing number of a knot $K$, which is the smallest $m$ such that $K$ is (smoothly or topologically) H-slice in $\#^m\mathbb{CP}^2$. This talk is based on arXiv:2112.14596.

3/8/22: Patricia Cahn (Smith College)

Title: Trisected geometrically simply-connected manifolds as 3-fold branched covers of S^4

Abstract: We prove that every (g;k_1,k_2,0)-trisected 4-manifold is a 3-fold simple cover of S^4, equipped with its standard genus-0 trisection, with branching set a surface in singular bridge position. A 4-manifold admits such a trisection if and only if it has a handle decomposition with no 1-handles; it is conjectured that all simply-connected 4-manifolds have this property. Our theorem can be viewed as a trisection-theoretic analogue of the following theorem of Hilden: Every 3-manifold, equipped with a Heegaard splitting, is a branched cover of S^3 with its standard genus-0 splitting, with branching set a bridge-position knot. This is joint work with Ryan Blair, Alexandra Kjuchukova, and Jeffrey Meier.

Link to recording

2/8/22: Gabe Islambouli (UC Davis)

Title: Relating two multisections on a given 4-manifold

Abstract: We give a set of moves which relate any two multisections on a given 4-manifold. By constructing multisections of 4-manifolds from loops in the pants complex or from loops of Morse functions on a surface, we also give a complete set of moves relating any two such loops yielding the same 4-manifold.

Link to recording

4/29/21 and 5/5/21: Peter Lambert-Cole (Georgia)

Title: Not a proof of the Schoenflies conjecture

Abstract: The alternate title for the talk is “Stein trisections and homotopy 4-balls.” It’s well known that if X is a compact Stein surface and B is a homotopy 4-ball embedded in X with pseudoconvex boundary, then B must be smoothly standard. In this talk, we will introduce Stein trisections and describe a compelling reembedding construction for homotopy 4-balls in C^2. In particular, if B is a homotopy 4-ball smoothly embedded in C^2, there is a diffeomorphic homotopy 4-ball Z that is the union of three pseudoconvex domains in C^2. Finally, we give some analytic criteria to deduce when this homotopy 4-ball is standard.

4/15/21 and 4/21/21: David Gay (Georgia)

Title: On the smooth mapping class group of the 4-sphere

Abstract: I'll spend two weeks discussing what conclusions can be drawn from the fact that every orientation preserving diffeomorphism of the 4-sphere is pseudoisotopic to the identity, starting by telling/reminding you what this means and why it is true. If I'm lucky I'll think of a way to sneak trisections into the discussion, but that part of the project might be left to the audience, The talk will be based extensively on my recent arXiv posting at https://arxiv.org/abs/2102.12890, in case you like to read ahead.

4/7/21: Jennifer Schultens (UC Davis)

Title: A new proof of Scharlemann's Strong Haken theorem

Abstract: Given a Heegaard splitting of a reducible 3-manifold, Haken's theorem tells us that the Heegaard splitting is reducible. Scharlemann's Strong Haken theorem tells us that more is true: Given an essential 2-sphere S in a 3-manifold M, any Heegaard splitting of M can be isotoped to meet S in a single simple closed curve. As it turns out, Scharlemann's proof can be simplified by referring to the sphere complex, which encodes the essential 2-spheres in a 3-manifold and some aspects of their positioning with respect to each other. This is joint work with Sebastian Hensel.

4/1/21: Jason Joseph (Rice)

Title: Distinguishing ribbon bridge trisections via Nielsen equivalence

Abstract: Ribbon surfaces in the 4-sphere are especially simple to understand because they can be very clearly imagined in 3-space. One such nice way to envision them comes from a ribbon presentation, i.e. an unlink of 2-spheres fused along tubes which can link them in interesting ways. These presentations are not unique, however, and one way to distinguish them is via Nielsen equivalence of the fundamental group presentations they induce for the complement of the surface. Nielsen classes also provide an invariant of a bridge trisection. In this talk we will present a method to bridge trisect these ribbon presentations, in such a way that the Nielsen class of the bridge trisection captures the same information as the Nielsen class of the ribbon presentation. This gives the first examples of inequivalent bridge trisections of the same surface (with the same, minimal parameters (b; c)), utilizing results of Funcke and Yasuda. This is joint work with Jeffrey Meier, Maggie Miller, and Alex Zupan.

3/24/21: Alex Zupan (Nebraska)

Title: Hexagonal lattice diagrams for complex curves in CP^2

Abstract: Peter Lambert-Cole and Jeff Meier revealed that bridge trisections of complex curves in CP^2 exhibit elegant structure: Every complex curve admits an inefficient shadow diagram (with respect to the standard genus one trisection) in which shadow arcs form a hexagonal lattice. Additionally, Lambert-Cole proved a combinatorial classification of symplectic surfaces in CP^2: A surface that minimizes genus in its homology class is symplectic if and only if it admits a transverse shadow diagram. We prove a complex version of Lambert-Cole's theorem, that a genus-minimizing surface in CP^2 is complex if and only if it admits a transverse hexagonal lattice diagram. In the process, we find infinite families of efficient hexagonal lattice diagrams for complex curves.

3/18/21: Nick Castro (Arkansas)

Title: Relative group trisections and 4-manifolds with boundary

Abstract: In this talk I will show how to extend the definition of a Group Trisection to the setting of compact 4-manifolds with boundary so that the one-to-one correspondence between relatively trisected groups and trisected 4-manifolds with boundary is preserved. This is a work-in-progress and is joint with Jason Joseph and Patrick McFaddin.

Since there are a lot of moving parts to a relative trisection, I will begin by discussing relative trisections of 4-manifolds with boundary and how to construct a manifold from a relative trisection diagram. Taking the fundamental group of each of these pieces will lead us naturally into the definition of a trisected group. I will also discuss relative stabilizations and their counterparts in the group setting. We will by discussing the group trisection statement of Abrams, Gay, and Kirby which is equivalent to the Smooth 4-dimensional Poincaré conjecture, and how one might try to mimic this approach to say something about smooth structures on the 4-ball.

3/10/21: Patrick Naylor (Waterloo)

Title: Trisections of non-orientable 4-manifolds

Abstract: In this talk, I’ll explain how to trisect non-orientable 4-manifolds, and explain how the theory differs from the orientable case. In particular, I will sketch a proof of an analogue of an often-used theorem of Laudenbach-Poénaru. This has applications to trisection diagrams and Kirby diagrams for non-orientable 4-manifolds.

2/24/21 and 3/4/21: Inanc Baykur (UMASS Amherst)

Title: Trisections via generic maps and fibrations - I (How to party with Thom, Levine, Lefschetz, Donaldson, Kirby, and Gay), and Trisections via generic maps and fibrations - II (Applications and curiosities)

Abstract: I will explain how to bridge between the theory of (broken) Lefschetz fibrations and that of trisections of 4-manifolds via classical singularity theory, based on my joint work with Osamu Saeki. The goal of these expository talks is to provide trisectors with a larger toolbox, and hopefully more inspiration to cater ideas and techniques from the theory of Lefschetz fibrations.

After we go over the basics, we will sketch several applications, like how to... get simplified trisections on all 4-manifolds, derive trisections via algebraic relations in the mapping class group, obtain trisections realizing any finitely presented group as their pi1, produce infinite families of exotic trisections, describe trisections on products, spuns and mapping tori of 3-manifolds, obtain natural 4-sections, get analogous results on non-orientable 4-manifolds... and maybe do more (or less, depending on our pace). I will then discuss some related questions and open problems.

2/10/21 and 2/18/21: Sarah Blackwell (Georgia), Vincent Longo (Nebraska) and Benjamin Ruppik (MPIM)

Title: Group trisections and smoothly knotted surfaces

Abstract: This is a project that started at the Summer Trisectors Workshop 2020 with Sarah Blackwell, Rob Kirby, Michael Klug, Vincent Longo, and Benjamin Ruppik. In 2016 Abrams, Gay, and Kirby showed, via trisections of 4-manifold groups, that the diffeomorphism classification of smooth, closed, oriented 4-manifolds is entirely group theoretical. That is, all the information of a trisection of a such a 4-manifold is contained in a collection of three epimorphisms from a surface group onto a free group.

Here we derive a similar statement for bridge trisected surfaces in the trisected 4-sphere (and hopefully in general trisected 4-manifolds). Three epimorphisms from the group of a punctured sphere onto a free group allow us to construct the trivial tangles of a bridge trisection and thus a knotted surface. Modding out by stabilization on the algebraic side sets up a correspondence between an algebraically defined set and embedded surfaces in the 4-sphere up to isotopy.

In the first talk we will provide relevant definitions, discuss our construction, and go through some examples. In the second talk we will sketch the proof of our main theorem and talk about possible future directions.

1/27/21 and 2/4/21: Jeffrey Meier (Western Washington)

Title: Bridge trisections for neatly embedded surfaces

Abstract: I will describe how to put a neatly embedded surface in bridge trisected position with respect to a trisection of the ambient compact four-manifold. I’ll first discuss the case when the ambient space is the four-ball and the bridge trisection structure leads to a tri-plane-diagrammatic theory. Next, I’ll discuss the general case, illustrating examples with shadow diagrams.

8/19/20: Slava Krushkal (Virginia)

Title: Embedding obstructions in 4-space from intersections of Whitney disks and from Goodwillie calculus

Abstract: This talk will outline an obstruction theory for embeddings into R^4 based on two approaches. Geometrically these obstructions measure intersections between surfaces and Whitney disks, and more generally intersections of Whitney towers studied by Conant-Schneiderman-Teichner. Homotopy-theoretically the obstructions are defined in terms of maps of configuration spaces, in the context of the Goodwillie-Weiss embedding calculus of functors. I will explain that the two approaches are in fact related. No prior knowledge of Whitney towers or of the embedding calculus will be assumed. This talk is based on joint work with Greg Arone.

8/13/20: Thomas Kindred (Nebraska)

Title: Symmetric efficient multisections of odd-dimensional tori

Abstract: In the PL category, Rubinstein--Tillmann show that every manifold of arbitrary dimension n has a multisection, i.e. a decomposition into k pieces with nice intersection properties, where n=2k-1 or 2k-2. I will describe multisections of the tori Tⁿ=[0,k]ⁿ/~, n=2k-1, which are determined by two symmetry conditions and a simple formula. I will also briefly address the even-dimensional case. In particular, I will describe an efficient symmetric trisection of T⁴ and will show pictures of the 3-, 4-, and 5-dimensional constructions.

8/5/20: Delphine Moussard (Aix-Marseille University)

Title: Uniqueness of the 4-manifold associated to a trisection diagram

Abstract: For closed smooth 4-manifolds, a trisection diagram determines a unique 4-manifold. This also holds for non-closed trisection diagrams of non-closed 4-manifolds. In the case of closed trisection diagram of non-closed 4-manifolds, David Gay pointed out that non-diffeomorphic 4-manifolds may share a common trisection diagram. We will show that the failure of uniqueness can occur only when the page on the boundary contains a 2-sphere; in this case, we will describe the set of 4-manifolds sharing a common diagram. In the other cases, including that of closed 4-manifolds, we will give a proof of the uniqueness of the 4-manifold associated to a trisection diagram which does not use Laudenbach and Poénaru's result. Joint work with Trent Schirmer.

7/30/20: Bill Olsen (Virginia)

Title: Trisections and Ozsvath-Szabo four-manifold invariants

Abstract: Given the data of a relative trisection map on a four-manifold with non-empty connected boundary, we'll give a procedure for computing the Ozsvath-Szabo cobordism maps in Heegaard Floer homology. Time permitting, we'll also speculate on a framework for recovering the closed invariants of Ozsvath and Szabo which may extend to other settings.

7/22/20: Tom Mark (Virginia)

Title: Some questions for trisectors

Abstract: The summer virtual trisectors meeting last month included a group studying questions in 4-manifold topology on which the best progress so far has come from gauge theory, but where is seems possible that trisection techniques may shed new light. The discussion was mainly on surfaces in 4-manifolds, particularly adjunction inequalities, the minimal genus problem, and the bounded negativity conjecture, both in symplectic manifolds and in 4-manifolds for which gauge theory has little to say, such as the connected sum of two projective planes. I’ll attempt to summarize the discussion, along with giving some background on these problems, but I hope audience members will help fill gaps in my exposition, memory, and understanding.

7/8/20: Hannah Schwartz (Princeton)

Title: Embedding concordances with Freedman and Quinn

Abstract: Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of the invariant fq of a pair of homotopic surfaces, first defined by Freedman and Quinn in the 90's. We will show that in any smooth 4-manifold with boundary, pairs of homotopic properly embedded disks with vanishing fq invariant are smoothly concordant. Our argument follows Section 10.9 of Freedman and Quinn's book, in which they prove a more general theorem that gives this result for spheres but not disks.

7/2/20: Puttipong Pongtanapaisan (Iowa)

Title: Bridge multisections of surfaces in 4-manifolds

Abstract: A multisection of a 4-manifold is a generalization of a trisection where more than three 4-dimensional 1-handlebodies are allowed in the decomposition. This notion of multisection allows us to see interesting 3-manifolds as cross-sections of a 4-manifold and plays an important role in Islambouli and Klug's elementary proof that every smooth, oriented, closed manifold is cobordant to connected sums of projective planes. In this talk, I will discuss ongoing work with Gabriel Islambouli on bridge multisection, which is a generalization of a bridge trisection that will allow us to see interesting knots as cross-sections of a multisected knotted surface.

6/18/20: Laura Starkston (UC Davis)

Title: Braided singular surfaces

Abstract: I'll talk about some ways to understand singular surfaces in C^2 and CP^2, particularly geared towards symplectic surfaces. I'll explain how singular braided surfaces can relate to bridge position and Weinstein trisections, as well as some other applications of singular braided surfaces I've been thinking about lately.

6/4/20: Jason Joseph (Rice)

Title: Meridional ranks of knotted surfaces

Abstract: The meridional rank of a knotted surface is the minimal number of meridians needed to generate the knot group. In ongoing work with Puttipong Pongtanapaisan, we investigate this quantity and an analogue of the meridional rank conjecture for knotted surfaces. We use Coxeter quotients and work of Baader, Blair, and Kjuchukova to compute the meridional ranks and verify the conjecture for a large family of knotted surfaces. We will also exhibit a potential counterexample to the conjecture, and show that there exist 2-knots with isomorphic knot groups but different meridional ranks. Applications to bridge trisections will be sprinkled in.

5/27/20: Martin Scharlemann (UCSB)

Title: Uniqueness in Haken’s Theorem

Abstract: (Joint with Michael Freedman) Given a Heegaard split 3-manifold M in which every sphere separates, and a reducing sphere or ∂-reducing disk S in M, it is now known that the Heegaard surface T can be isotoped so that it intersects S in a single circle. Here we show that when this is achieved by two different positionings of T, one position can be moved to the other by a sequence of

• isotopies of T rel S

• pushing a stabilizing pair of T through S and

• eyegelass twists of T.

This last move is inspired by one of Powell’s proposed generators for the Goeritz group.

5/21/20: Mark Hughes (BYU)

Title: Braided ribbon surfaces and bounds on the band rank

Abstract: Rudolph showed that any properly embedded ribbon surface S in B^4 can be braided, or in other words, situated in such a way that each level set of S is a (possibly singular) closed braid. A choice of a braided ribbon surface S induces a special factorization of the boundary braid \partial S, called a band decomposition. The minimal length of a band decomposition (called the braid rank) provides information about the ribbon genus of it's braid closure, but is difficult to compute except in a few instances. Lower bounds on the braid rank can be found using annular Khovanov homology, Dehornoy braid orderings, and the Hurewitz representation. In this talk I will describe some of these bounds.

5/13/20: Juanita Pinzon Caicedo (Notre Dame/MPIM)

Title: The contact invariant of covers of S^3

Abstract: Relative trisections induce open book decompositions on the bounding 3–manifolds and can thus be regarded as fillings of open book decompositions. A theorem of Thurston-Winkelnkemper shows that every open book decomposition supports a contact structure, and a theorem of Honda-Kazez-Matic shows that the monodromy of an open book decomposition defines a contact invariant in the Heegaard Floer homology of the boundary 3–manifold. This suggests a possible connection between 3–manifolds with an open book induced by a trisection, and the Heegaard-Floer invariant of contact 3-manifolds. In this talk I will explain a method to produce open book decompositions of branched covers, in a way that makes a systematic computation of their contact invariants possible.

5/7/20: David Gay (Georgia/MPIM)

Title: "Isotopic" vs "diffeomorphic" in trisection-land, and related nuggets

Abstract: Start with this annoying homework assignment: prove that there is a unique genus zero trisection of S^4. Well, what do you mean by unique exactly? It gets worse from there. I hope to add some clarity rather than increase confusion, and present a few lemmas and examples and lots of questions. Perhaps the most concrete result I will prove, which you might be able to prove on your own in advance, is that any two trisections of S^4 which are diffeomorphic are in fact isotopic. To do this you do not need to know whether the smooth mapping class group of S^4 is trivial, you just need to know that the complement of a ball in S^4 is a ball.

4/29/20: Jeffrey Meier (Western Washington)

Title: Trisections and homotopy four-spheres

Abstract: I will describe how to obtain a trisection of a homotopy four-sphere that is built without 1-handles, emphasizing the relevance of R-links and fibered, homotopy-ribbon knots. Connections between the 4-dimensional Waldhausen Conjecture, the Andrews-Curtis Conjecture, the Generalized Property R Conjecture, and the Slice-Ribbon Conjecture will be discussed. This talk is based on joint works with Zupan and with Schirmer and Zupan.

4/23/20: Román Aranda (Iowa)

Title: Diagrams of $\star$-trisections.

Abstract: A trisection is described by a trisection diagram: three sets of curves in a surface satisfying some properties. In general, it is not evident whether two trisection diagrams represent the same decomposition or even what 4-manifold they depict.

In this talk, I will explain how to soften the definition of a trisection to prove that a large family of genus three trisections is standard. If time permits, I will describe how to draw trisection diagrams for 4-manifolds after surgery along embedded surfaces. This is joint work with Jesse Moeller.

4/15/20: Scott Taylor (Colby College)

Title: Kirby-Thompson Invariants for Genus 0 Bridge Trisections

Abstract: Given a genus 0 bridge trisection of a (smooth) surface in S^4, I’ll describe an nonnegative integer invariant measuring its complexity, modeled on Kirby and Thompson’s invariant for trisections of closed 4-manifolds. This invariant has a curious relationship with the bridge number of the trisection; it all comes down to a property that a bridge trisection may or may not have which is analogous to the (ir)reducibility of Heegaard splittings. The invariant can also be applied to bridge trisections of surfaces in the 4-ball and I’ll spend some time discussing our definition of such bridge trisections. This all joint work with Ryan Blair, Marion Campisi, and Maggy Tomova.

4/9/20: Maggie Miller (Princeton University)

Title: Uniqueness of Relative Trisections

Abstract: Gay and Kirby showed that two relative trisections of a 4-manifold with boundary are equivalent under interior stabilization if they have the same boundary data -- i.e. an open book on the boundary 3-manifold. Castro showed how to achieve Hopf stabilization on the boundary of a trisected manifold. Here's a motivating 3D fact: Piergallini and Zuddas proved that any two open books of a 3-manifold are related by Hopf stabilization and one extra move, "dU."

Last fall, Nick Castro, Gabriel Islambouli, Maggy Tomova and I figured out a stabilization operation on relative trisections that induces a dU move on the boundary -- giving a uniqueness statement for relative trisections of a fixed 4-manifold. I'll talk about that.

4/1/20: Gabriel Islambouli (University of Waterloo)

Title: Multisections of 4-manifolds

Abstract: Multisections are the obvious generalization of a trisection to a decomposition with more than 3 sectors. At the expense of a few more cut systems on the central surface, we gain the ability to clearly see many important submanifolds of a 4-manifold as subsections of a multisection. In particular, we show how corks fit very nicely into quadrisections of 4-manifolds, and give a procedure to produce genus 3 multisection diagrams of E(n)_{p,q}. We also show how a correspondence between multisections and loops in the pants/cut complex guide us towards nullcobordisms of signature 0 4-manifolds. This is joint work with Mike Klug and Pat Naylor.

3/26/20: Alex Zupan (Nebraska)

Title: The realization problem for Tait colored cubic graphs

Abstract: Every bridge trisection of an embedded surface S in a 4-manifold induces a cell decomposition of S, the 1-skeleton of which is a cubic graph equipped with a Tait coloring; that is, each edge can be assigned one of three colors such that every vertex is incident to exactly one edge of each color. In the reverse direction, we prove that every Tait colored cubic graph can be realized as the 1-skeleton of a surface S in the 4-sphere with a planar bridge trisection. Moreover, we show that S can be chosen to be unknotted, and if S is nonorientable, it can be constructed to have any possible normal Euler number. As a corollary, every tri-plane diagram can be converted via crossing changes to one for an unknotted surface. This is joint work with Jeff Meier and Abby Thompson.