# Spring 2018 Redbud conference - Triangulations and Trisections

## Speakers

• Nick Castro, UC Davis
• Title: Relative Trisections and Bounding Open Book Decompositions
Abstract: A relative trisection of a 4-manifold with non-empty boundary is a decomposition into three diffeomorphic pieces with nice pairwise and triple intersections, which induces an open book decomposition on the bounding 3-manifold(s). Moreover, given a relative trisection diagram there is an algorithm to determine this induced open book via a cut system of properly embedded arcs. Under appropriate conditions, relatively trisected manifolds can be glued together along diffeomorphic boundaries which preserves the structure of a trisection. In this talk I will discuss the basics of relative trisections; namely, the algorithm for determining the monodromy algorithm and the gluing theorem. Additionally, I will discuss recent results which allow us to modify given relative trisections so as to induce certain open books on the resulting boundary. This is a step towards understanding the complexity of a relative trisection and the complexity of the induced open book. Portions of this talk is based on joint works with D. Gay and J. Pinzón, and M. Miller, J. Pinzón, and M. Tomova.

• Matthias Goerner, Pixar
• Title: Geometric cell decompositions
Abstract: It is an unsolved question whether each cusped finite-volume hyperbolic 3-manifold has a geometric triangulation by ideal non-flat tetrahedra. Starting with a review of Epstein-Penner decompositions, I will prove a result by Luo, Schleimer and Tillmann that geometric triangulations (which is a Epstein-Penner decomposition) exists virtually and a result by me that each geometric decomposition into Platonic solids can be subdivided into a geometric triangulation.

• Sergei Gukov, CalTech
• Title: Constructing 4-manifold invariants via trisections
Abstract: In this talk I will try to explain a general framework, motivated by physics, that allows to define 4-manifold invariants via trisections of Gay and Kirby. In this approach, an invariant of a 4-manifold is constructed starting from the data of a triangulated category whose group of autoequivalences is a mapping class group of the genus-g surface. Examples of such categories are not easy to find, and each new one leads to an invariant of smooth 4-manifolds. I will illustrate this somewhat abstract construction with concrete and simple examples.

• Efstratia Kalfagianni, Michigan State
• Title: On the growth of Turaev-Viro 3-manifold invariants
Abstract: Around 1990 Turaev and Viro defined a family of real-valued 3-manifold invariants as state sums on triangulations of 3-manifolds. The family is indexed by an integer (the level) and at each level the invariant depends on a certain root of unity. We will discuss recent results making progress in understanding the `large-level” asymptotic behavior of Turaev-Viro invariants and some applications of these results.

• Jeff Meier, Georgia
• Colloquium Talk: The Theory of Trisections
Abstract: In their foundational 2012 paper, David Gay and Robion Kirby showed that four-dimensional smooth manifolds can be studied via decompositions called trisections. A trisection is a decomposition of a four-dimensional object into three pieces, each of which is as simple as possible. Since its inception, the theory of trisections has developed and expanded in a number of ways. In this talk, I'll give a gentle introduction to this theory, highlighting emerging techniques in the area and giving many examples along the way. This talk should be accessible for beginning graduate students.

(Conference talk) Title: Bridge trisections of knotted surfaces in four-manifolds

Abstract: In this talk, we will examine the theory of bridge trisections for knotted surfaces in four-manifolds. The main result is that any smooth surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambient four-manifold. In the setting of knotted surfaces in the four-sphere, this gives a diagrammatic calculus that offers a promising new approach to four-dimensional knot theory. However, the theory extends to other ambient four-manifolds, and we will pay particular attention to the setting of complex curves in simple complex surfaces, where the theory produces surprisingly satisfying pictures and leads to interesting results about trisections of complex surfaces. This talk is based on various joint works with Dave Gay, Peter Lambert-Cole, and Alex Zupan.

• Hyam Rubinstein, University of Melbourne
• Title: Minimal triangulations of 3-manifolds
Abstract: An old problem is to determine the smallest number of tetrahedra in a triangulation of a given 3-manifold. In previous work we have described several infinite families of lens spaces and spherical space forms for which minimum triangulations can be determined. We further extend this work finding more lens spaces and also the first infinite family of examples with the geometry $\tilde SL(2,R)$ in the closed case. For ideal triangulations we are currently finishing a paper showing that all hyperbolic once punctured surface bundles over a circle have minimum ideal triangulations which are the canonical hyperbolic ones. Moreover we are also able to show that a related family of hyperbolic semi bundles also has this property. This is joint work with Bus Jaco, Jonathan Spreer and Stephan Tillmann.

• Abby Thompson, UC Davis
• Title: Trisections and link surgeries
Abstract: I’ll discuss some natural conjectures that arise about integral surgeries on the links generated by trisected 4-manifolds.

• Maria Trnkova, UC Davis
• Title: Ideal Triangulations of closed hyperbolic 3-manifolds
Abstract: W.Thurston showed that every hyperbolic 3-manifold can be decomposed into a set of ideal tetrahedra (and also simple closed geodesics in the case of closed manifolds). Unfortunately such a decomposition sometimes produces overlapping tetrahedra, a so called non-geometric ideal triangulation. There is a conjecture that every cusped hyperbolic 3-manifold admits a geometric ideal triangulation. We are interested in ideal triangulations of closed hyperbolic 3-manifolds. SnapPy's computations provide several examples of non-geometric ideal triangulations of closed manifolds. The smallest of them is known as Vol3. In this talk we show that Vol3 does not have any geometric ideal triangulation of small complexity. The main techniques that we use are Thurston's Dehn surgery theorem, Dehn parental test and a quantum invariant called 3D index. This is a joint work with Feng Luo.

• Gabriel Islamboul, Virginia
• Title: Nielsen equivalence and trisections
Abstract: We show how to construct an interesting group theoretic invariant of a trisection which parallels an established invariant of Heegaard splittings. We then apply this invariant to the spun trisections of Meier to construct non-diffeomorphic trisections on spun Seifert fiber spaces.

• Sanjay Kumar, Michigan State
• Title: Volume Conjecture for Turaev-Viro Invariants
Abstract: In this talk, we will construct an infinite family of 3-manifold invariants from a triangulation of a given manifold originally defined by Turaev and Viro. These invariants appear to be closely related to the geometry of the manifold. In particular, when the manifold is hyperbolic, we will see there is evidence to support that the asymptotics of this family of invariants predict the volume of the manifold.

• Maggie Miller, Princeton
• Title: Relative trisections of surface complements
Abstract: In this talk, I will first give some exposition on relative trisections (trisections of 4-manifolds with boundary). We will see how one can draw a surgery diagram from a relative trisection. Then I will show how to produce a trisection of the complement of a surface embedded in a 4-manifold, with emphasis on the illustrative case of an embedded $RP^2$. This is joint work with Seungwon Kim.

• Eric Samperton, UC Davis
• Title: Finding covers is hard
Abstract: Covering spaces are a useful way to understand a space, especially when the space is a 3-manifold. In particular, given a fixed integer $k$, one might want to decide when 3-manifolds admit a $k$-sheeted cover. Our main result is that whenever $k \ge 5$, this question is $\mathsf{NP}$-complete. Thus, assuming $\mathsf{P} \neq \mathsf{NP}$, there exist 3-manifolds where it is effectively impossible to decide if they have 5-sheeted covers. The proof is a combination of ideas involving reversible circuits, finite group theory, and the action of the mapping class group of a surface $\Sigma$ on the set of homomorphisms $\{ \pi_1(\Sigma) \to S_n \}$.

• William Worden, Temple
• Title: Random veering triangulations
Abstract: For certain fibered hyperbolic 3-manifolds, Agol gives a construction for a layered veering triangulation, which is canonically associated to the conjugacy class of the fiber bundle monodromy. We will present some experimental results which suggest that combinatorial data of the veering triangulation carries some geometric information about the underlying manifold. On the other hand, experiments also provide strong evidence that these triangulations are typically non-geometric, and we will present a theorem that confirms this. Joint with Sam Taylor and Dave Futer.