Schedule for: 21w5094 - Knots, Surfaces and 3-manifolds (Online)

Given a 3-manifold with finite group of diffeomorphisms, there is an equivariant Heegaard splitting of the 3-manifold. The Equivariant Sphere Theorem says that if the manifold is reducible, then there is also an equivariant sphere. Scharlemann recently showed that every Heegaard surface in a reducible manifold can be isotoped to intersect a given reducing sphere in a single curve. So it is natural to ask, if an Equivariant Heegaard surface for a reducible manifold necessarily intersects an equivariant reducing sphere in a single closed curve. I’ll sketch how recent advances in the theory of bridge position of spatial graphs apply to studying the nonadditivity of equivariant Heegaard genus.

Every 3-manifold can be triangulated, i.e. decomposed into tetrahedra. If the 3-manifold has geometry, we would like the corresponding tetrahedra to have geometry. For example, if the 3-manifold is hyperbolic, we would like the tetrahedra to be convex hyperbolic tetrahedra, with positive volume; this is called a geometric triangulation. However, it is still unknown whether every cusped hyperbolic 3-manifold admits such a triangulation. In this talk, I will show that the complement of every knot in the 3-sphere admits a geometric triangulation, provided it is sufficiently highly twisted. This is joint work with Sophie Ham.

The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds. One measure of the non-triviality of a three- manifold is the existence of non-trivial SU(2)-representations. In this talk I will show that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)- representations. This is joint work with Tye Lidman and Raphael Zentner.

Circles in the boundary of genus two handlebodies naturally appear in definitions or properties of knots in the 3-sphere. In this talk we discuss how primitive, power and Seifert circles are used to find information about hyperbolic genus one knots and primitive/Seifert knots.

Curve graphs are combinatorial objects which encode intersection patterns of simple closed curves on surfaces. Inspired by this, such graphs have been built in many other topological settings -- and it turns out that these objects are surprisingly effective tools in geometric group theory, hyperbolic geometry, and topology. In this talk, I will try to give an overview of these methods, and try to convince you that curve graphs and their relatives can be both fascinating in their own right and very much usable. I will present two recent results as evidence: the unboundedness of the fragmentation norm on isotopically trivial homeomorphism groups of surfaces (joint with Jonathan Bowden and Richard Webb), and a quick proof of the Strong Haken Theorem (joint with Jennifer Schultens)

There are several computationally hard problems about triangulated 3-manifolds that admit an efficient algorithmic solution, provided the input triangulation is sufficiently "thin." Exhibiting such triangulations is therefore a compelling task, however, this can be limited by the topology of the underlying manifold. In this talk I give an overview of recent results that link the key combinatorial parameters in the above context to classical topological invariants of 3-manifolds in a quantitative way. We establish these results through constructions involving generalized Heegaard splittings and layered triangulations of 3-manifolds. Joint work with Jonathan Spreer and Uli Wagner.

Given an oriented minimal genus Seifert surface F′ for a (1,1)-knot K it is possible to surger F′ along annuli to obtain a simple minimal Seifert surface F. Such a surface can be put in a very nice position with respect to the (1,1)-position of the knot K. Using this kind of surfaces we give a description of a (1,1)-knot of genus g as a vertical banding of (1,1)-knots of genus smaller than g. In addition, we show that any rational knot of genus g is obtained as a vertical banding of g genus one rational knots. Joint work with Mario Eudave-Muñoz (UNAM) and Enrique Ramírez-Losada (CIMAT).

The crease set of a surface embedded in R^3 captures where the surface folds under a choice of projection to R^2. In forthcoming work with W. Menasco, we develop tools to characterize the crease set of an embedded sphere and its behavior under isotopy of the embedding. In this talk, we introduce some of these tools, explore a few examples, and discuss potential applications.