UB Geometry and Topology Seminar 

Abstract: We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more generally for link diagrams that are adequate, and the proof involves a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. The main result is used to establish the first and second Tait conjectures for links in thickened surfaces and for virtual links.This is joint work with Homayun Karimi.

Abstract: Problem 3.17 in Kirby's problem list is: which immersed 2-spheres in 3-Euclidean space bound 3-balls (and in which inequivalent ways)? The question of which immersed planar circles bound immersed surfaces and in which inequivalent ways was accomplished by S. Blank, V. Poenaru, G. Francis, K. Bailey, M. Marx, etc. in different cases. But their approach can't be generalized to 3-dimensional case. We have 2 works in dimension 2 and 3 respectively. In 2-dimensional case, we determine all inequivalent immersed surfaces bounded by an arbitrary immersed circle in a compact oriented surface (or 2-plane), and give a hierarchy of immersed surface decomposition. Our 2-dimensional work is straightforward to the final case and completely different from the former works. In 3-dimensional case, we determine all immersed 3-manifold bounded by an arbitrary immersed surface in 3-Euclidean space, and also give a hierarchy of immersed 3-manifold decomposition. This implies the Problem 3.17 in Kirby's problem list by means of applying a detecting algorithm. These works were basically accomplished in my undergraduate.

Abstract: I will discuss a Riemannian metric on the Culler-Vogtmann outer space arising from the tools of thermodynamic formalism. In the context of surfaces, McMullen showed that this metric is proportional to the Weil-Petersson metric on Teichmüller space. This is joint work with Tarik Aougab and Yo’av Rieck.

Abstract: Geodesic currents on surfaces are measure theoretic generalizations of closed curves on surfaces and they play an important role in the study of the Teichmuller spaces. I will talk about their analogs in the setting of free groups, and try to illustrate how the dynamics and geometry of the $\textit{Out}(F_N)$ action reflects on the algebraic structure of $\textit{Out}(F_N)$.

Abstract: A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’. One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such? We explore the case of sutured handlebodies, and see even among the simplest class of these, twisting is required. We give examples that, when restricted to solvable representations, the twisting representation cannot be ‘too simple’.

Abstract: Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in $S^3$ by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.

Abstract: Exponential growth of groups was studied by Milnor and Schwarz as a group theoretic certificate of nonpositive curvature in Riemannian manifolds. Uniform bounds on this growth rate are closely related to detecting free sub-semigroups as well as dynamical and probabilistic characterizations of group elements. In this talk we will describe how nonpositive curvature has been leveraged to show uniform exponential growth for groups with several different notions of nonpositive curvature. Including new results on hierarchically hyperbolic groups, CAT(0) cubical groups, and free-by-cyclic groups based on joint work with Carolyn Abbott and Davide Spriano, Radhika Gupta and Kasia Jankiewicz, and Robert Kropholler and Rylee Lyman respectively.

Abstract: Given a hyperbolic 3- manifold which fibers over the circle with hyperbolic surface fiber, the inclusion map between the fiber and the manifold can be extended continuously to a map, resulting in a space-filling Peano curve. This type of continuous extension of a map to between corresponding boundaries is called a Cannon-Thurston map. Using Birman exact sequence for mapping class groups, we will explain how to construct a Cannon-Thurston map for the boundary of ’surviving’ curve complex of a surface with punctures. This is a joint work with Christopher J. Leininger and Witsarut Pho-on.