Redbud Topology Conference

The talks are on the University of Arkansas campus in SCEN 408. Abstracts are below.

Time	Speaker	Talk
9:00 – 9:50	Birch Bryant	Inflating along spun-normal surfaces
10:05 – 10:55	Max Forester	Commensurability of lattices in Baumslag–Solitar complexes
11:10 – 12:00	Sümeyra Sakalli	Slicing knots in definite 4–manifolds
12:00 – 2:00	lunch
2:00 – 2:50	Jonathan Johnson	Non-standard orders on torus bundles with one boundary
3:05 – 3:55	Katherine Raoux	Rational 4–genus and knot Floer homology
4:10 – 5:00	Jing Tao	Genericity of pseudo-Anosov maps

Abstracts:

Title: Inflating along spun-normal surfaces
Abstract: From its inception, normal surface theory has been employed to algorithmically study 3–manifolds, both in terms of overall structure such as the JSJ–decomposition and regarding the existence of embedded surfaces. In unpublished notes Thurston developed spun-normal surfaces which gave a place for embedded surfaces with boundary to live in ideal triangulations. Unlike normal surfaces, spun-normal surfaces are not supported by so many existence theorems. Through the techniques of crushing and inflating, algorithms to construct triangulations with instructions given by normal surfaces developed by Jaco and Rubinstein, there are pleasing connections between closed normal surfaces in ideal triangulations and normal surfaces in inflations. Following my thesis work, we will briefly explore some analogous (if not as robust) results connecting spun-normal surfaces with counterparts in inflated triangulations.

Title: Commensurability of lattices in Baumslag–Solitar complexes
Abstract: I will talk about lattices in the automorphism groups of Baumslag–Solitar complexes. The classical Baumslag–Solitar groups are examples but there are others too. I will discuss a new commensurability invariant and a construction of incommensurable lattices.

Title: Non-standard orders on torus bundles with one boundary
Abstract: Consider a torus bundle over the circle with one boundary. Perron-Rolfsen shows that having an Alexander polynomial with real positive roots is a sufficient condition for the bundle to have bi-orderable fundamental group. This is done by showing the action of the monodromy induced on the fundamental group of the surface preserves a "standard" bi-ordering. In this talk, we discuss if there are other ways to bi-order the fundamental group of a torus bundle with one boundary component. This work is joint with Henry Segerman.

Title:
Abstract: Knots that represent torsion homology classes don't bound Seifert surfaces. But, some multiple of the knot does bound a surface. The rational genus measures the minimal complexity of one of these surfaces.

In this talk we explore a related 4–dimensional question: what if we allow surfaces in the ambient 3–manifold cross interval? For knots in the 3–sphere, minimal complexity surfaces realize the 4–genus or slice genus of the knot. We define "rational 4–genus" for knots representing nontrivial torsion classes in general 3–manifolds. We show that generalized Heegaard Floer τ-invariants give a lower bound for the rational 4–genus and that the class of Floer simple knots have equivalent 3 and 4–dimensional rational genus.

Title: Slicing knots in definite 4–manifolds
Abstract: We will talk about the smooth (resp. topological) ℂℙ²–slicing number of knots, i.e., the smallest m≥0 such that a knot K in S³ bounds a smooth (resp. locally flat), properly embedded, null-homologous disk in a punctured connected sum of m copies of ℂℙ². We will give a lower bound on the smooth ℂℙ²–slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth ℂℙ²–slicing numbers. We will also give the first examples of knots where the smooth and topological ℂℙ²–slicing numbers are finite, distinct and simultaneously nonzero. This is a joint work with A. Kjuchukova, A. N. Miller, and A. Ray.

Title: Genericity of pseudo-Anosov maps
Abstract: By Nielsen–Thurston classification, every homeomorphism of a surface is isotopic to one of three types: finite order, reducible, or pseudo-Anosov. While there are these three types, it is natural to wonder which type is more prevalent. In any reasonable way to sample matrices in SL(2,ℤ), irreducible matrices should be generic. One expects something similar for pseudo-Anosov maps. In joint work with Erlandsson and Souto, we define a notion of genericity and show that pseudo-Anosov maps are indeed generic. More precisely, we consider several "norms" on the mapping class group of the surface, and show that the proportion of pseudo-Anosov maps in a ball of radius r tends to 1 as r tends to infinity. The norms can be thought of as the natural analogues of matrix norms on SL(2,ℤ).