Claremont Topology Seminar

2012-13 Schedule

Fall, 2012

Date	Speaker	Title and Abstract
Tuesday Sept 11 3:00 pm	Organizational Meeting
Tuesday Sept 18 3:00 pm	Dave Bachman Pitzer College	Title: Double covers of strongly irreducible surfaces Abstract: We show that double covers of strongly irreducible surfaces are topologically minimal. This is a first step in the confirmation of two larger conjectures, which together give a purely topological proof of the virtually Haken theorem. This is joint work with Yoav Moriah.
Tuesday Sept 25 3:00 pm	Helen Wong Carlton College	Title: A Quantum Trace Map For a surface S, we describe two seemingly unrelated combinatorial objects, the Kauffman skein algebra K(S) and the quantum Teichmuller space T(S). One, the Kauffman skein algebra K(S), was inspired by the Jones polynomial of knots and links, whereas the other, the quantum Teichmuller space T(S), is a non-commutative version of a well-studied object from hyperbolic geometry. In this talk, we'll describe an injective ``quantum trace'' map from K(S) and T(S) and also mention why such a map might be useful. This work is joint with Francis Bonahon.
Tuesday Oct 2 3:00 pm	Sam Nelson Claremont McKenna College	Title: Quantum Enhancements of Birack Counting Invariants Abstract: A quantum enhancement of the birack counting invariant is a quantum invariant of birack-labeled knots and links. We will examine various schemes for finding such enhancements and using the results to enhance the birack counting invariant.
Tuesday Oct 9 3:00 pm	Grace Kennedy UCSB	Title: A Diagrammatic Multivariate Alexander Invariant of Tangles Abstract: I will present my multivariate Alexander polynomial, which also generalizes to a tangle invariant. I'll discuss the algorithm and proof that our algorithm is in fact the multivariate Alexander polynomial defined by Conway in 1970. This multivariate calculation generalizes Professor Stephen Bigelow's diagrammatic method for calculating the single variable Alexander polynomial of a knot or link.
Tuesday Oct 16 3:00 pm	Emille Davie Lawrence University of San Francisco	Title: The sigma-ordering of the braid groups Abstract: The braid groups have been an interesting field of study in low-dimensional topology and algebra since Emil Artin introduced the notion of a braid in the 1920s. Over the years, it has been discovered that the braid groups play a useful role in knot theory, robotics, theoretical physics, and a variety of other areas. In 1992 Patrick Dehornoy proved that the braid groups were left-orderable, providing a long overdue merger between braid groups and orderable. We will give an introduction to the braid groups and discuss a new distinguished form for 3-braids. We will also define Dehornoy’s sigma-ordering of the braid groups, and show how our distinguished form allows us in most cases to determine positivity in this ordering.
Tuesday Oct 23	Fall Break
Tuesday Oct 30 3:00 pm	David Rose USC	Title: Quantum link invariants and (higher) representation theory via skew Howe duality Abstract: Quantum link invariants (e.g. the Jones polynomial) arise due to structures on the category of finite-dimensional representations of quantum groups. These categories often have diagrammatic descriptions which give the skein-theoretic definitions of the link invariants. We will discuss the relation between the diagrammatic framework and skew Howe duality, a representation-theoretic construction which is intimately connected to link invariants. Time permitting, we'll also discuss recent work of the speaker (joint with A. Lauda and H. Queffelec) where we sort out the categorified version of this picture, showing that Khovanov homology is a 2-representation of the categorified quantum group.
Tuesday Nov 6 3:00 pm
Tuesday Nov 13 3:00 pm	Zhongtao Wu California Institute of Technology	Title: An introduction to the rational genus of a knot Abstract: What is the "simplest" knot in a given three-manifold $Y$? We know that the answer is the unknot when $Y=S^3$, as the unknot happens to be the only knot in the three-sphere with the smallest genus (=0). In this talk, we will discuss the more general notion of the rational genus of knots. In particular, we will show that the simple knots are really the "simplest" knots in the lens spaces in the sense of being a genus minimizer in its homology class. This is a joint work with Yi Ni.
Tuesday Nov 20 3:00 pm
Tuesday Nov 27 3:00 pm	Hans Boden McMaster University	Title: Metabelian SL(n,C) representations of knot groups Abstract: This talk will focus on applications of group representation theory in low-dimensional topology. We will focus attention on metabelian representations into SL(n,C), and we report on joint work with S. Friedl on the classification problem of such representations (up to conjugacy) and on the problem of constructing deformations within the larger space of all SL(n,C) representations. We use this approach to establish the existence of large families of irreducible SL(n,C) representations under certain mild conditions on the knot which are easily expressed in terms of its (twisted) Alexander polynomial.
Thursday Dec 6 3:00 pm	Helen Wong Carlton College	The Kauffman Skein Algebra and hyperbolic geometry The Kauffman skein algebra of a surface $S$ was originally defined as a generalization of the Jones polynomial, but it was later found to have connections with hyperbolic geometry. Here, we'll describe how it is related both to the character variety, consisting of homomorphisms from $\pi_1 S$ to $\mathrm{SL}_2 \mathbb C$, and to a quantizations thereof. If time permits, we will also discuss ways this might be exploited, for instance to interpret quantum invariants of 3-manifolds in terms of hyperbolic geometry. This work is joint with Francis Bonahon.
Tuesday Dec 11 3:00 pm	Mike Williams UC Riverside	Title: Advanced Positions of Knots in the Three Sphere Abstract: In this talk, we introduce the notion of advanced position of a knot on a Heegaard surface and present several examples. This is joint work with Jesse Johnson and Alice Stevens.

Spring 2013 Schedule

Date	Speaker	Title and Abstract
Tuesday Jan 29 3:00 pm	Rena Levitt Pomona College	Title: Graphs and Geometry: A Combinatorial Version of Area and Perimeter Abstract: First introduced by Leonhard Euler to solve the famous Bridges of Konigsberg Problem in 1736, graphs have emerged as an important concept in modern mathematics. We use graphs to model everything from links among websites, to bonds between atoms in molecules, or the evolutionary relationships among related species. In this talk I will introduce the notion of a graph, and present some applications of graph theory. I will then introduce combinatorial versions of length and area that allow us to add geometric structure to an abstract graph. Later in the talk, I will focus on the class of bridged graphs, and show that the area of a disk in a bridged graph is bounded quadratically by the length of its perimeter. This is joint work with my students from the Fletcher Jones Fellowship Summer Research Program: Nicholas Bosviel, Gillian Grindstaff, Lingge Li, Patrick Meehan, and Matthew Owen.
Tuesday Feb 5 3:00 pm
Thursday Feb 14 3:00 pm	Danny Ruberman Brandeis University	Title: Embeddings of non-orientable surfaces in 4-manifolds Abstract: Two-dimensional surfaces are classified by two characteristics: their orientability, and their genus (a non-negative integer). Low dimensional topology has long centered around the problem of finding the least complicated (meaning lowest genus) oriented surface carrying a given 2-dimensional integral homology class in a 4-manifold. The Thom conjecture about homology classes in complex projective space, solved by Kronheimer-Mrowka using Seiberg-Witten gauge theory, was the most famous such problem. I will discuss joint work with Adam Levine and Saso Strle about embeddings of non-orientable surfaces in 4-manifolds of the form (3-manifold x I).
Tuesday Feb 19 3:00 pm	Chad Musick Nagoya University	Title: A method of encoding generalized link diagrams Abstract: We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots, the minimal number of such marks is twice the bridge index, and a classical knot diagram in minimal bridge form with bridge index b may be encoded in order b^2 integers. A set of moves on the notation is defined. Some uses of the notation are discussed.
Tuesday Feb 26 3:00 pm	Ryo Nikkuni Tokyo Women's Christian University	Title: A homotopy classification of two-component spatial graphs up to neighborhood equivalence Abstract: A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to the first homology group of each of the connected components. This also leads a kind of homotopy classification of 2-component handlebody-links. This is a joint work with Atsuhiko Mizusawa.
Tuesday Mar 5 3:00 pm	Dave Bachman Pitzer College	Title: Normalizing Topologically Minimal surfaces Abstract: Topologically minimal surfaces generalize several well-studied classes of surfaces in 3-manifolds, and provide a topological analogue to geometrically minimal surfaces. We will discuss recent progress in obtaining a normal form for any such surface with respect to a fixed triangulation. This provides striking analogues with results of Colding and Minicozzi, and establishes finiteness results which are crucial to understanding how Heegaard splittings are effected by Dehn surgery.
Tuesday Mar 12 3:00 pm	Yi Liu Cal Tech	TITLE: Representation volume of 3-manifolds ABSTRACT: In this talk we discuss volume of 3-manifold arise from representations into PSL(2,C) and \widetilde{SL}_2(R). Recent techniques of Przytycki and Wise allows us to construct certain interesting representations after passing to a finite index subgroup of the fundamental group. In particular, one can show the virtual representation volume to be positive if a corresponding geometric piece presents. This is joint work with Pierre Derbez and Shicheng Wang.
Tuesday Mar 19	Spring Break
Tuesday Mar 26 3:00 pm	Scott Carter University of South Alabama	Title: Braiding branched covers of spheres over knots Classical theorems (Alexander, Hilden, Montesinos) indicate that any 3-manifold can be realized as a 3-fold branched covering of the 3-sphere with branched set a knot or link. By generalizing Kamada's braid charts to one higher dimension, we show how to embed and immerse these coverings in $S^3 \times D^2$ such that the projection onto the first factor is the covering. Similarly, every 4-manifold is a 5-fold branched cover of $S^4$ with branched set an embedded or linked surface. In some cases, we can also construct analogous embeddings and immersions in $S^4 \times D^2$. The methods for doing so are very detailed. Lots of examples will be given.
Tuesday Apr 2 3:00 pm
Tuesday Apr 9 3:00 pm	Emily Hamilton Cal Poly SLO	Title: Separability of Double Cosets and Conjugacy Classes in 3-Manifold Groups Abstract: A subset $X$ of a group $\Gamma$ is {\it separable} in $\Gamma$ if for every element $\gamma \in \Gamma - X$ there is a homomorphism $\phi$ from $\Gamma$ to a finite group such that $\phi(\gamma) \notin \phi(X)$. A group $\Gamma$ is {\it residually finite} if the trivial subgroup is separable, {\it subgroup separable} if every finitely generated subgroup of $\Gamma$ is separable, and {\it conjugacy separable} if every conjugacy class in $\Gamma$ is separable. Separability has applications in group theory and geometric topology. If a finitely presented group $\Gamma$ is residually finite, then there exists an algorithm to decide if a given word in the presentation of $\Gamma$ is trivial. If $G$ is subgroup separable, then one can solve more generalized word problems. In the context of geometric topology, subgroup separability has been used to solve immersion to embedding problems. For example, in $3$-manifold topology it is well known that subgroup separability allows passage from immersed incompressible surfaces to embedded incompressible surfaces in finite covers. In this talk we consider separability of double cosets and conjugacy classes in $3$-manifold groups. Let $M = {\Bbb H}^3 / \Gamma$ be a hyperbolic $3$-manifold of finite volume. We show that if $H$ and $K$ are abelian subgroups of $\Gamma$ and $g \in \Gamma$, then the double coset $HgK$ is separable in $\Gamma$. As a consequence, we prove that if M is a closed, orientable Haken $3$-manifold and the fundamental group of every hyperbolic piece of the torus decomposition of $M$ is conjugacy separable then so is the fundamental group of $M$. Invoking recent work of Agol and Wise, it follows that if $M$ is a compact, orientable $3$-manifold, then $\pi_1(M)$ is conjugacy separable.
Tuesday Apr 16 3:00 pm	Katie Walsh UC San Diego	Title: Patterns in the Coefficients of the Colored Jones Polynomial The colored Jones polynomial assigns to each knot a sequence of Laurent polynomials. We will discuss the various ways of calculating the colored Jones polynomial and formulas that allow us to calculate many of the polynomials in the sequence for certain knots. These formulas allow us to look at patterns in the coefficients. A few conjectures relating these coefficients to the hyperbolic volume conjecture will be discussed.
Tuesday Apr 23 3:00 pm	Danny Stoll Oakland Technical High School Cliff Stoll Acme Klein Bottle	Title: Low Dimensional Topology for Fun and Profit and 17 ways to Extract Lucre from R4 Space and The Bad Pants Homology Abstract: For over a decade, Acme Klein Bottle has supplied nonorientable manifolds to math folk. Like much of mathematics, it's marginally profitable, but endlessly entertaining. There are thousands of computer models of the Klein Bottle and its bounded friend, the Moebius loop. But physical models are rarely made. So how do you turn a set of parameterized equations for a manifold into a glass Klein Bottle? When you immerse a manifold into R3, what's lost? How about glass models of the projective plane, Boy's surface, the torus, and other manifolds? Recently, Kahn and Markovic have used the good pants homology to prove the Ehrenpreis conjecture. Inspired by this, we have developed the bad pants homology to create a certain nonorientable Riemannian manifold. As a door prize, we will give away a Hausdorffian, unbounded, affine, closed, rustproof, self-intersecting, compact, microwave-safe borosilicate manifold that's locally Euclidean and homeomorphic to a sphere with two crosscaps.
Tuesday Apr 30 3:00 pm	Matt Owen Pitzer College	Title: Construction between partially-ordered sets and CAT(0) cube complexes Abstract: Hyperplanes in a cube complex X allow us a nice way of inducing the order of inclusion on the vertices of X. We first provide an introduction to cube complexes and partially-ordered sets (posets). We then examine a construction under which one can build CAT(0) cube complexes from posets, and posets from CAT(0) cube complexes. We conclude by considering consequences of this construction, such as its domain and range, how the dimension of the cube complex is affected, what the degree of the fixed vertex implies.
Tuesday 3:00 pm May 7	No Seminar