Claremont Topology Seminar

2010-11 Schedule

Fall, 2010

Date	Speaker	Title and Abstract
Tuesday Sept 7 3:00 pm	Organizational Meeting
Tuesday Sept 14 3:00 pm	Azadeh Rafizadeh UC Riverside	Title: Twisted Alexander Polynomials and Fiberability Abstract: D. Eisenbud and W. Neumann have developed a theory to determine fiberability of graph links. We use twisted Alexander polynomials to investigate this problem.
Friday Sept 24 4:00 pm	Danny Ruberman Brandeis University	Title: Smooth versus topological concordance of knots Abstract: It has been known since the early 1980's that there are knots that are topologically (flat) slice, but not smoothly slice. These result from Freedman's proof that knots with trivial Alexander polyno- mial are topologically slice, combined with gauge-theory techniques originating with Donaldson. In joint work with C. Livingston and M. Hedden, we answer the natural question of whether Freedman's result yields all topologically slice knots. We show that the group of topologically slice knots, modulo those with trivial Alexander polynomial, is infitely generated. The proof uses Heegaard-Floer theory.
Tuesday Sept 28 3:00 pm	Sam Nelson CMC	Title: Rack algebras and beyond Abstract: In 2002, Andruskiewitsch and Grana defined an associative algebra associated to a finite quandle X, representations of which were used in 2003 by Carter, Grana, Elhamdadi and Saito to define enhancements of the quandle counting invariant. In summer 2010, my REU team defined a modification of the quandle algebra appropriate for enhancing the rack counting invariant. In this talk we will see how the rack algebra arises from a modification of the (t,s)-rack structure, how to define new link invariants using rack modules, and preview four current projects extending these idea in various directions.
Tuesday Oct 5 3:00 pm
Tuesday Oct 12 3:00 pm	Michael Yoshizawa UCSB	Title: Hempel Distance 3 on Heegaard splittings Abstract: If a Heegaard splitting of a 3-manifold satisfies the Casson-Gordon rectangle condition, then it will have Hempel distance greater than or equal to 2. Furthermore, John Berge developed a criterion that would ensure a genus 2 Heegaard splitting has Hempel distance greater than or equal to 3. I will review these two results and then discuss my attempts to generalize Berge's criterion to Heegaard splittings of arbitrary genus.
Tuesday Oct 19 3:00 pm	Fall Break
Tuesday Oct 26 3:00 pm	Kanako Oshiro Hiroshima University, Japan	Title: On pallets for coloring invariants of spatial graphs A Fox $p$-coloring for a spatial graph diagram is an assignment of an element of $\mathbb Z_p =\{ 0, \cdots ,p-1 \}$ to each arc. At each crossing, the well-known coloring condition is satisfied. An $n$-pallet of $\mathbb Z_p$ is a subset of $(\mathbb Z_p)^n$ satisfying some condition. It gives a coloring condition for $n$-valent vertices. In this talk, we consider what kind of pallets can be obtained for some integers $p\geq 3$ and $n\geq 2$.
Tuesday Nov 2 3:00 pm	Dave Bachman Pitzer College	Title: Victories and defeats from the front-lines of normal surface theory Abstract: Normal surfaces are a classical tool used to study 3-manifolds. The analogy with minimal surfaces motivates us to abstract this theory in new ways. We will discuss several recent (pleasant) surprises that have arisen from this endeavor, as well as a few new difficulties.
Tuesday Nov 9 3:00 pm	Vin de Silva Pomona College
Tuesday Nov 16 3:00 pm	Rob Sulway UCSB	Title: The Artin group of type $\tilde{B}_3$ is torsion-free and has decidable word problem. Abstract: Coxeter groups are discrete groups of symmetries of various spaces and each Artin group can be though of as a braided version of a Coxeter group. I will explain how this relationship arises in two different ways, one of which is via a process known as "pulling apart". This process also provides very nice geometric pictures of the groups and can give rise to groups which are "Garside", which means they have nice properties, in particular are torsion-free and have decidable word problem. Although many affine Artin groups are not Garside, I will explain how they can be embedded in groups that are, using $\tilde{B}_3$ as the primary example.
Tuesday Nov 23 3:00 pm	Ko Honda USC	Title: HF=ECH via open book decompositions Abstract: The goal of this talk is to sketch a proof of the equivalence of Heegaard Floer homology (due to Ozsvath-Szabo) and embedded contact homology (due to Hutchings). This is joint work with Vincent Colin and Paolo Ghiggini.
Tuesday Nov 30 3:00 pm	Helen Wong Carleton College	Title: Representations of the Kauffman Bracket Skein Algebra Abstract: The Kauffman bracket skein algebra was originally defined as a means of generalizing the Jones polynomial and the related Witten-Reshetikhin-Turaev 3-manifold invariants. As such, it has a simple, combinatorial definition, though surprisingly it also has an interpretation via hyperbolic geometry. In order to further understand its structure, we study the representations of the Kauffman bracket skein algebra and provide a conjectural classification of them.
Tuesday Dec 7 3:00 pm	Barbara Herzog UCR	Title: Toward a Notion of Index for Critical Points of Distance Functions Abstract: The index of a smooth function at a critical point is the dimension of the largest subspace on which the Hessian is negative definite. Morse Theory uses critical points (or lack of critical points) of a smooth function as well as index to describe the topology of a space. In Riemannian geometry distance functions are not smooth meaning that both critical points and the Hessian cannot be defined in the usual way. In 1977 Grove and Shiohama created a definition of critical point for distance functions and used it to describe the topology of a space in the absence of a critical point. This generalization of Morse Theory has had far reaching consequences. Currently we are working to create a definition of index for distance functions in order to describe the topology of a space at a critical point. A notion of lower index will be presented.

Spring 2011 Schedule

Date	Speaker	Title and Abstract
Tuesday Jan 25 Cancelled!	Ellie Grano UCSB	Title: The jellyfish algorithm Abstract: Algorithms for evaluating closed diagrams are ubiquitous in topology, e.g. the HOMFLY polynomial and the Kauffman bracket. In 2009, Stephen Bigelow defined the jellyfish algorithm to evaluate closed diagrams for the ADE planar algebras. I will introduce the algorithm and then show the algorithm is well defined for certain planar algebras. The main result is that these planar algebras are not trivial. This follows the Kuperberg program: Give a presentation for every interesting planar algebra, and prove as much as possible about the planar algebra using only its presentation.
Tuesday Feb 1 3:00 pm	Adam Lowrence U of Iowa	Title: Turaev genus and knot homologies Abstract: The Turaev surface of a link diagram is a certain Heegaard surface of S^3 on which the link has an alternating projection. The Turaev genus of link is the minimum genus of any Turaev surface for the link. In this talk, I will discuss a relationship between Turaev genus and two knot homologies: Khovanov homology and knot Floer homology.
Tuesday Feb 8 3:00 pm	Liam Watson UCLA	Title: Left-orderability and Dehn surgery. Abstract: Left-orderability of groups is a property that seems to have a geometric flavour. In the context of the fundamental group of a 3-manifold, this property is related to coorientable taut foliations, for example. It is known that the fundamental group of a knot complement is always left-orderable, however the result of Dehn surgery along the knot need not inherit this property. This talk investigates the relationship between left-orderability and Dehn surgery, and exhibit knots having the property that all sufficiently positive surgeries yield manifolds with non-left-orderable fundamental groups. These knots turn out to all be L-space knots (arising in Heegaard Floer homology), and in light of the fact that all known examples of L-spaces have non-left-orderable fundamental group, it is interesting to compare this result with the fact that L-space surgeries behave in an analogous manner. This is joint work with Adam Clay.
Tuesday Feb 15 3:00 pm	Jim Hoste Pitzer College	Title:Upper bounds in the Ohtsuki-Riley-Sakuma partial order on 2-bridge knots Abstract: We use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot $K_1$ we characterize all other 2-bridge knots $K_2$ such that $\{K_1,K_2\}$ has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots. This is joint work with Pat Shanahan and Scott Garrabrant.
Tuesday Feb 22 3:00 pm	Matthias Goerner UC Berkeley	Title: Representations of 3-Manifold Groups Abstract: I present how to algorithmically find all parabolic representations of 3-manifolds $\pi_1(M)\rightarrow \mathrm{SL}(N,\mathbb{C})/(-1)^{N+1} I$ using a new technique to parametrize representations due to Christian Zickert, Dylan Thurston and Stavros Garoufalidis. I have implemented these algorithms and computed the resulting invariants such as the complex volume (regular volume and Chern Simons invariant) and the induced element in the Bloch group for these representations, giving rise to examples for Walter Neumann's conjecture on the Bloch group.
Tuesday Mar 1 3:00 pm	Eamonn Tweedy UCLA	Title: Knot invariants in Heegaard Floer homology arising from symplectic geometry Abstract: Given a knot K in the 3-sphere, one can draw a correspondence between Seidel and Smith's fixed-point symplectic Khovanov cochain complex and the Heegaard Floer chain complex of the branched double cover. This relationship induces a filtration on the Heegaard Floer complex, and we'll discuss its construction, an invariance result, and some properties. In particular, it provides a spectral sequence connecting the two theories, a new family of knot invariants, and (conjecturally) an invariant of the smooth knot concordance class of K.
Tuesday Mar 8 3:00 pm
Tuesday Mar 15 3:00 pm	Spring Break
Tuesday Mar 22 3:00 pm	Allison Henrich Seattle Univ.	Title: Knot Games Abstract: We examine various games that can be played on knot projections with players taking turns changing double points into crossings.
Saturday Mar 26	N+6th Southern California Topology Colloquium
Tuesday Mar 29 3:00 pm	Michael Williams UC Riverside	Title: On nonhyperbolic handle number one links Abstract: The handle number of a link in the three sphere is the least number of disjoint proper arcs needed to be attached to the link so that resulting (possibly disconnected) graph is unknotted. In particular, the exteriors of handle number one links admit genus 2 Heegaard splittings. In this talk, some results on nonhyperbolic handle number links will be presented. This is joint work with Abby Thompson (UC Davis).
Tuesday Apr 5 3:00 pm	Bus Jaco Oklahoma State University	Title: Complexity of 3-manifolds Abstract: (work with H. Rubinstein and S. Tillmann) The complexity of a 3-manifold is the minimal number of tetrahedra taken over all (pseudo-simplicial) triangulations of the 3-manifold. Prior to this work the only results on the complexity of 3-manifolds were for those manifolds appearing in various computer generated census (finitely many). We determine the complexity of several infinite families of 3-manifolds. The proof employs results and methods from 0-efficient triangulations, layered triangulations, normal surfaces and barrier surface theory.
Tuesday Apr 5 4:00 pm	Lorenzo Sadun UT Austin	Title: A relative cohomology theory for dynamical systems Abstract: Relative homology is based on inclusions of spaces, but factor maps between dynamical systems are typically surjective, not injective. I will present a variant of relative cohomology, called "quotient cohomology", that is adapted to this case, and show how it can be used to better understand dynamical systems with R^d actions, such as tiling spaces. This is joint work with Marcy Barge.
Tuesday Apr 12 3:00 pm	Ellie Grano UCSB	Title: Investigating the Closed Diagrams of Planar Algebras Abstract: Topologists are often interested in algorithms for simplifying closed diagrams, e.g. the HOMFLY polynomial and the Kauffman bracket. We will discuss the D_{2n} planar algebras and how their closed diagrams can be evaluated using the Stephen Bigelow's jellyfish algorithm. Then we will discuss a version Bigelow's new planar algebra, the Disambiguated Temperley-Lieb, and give a full description of its closed diagrams.
Tuesday Apr 19 3:00 pm
Tuesday Apr 26 3:00 pm	Sam Nelson Claremont McKenna College	Title: Bikei and unoriented link invariants Abstract: Bikei or involutory biquandles are algebraic structures we can use to label the semiarcs of an unoriented link diagram to obtain link invariants. As an application, we use a non-involutory biquandle to detect the non-invertibility of a certain knot, answering in the affirmative a question of Xiao-Song Lin.
Tuesday May 3 3:00 pm	Rena Levitt Claremont McKenna College	Title: Semi-Automatic Groups Abstract: A motivating principle of geometric group theory is the strong connection between the intrinsic geometry of a topological space and the computation properties of its fundamental group. A prototypical example of such a connection is the fact that a closed, compact n-dimensional Riemannian manifold with strictly negative sectional curvatures has a contractible universal cover, and a fundamental group whose word problem can be solved in linear time. These properties are not independent of each other; the linear solution of the word problem is a consequence of the course geometry of the universal cover. Both the geometric and computational properties of these spaces have been generalized. The geometric properties lead to the study of CAT(0) spaces, and the computational properties inspired the theory of automatic groups. Epstein and Thurston proved that the fundamental group of of a 3-manifold M is automatic if and only if none of the factors of the prime decomposition of M is closed and modeled on nilgeometry or solvgeometry, and Bridson and Gilman later defined a family of groups that extended automaticity and included the fundamental groups of all compact 3-manifolds. In this talk I will give a brief overview of the development of the theory of automatic groups, and its extension a la Bridson and Gilman to semiautomatic groups, with the goal of proving the fundamental groups of cell complexes satisfying combinatorial non positive curvature conditions are semiautomatic.