Claremont Topology Seminar

Fall 2008 Schedule

Date	Speaker	Title and Abstract
Tuesday Sept 9 3:00 pm	Organizational Meeting
Tuesday Sept 16 3:00 pm	Sam Nelson Claremont McKenna College	Title: Semiquandles and flat knots Abstract: We define a new algebraic structure we call a semiquandle and use it to define invariants of Kauffman's flat singular knots. This type of object has potential applications to link homotopy and knotted graphs. This is joint work with Allison Henrich.
Tuesday Sept 23 3:00 pm	Jiajun Wang Cal Tech	Title: On combinatorial Floer homology Abstract: Heegaard Floer homology is introduced by Peter Ozsvath and Zoltan Szabo, which is defined from the Heegaard splitting of a three-manifold. On a "nice" Heegaard diagram, Heegaard Floer homology can be computed combinatorially. In this talk, we discuss how to make it a combinatorial theory.
Tuesday Sept 30 3:00 pm	Dongping Zhuang Cal Tech	Title: Large scale geometry of commutator subgroups Abstract: Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary word-hyperbolic group, C is one-ended. Hence (in this case), the asymptotic dimension of C is at least 2.
Tuesday Oct 7 3:00 pm	Emily Stark Pomona College	Title: Intrinsically Linked and Triple-Linked graphs in RP^3 Abstract: It has been shown that the complete set of minor-minimally intrinsically linked graphs in arbitrary 3-manifolds are characterized by the seven Petersen-family graphs. We explore a weaker definition of unlinks in projective space, which leads to two distinct two-component unlinks. The number of minor-minimal intrinsically linked graphs increases and we begin to characterize such graphs up to connectivity 3. In particular, K6 , the complete graph on 6 vertices is unlinked in RP^3 , and 7 is the smallest n for which Kn is intrinsically linked in RP^3 . In contrast, Flapan, Naimi, and Pommersheim showed that every spatial embedding of K10 contains a non-split three-component link, and 10 is also the smallest n for which Kn is intrinsically triple-linked in RP^3.
Tuesday Oct 14 3:00 pm	Sandra Ritz USC	Title: A Categorification of the Burau Representation via Contact Geometry Abstract: We will begin with an overview of the Burau representation of the braid group. This will be followed by an introduction to a contact category on 3-manifolds, with a brief discussion of its relation to the braid group.
Tuesday Oct 21	No Seminar Fall Break
Tuesday Oct 28 3:00 pm	Roman Golovko USC	Title: The Sutured Embedded Contact Homology of S^1 x D^2 Abstract: We will define the sutured embedded contact homology for contact oriented 3-manifolds with convex boundary. After that, we will show that the sutured embedded contact homology of S^1 x D^2, equipped with 2n sutures of integral or infinite slope on the boundary, coincides with the sutured Floer homology.
Tuesday Nov 4 3:00 pm	Davie Bachman Pitzer College CANCELLED	Title: Topological index theory for surfaces in 3-manifolds Abstract: We define an isotopy invariant index of a surface in a 3-manifold, and show that it mimics the index of a (unstable) minimal surface. This allows us to find purely topological analogues to familiar techniques from geometry, such as barrier arguments. Deep results follow concerning Heegaard splittings, bridge positions of knots, and normal surfaces. In addition, this new viewpoint opens up a host of exciting new questions for the field of 3-manifold topology.
Tuesday Nov 11 3:00 pm	Alex Hoffnung UC Riverside	Title: Multisymplectic Geometry Abstract: A Lie 2-algebra is a "categorified" version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an "n-plectic manifold": a finite-dimensional manifold equipped with a closed nondegenerate (n+1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables.
Tuesday Nov 18 3:00 pm	Cornelia Van Cott University of San Francisco	Title: Obstructions to slicing Bing doubles Abstract: We will begin by introducing slice knots and links and reviewing several important related results. Next we will introduce a particular family of links called Bing doubles. Much recent attention has focused on characterizing when a Bing double is slice. We will discuss recent progress toward achieving this goal using a new tool called covering link calculus.
Tuesday Nov 25 3:00 pm	Juan Ortiz-Navarro	Title: A volume form on the Khovanov Invariant Abstract: The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This denes a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this talk, I will start with some basic facts about knot theory and algebraic topology which will lead us to Khovanov Homology and the construction and invariance of this volume form. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov Chain complex when homology is computed over Z is recovered.
Tuesday Dec 2 3:00 pm
Tuesday Dec 9 3:00 pm	Liam Watson Universite du Quebec a Montreal, et Centre Interuniversitaire de Recherches en Geometrie et Topologie	Title: Involutions on 3-manifolds and Khovanov homology Abstract: Given a manifold with torus boundary, together with an appropriate involution, it is possible give obstructions to certain exceptional surgeries using Khovanov homology. In particular, obstructions to lens space surgeries, as well as obstructions to finite fillings may be obtained. This talk will explain how these obstructions arise, and attempt to compare them with strong obstructions arising in Heegaard-Floer homology.

Spring 2009 Schedule

Date	Speaker	Title and Abstract
Tuesday January 27 3:00 pm	Sangbum Cho UC Riverside	Title: Goeritz groups for lens spaces Abstract: Given a lens space (or the 3-sphere) with its Heegaard splitting of genus two, the Goeritz group is defined to be the group of isotopy classes of orientation preserving homeomorphisms of the space that preserve the splitting. In this talk, we consider each of the Goeritz groups for lens spaces L = L(p, 1) and the 3-sphere. We construct a tree on which the group acts, and give an explicit presentation of each group.
Tuesday February 3 3:00 pm	Julie Bergner UC Riverside	Title: Model categories and homotopy theories Abstract: In classical homotopy theory, two topological spaces are considered to be the same if they have a weak homotopy equivalence, or map inducing isomorphisms on homotopy groups, between them. Thus, we have a notion of equivalence which is weaker than isomorphism. A similar phenomenon occurs when we consider quasi-isomorphisms between chain complexes. The structures possessed by the category of topological spaces and by the category of chain complexes can be axiomatized via a model category structure, thus enabling us to consider more general "homotopy theories". In this talk, we'll consider the advantages and difficulties with model categories as well as more modern approaches to the study of homotopy theories.
Tuesday February 10 3:00 pm	Dave Bachman Pitzer College	Title: Topological index theory for surfaces in 3-manifolds Abstract: We define an isotopy invariant index of a surface in a 3-manifold, and show that it mimics the index of a (unstable) minimal surface. This allows us to find purely topological analogues to familiar techniques from geometry, such as barrier arguments. Deep results follow concerning Heegaard splittings, bridge positions of knots, and normal surfaces. In addition, this new viewpoint opens up a host of exciting new questions for the field of 3-manifold topology.
Tuesday February 17 3:00 pm	Fangyun Yang UC Riverside	Title: An Index Theorem for Singular Dirac Operators Abstract: We study Dirac Operators with singularities on a codimension $2$ submanifold. Suppose $M$ is a manifold of dimension $2n$, $B$ is a submanifold of dimension $2n - 2$, and $M€backslash B$ has a spin structure, which can not be extended to $M$. The metric we consider on $M€backslash B$ is the restriction of a smooth metric on $M$. Then we show that if we put some APS-type boundary conditions along the submanifold $B$, the associated Dirac operator is Fredholm. We also obtain an explicit Index formula for the Dirac operator. One immediate application of the Index formula is that it gives us an analytic proof of the generalized Rohklin congruence formula.
Tuesday February 24 3:00 pm
Tuesday March 3 3:00 pm	Julien Roger USC	Title: Quantum Teichmuller Spaces and Modular Functors Abstract: Let S be a surface with punctures. The quantum Teichmuller space T^q(S) is a deformation of the algebra of rational functions on the classical Teichmuller space T(S). Bonahon and Liu obtained a classification of the representations of T^q(S). Using this classification one can construct a vector bundle over the moduli space M(S). I will describe the first steps in trying to extend this construction to the Deligne-Mumford compactification of M(S). It involves looking at families of hyperbolic metrics where the length of a finite number of geodesics tends to 0, and the way this affects ideal triangulations on S. I will also explain how it relates to the notion of a modular functor.
Tuesday March 10 3:00 pm	Emille Davie UC Santa Barbara	Title: Detecting Veering Information of 3-Braids using the Burau Representation Abstract: We will give a brief introduction to the braid group, to the Burau representation of the braid group, and to right-veering diffeomorphisms of surfaces with boundary. We will also discuss how the Burau representation can be used to detect right-veering 3-braids.
Tuesday March 17	No Seminar Spring Break
Tuesday March 24 3:00 pm	Carmen Caprau CSU Fresno	Title: On the universial sl(2) foam cohomology Abstract: We introduce and describe the features of the euniversalf sl(2)-link cohomology via foams, also called singular cobordisms, modulo local relations. If time permitting, we will also discuss the tools that lead to efficient computations of the corresponding cohomology groups, even for big knots and links.
Tuesday March 31 3:00 pm	Matt Day Cal Tech	Title: Topological interpretation and extension of Morita's homomorphisms Abstract: In 1993, S. Morita defined a series of homomorphisms from subgroups of the mapping class group of a surface to abelian groups. This definition is entirely algebraic. We will discuss a topological interpretation of these homomorphisms due to A. Heap. Then we will use this topological interpretation to extend Morita's homomorphisms to crossed homomorphisms on the entire mapping class group.
Tuesday April 7 3:00 pm	Alex Coward UC Davis	Title: Unknotting, algorithms and genus one knots. (Joint work with Marc Lackenby) Abstract: There is no known algorithm for determining whether a knot has unknotting number one, practical or otherwise. Indeed, there are many explicit knots that are conjectured to have unknotting number two, but for which no proof of this fact is currently available. For many years the knot 8-10 was in this class, but a celebrated application of Heegaard Floer homology by Ozsvath and Szabo established that its unknotting number is in fact two. In this talk we will see how tools from sutured manifold theory can be used to characterize unknotting crossing changes. We shall use these tools to address the following question: 'If a knot has unknotting number one, are there finitely many unknotting crossing changes and, if so, can one find them?'. We completely answer this question for genus one knots. Apart from sutured manifold theory, the proof utilizes some classical topology and an analysis of the arc complex of the once punctured torus.
Tuesday April 14 3:00 pm	Tim Carrell Pomona College	Title:The Surface Biquandle Abstract: Knot invariants are an important technique of knot theory that associate comparatively easy to study algebraic objects with knots. One such invariant, the biquandle, is constructed as the minimal structure needed to label the semiarcs of a knot. In this talk, we construct an invariant of surface knots similar to the biquandle by labeling semisheets. We also examine the relationship between such a surface biquandle and the classical biquandle.
Tuesday April 21 3:00 pm
Tuesday April 28 3:00 pm	Dorothy Buck	Title: Knots and Links arising from Protein Action on DNA Abstract: The central axis of the famous DNA double helix is often topologically constrained or even circular. The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function is to change the DNA axis topology -- for example converting a torus link into an unknot. Additionally, there are several protein families that change the axis topology as a by-product of their interaction with DNA. This talk will describe two areas where 3-manifold topology methods have been useful in predicting which types of knots and links arise from the action of these proteins on DNA. The first is joint work with Erica Flapan, and the second is joint work with Ken Baker and Andrew Lobb.
Tuesday May 5 3:00 pm	Alice Stevens UC Davis	Title: K-stable equivalence for knots in Heegaard surfaces Abstract: Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. Two pairs (S, K) and (S', K') are equivalent in M if there is an ambient isotopy of M that maps (S, K) onto (S', K'). We define K-stabilization, which is essentially the addition of an unknotted tube to the surface S-K, and we prove that any two pairs (S, K) and (S', K') are K-stably equivalent in M if they have the same surface slope.