Claremont Topology Seminar

Fall 2006 Schedule

Date	Speaker	Title and Abstract
Tuesday Sept 5	Organizational Meeting
Tuesday Sept 12	Jim Hoste Pitzer College	Title: Lissajous Knots and Fourier Knots Abstract: A Lissajous knot is one that can be parameterized by a single cosine funtion for each of the three coordinates. Equivalently, it is a knot that is traced out by a particle whose motion in each of the coordinate directions is simple harmonic. A Fourier knot is more general: we allow the sum of any number of cosine functons to parameterize each coordinate function. In this talk I will review essentially everything that is known about Lissajous and Fourier knots to date.
Tuesday Sept 19	Sam Nelson Whittier College	Title: Biquandles and Yang-Baxter cocycle invariants Abstract: Much like quandles, biquandles are algebras which satisfy axioms derived from the Reidemeister moves; such algebras are naturally useful for defining invariants of knots and links. Yang-Baxter cohomology is for biquandles what group cohomology is for groups; in particular, for every Yang-Baxter 2-cocycle there is a corresponding link invariant. In this talk I will talk about some recent work I've done with students on biquandles and Yang-Baxter cohomology.
Tuesday Sept 26	Dave Bachman Pitzer College	Title: Heegaard splittings of amalgamated 3-manifolds, a survey. Abstract: An amalgamated 3-manifold is one which is constructed by gluing component 3-manifold along their boundaries. By examining the genus of the gluing surface and the complexity of the gluing map one can say a lot about the Heegard splittings of the resulting 3-manifold. Recently these results were surveyed jointly with R. Derby-Talbot, and a beautiful picture emerged. This talk will be an overview of these results, related open problems, and their relationship to conjectures in knot theory.
Tuesday Oct 3	Terry Fuller CSU Northridge	Title: Realizing 4-manifolds as achiral Lefschetz fibrations Abstract: Work of Donaldson and Gompf in the late 90s characterized symplectic 4-dimensional manifolds as those admitting a particular sort of fibration by surfaces known as a Lefschetz fibration. Following this, it was natural to ask to what extent arbitrary smooth 4-manifolds admit similar structures. In this talk, I will discuss joint work with John Etnyre which demonstrates that any smooth 4-manifold contains an embedded curve such that the manifold obtained by surgery along that curve admits an achiral Lefschetz fibration.
Tuesday Oct 10	Scott Taylor UCSB/Westmont College	Title: Unique Heegaard Splittings of Deleted-Boundary 3-Manifolds. Abstract: Non-compact 3-manifolds have Heegaard splittings, just as compact 3-manifolds do. I will define and give examples of Heegaard splittings of some favorite non-compact 3-manifolds: R^3, (closed surface) x R, and the Whitehead manifold. While very few compact 3-manifolds are known to have "unique" Heegaard splittings, every open 3-manifold which is obtained by removing boundary components from a compact 3-manifold has a "unique" Heegaard splitting. I will describe the proof of this theorem, including notions of uniqueness, and its connection to a celebrated theorem of Casson and Gordon.
Tuesday Oct 17	No Meeting, Fall Break
Tuesday Oct 24	Vin de Silva Pomona College	Title: Persistent cohomology: its not just a ring
Tuesday Oct 31
Tuesday Nov 7	Swatee Naik Univ of Nevada, Reno	Title: Torsion in the concordance group Abstract: Concordance classes of knots form an abelian group under connected sum. Very little is known about torsion in this group other than the fact that there are order two classes represented by negative amphicheiral knots. It is known that the group maps onto the algebraic concordance group which is a direct sum of countably infinite copies of the infinite cyclic group and of finite cyclic groups of orders 2 and 4. This map has an infinitely generated kernel which contains a summand isomorphic to the direct sum of three copies of the infinite cyclic group. Previously Casson-Gordon invariants have been used to show that certain knots which map to order four algebraic concordance classes are of infinite order. Twisted Alexander polynomials, which are related to discriminants of Casson-Gordon invariants were used to eliminate concordance order two for low crossing knots which are order two in the algebraic concordance class. I will discuss joint work with Stanislav Jabuka in which we have used the Heegaard-Floer homology correction term to improve on lower bounds on the order of small crossings knots in the smooth knot concordance group.
Tuesday Nov 14	Daniel Stevenson UC Riverside	Title: Schreier theory for Lie 2-algebras Abstract: The notion of a connection on a principal bundle is ubiquitous in geometry and physics. Recently, motivated by string theory, there has been an interest in generalisations of principal bundles, in which groups are replaced by groupoids. Developing the geometry of these generalised principal bundles, known as `gerbes', is an important problem. In the case of ordinary principal bundles, Atiyah gave an elegant formulation of the notion of a connection in terms of a splitting of an exact sequence of Lie algebras. In this talk we will present a classification theorem for extensions of `Lie 2-algebras' - certain categories whose sets of objects and morphisms have the structure of Lie algebras - and show how the geometry of gerbes can be understood in this framework.
Tuesday Nov 21	Bob Pelayo Cal Tech	Title: Complexes of Seifert Surfaces in S^3 Abstract: This talk focuses on Kakimizu's complexes of incompressible and minimal genus Seifert surfaces for a given link L. These simplicial complexes are defined using a disjointness property and are conjectured to be contractible. While global properties of these complexes are relatively scarce, many local properties have been discovered by Gabai, Sakuma, and Kakimizu. We will discuss these properties, as well as compute several explicit examples for certain classes of links. Further, we will utilize geometric concepts like minimal surfaces (those with mean curvature zero) to obtain diameter bounds on these complexes for hyperbolic links.
Tuesday Nov 28	Ramin Naimi Occidental College	Title: Intrinsically highly-linked graphs Abstract: During 1981-83 it was shown by Conway and Gordon, and independently by Sachs, that every embedding of K_6 (the complete graph on six vertices) in R^3 contains a 2-component link with nonzero linking number. Since then various authors have extended this to construct graphs that are "intrinsically n-linked": every embedding of the graph in R^3 contains a non-split n-component link. We (Flapan, Mellor, Naimi) extend these results by constructing for each n a graph G such that every embedding of G in R^3 contains an n-component link all of whose components have nonzero pairwise linking numbers; in fact, given any m, G can be constructed so that all the pairwise linking numbers are greater than m.
Tuesday Dec 5	Tevian Dray Oregon State U.	Title: The geometry of special relativity Abstract: The geometry of hyperbolas is the key to understanding special relativity. The Lorentz transformations of special relativity are just hyperbolic rotations, yet this point of view has all but disappeared from the standard physics textbooks. This approach replaces the ubiquitous $\gamma$ symbol of most standard treatments by the appropriate hyperbolic trigonometric functions. In most cases, this simplifies the resulting formulas, while emphasizing their geometric content.

Spring 2007 Schedule

Date	Speaker	Title and Abstract
Tuesday Jan 23	Ilesanmi Adeboye USC	Title: Volumes of hyperbolic orbifolds Abstract: Various authors have previously established explicit lower bounds on the volume of hyperbolic orbifolds of dimension three and of hyperbolic manifolds of any dimension. A hyperbolic $n$-orbifold is a quotient of hyperbolic $n$-space by a discrete group $\Gamma$ of isometries of $\mathbb{H}^{n}$. In pursuit of volume bounds for hyperbolic orbifolds, the presence of torsion elements in the group $\Gamma$ raise challenges that cannot be handled purely by the techniques developed in the manifold case. We will establish lower bounds on the volume of hyperbolic orbifolds that depend on dimension and the maximum order of the elliptic elements of $\Gamma$. We also discuss progress in refining this result as well as applications.
Tuesday Jan 30 2:40 pm	Helen Wong Yale University	TItle: Quantum invariants and Heegaard genus Abstract: The SU(2)- and SO(3)- quantum theories provide both a family of three-manifold invariants and a family of projective unitary representations of a surface's mapping class group. These objects can be viewed as a generalization of the Jones polynomial for knots to 3-manifolds, and much effort has been expended unearthing their topological significance and applications. We will introduce the relevant constructions and describe some applications to the study of Heegaard genus.
Tuesday Feb 6 3:00 pm	Sam Nelson Whittier College	Title: Quandles and linking number Abstract: The knot quandle, introduced by Joyce in 1980, is a complete invariant of knots in S^3 up to orientation-reversing homeomorphism of topological pairs (K,S^3). Thus, in principle, it should be possible to derive many known knot invariants from the fundamental quandle. We will show how the quandle of a 2-component link determines the linking number up to sign. This is joint work with Natasha Harrell of UCR.
Tuesday Feb 13 3:00 pm	"No Speaker Seminar"	Meet at Some Crust Bakery in the Claremont Village at 3 pm for coffee and informal math discussion.
Tuesday Feb 20 3:00 pm	Akira Yasuhara Tokyo Gakugei University	Title: Claspers and link-homotopy classification of string links Abstract: It is known that the link-homotopy classification of string links is given by Milnor's $\mu$-invariants. In this talk, we will give an alternate proof of this classification via clasper theory.
Tuesday Feb 27 3:00 pm	Jean-Baptiste Meilhan UC Riverside	Title: Borromean surgery and the Casson invariant Abstract: Every oriented integral homology 3-sphere is obtained from S^3 by some Borromean surgery moves. We give an explicit formula for the Casson invariant of an integral homology sphere given by such a surgery presentation. The formula involves simple classical invariants, namely the framing, linking number and Milnor's triple linking number.
Tuesday Mar 6 3:00 pm
Tuesday Mar 13 3:00 pm	No Meeting, Spring Break
Tuesday Mar 20 3:00 pm	Jon McCammond UCSB	Title: Pulling apart orthogonal groups to find continuous braids Abstract: Suppose you were asked to complete the following analogy - symmetric groups are to braids groups as the orthogonal groups O(n) are to (blank). In this talk I'll present one possible answer to this question. Other answers are possible since it depends on how one envisions the braid groups being constructed out of symmetric groups, but most of the standard constructions do not extend to continuous groups such as O(n). The groups I'll discuss are slightly odd in that they have a continuum of generators, a continuum of relators but nevertheless have a decidable word problem (in a suitable sense), a finite dimensional Eilenberg-Maclane space (that is not locally finite) and many other nice properties. In particular, if we view Sym_n as the subgroup of O(n) that permutes the coordinates, then Braid_n naturally embeds as a subgroup of the pulled apart version of O(n). Finally, since this is work in progress, the emphasis will be on explaining these constructions for particular examples with lots of pictures. The talk should be very accessible to graduate students.
Tuesday Mar 27 3:00 pm	Rollie Trapp CSUSB	Title: Knot Energies Abstract: This talk will be a brief introduction to knot energies. It will start with an overview of several examples of energies and their properties--from Milnor's curvature of a knot to O'Hara's Mobius invariant energy and beyond. A new polygonal knot energy, dubbed the crossing probability energy, will be introduced and described in some detail. Finally, the crossing probability energy will be generalized to suggest a rubric for defining other polygonal knot energies. Open questions will be discussed, with the talk being accessible to advanced undergraduates. The crossing probability energy is joint work with Blake Mitchell conducted during the summer 2006 REU at CSUSB.
Tuesday Apr 3 3:00 pm	Alex Bene USC	Title: Canonical lifts of the Johnson homomorphisms to the Torelli groupoid. Abstract: The Torelli group of a surface S with one boundary component is defined as the subgroup of the mapping class group of S which acts trivially on the first homology of S. This is in fact just the first of a sequence of nested "higher Torelli" subgroups which serves as an approximation to the mapping class group itself. The study of the Torelli groups often involves analysis of the Johnson homomorphisms which are certain abelian quotients of the Torelli groups. Morita has shown that the first Johnson homomorphism lifts to a crossed homomorphism of the whole mapping class group, while recently Morita and Penner have shown by use of homology marked fatgraphs that it in fact lifts canonically to the Torelli groupoid. In this talk, I will report on recent work (joint with R. Penner and N. Kawazumi) in which canonical lifts of the higher Johnson homomorphisms to the Torelli groupoid have been constructed. The construction relies on Kawazumi's interpretation of the Johnson homomorphisms in terms of Magnus expansions adapted to the Morita-Penner perspective using fatgraphs.
Tuesday Apr 10 3:00 pm
Tuesday Apr 17 3:00 pm	Morwen Thistlethwaite University of Tennessee, Knoxville	Title: Representations of manifold and orbifold groups as groups of isometries of complex hyperbolic space Abstract: A real hyperbolic structure on an n-dimensional orientable manifold M is determined by the associated holonomy representation of its fundamental group into the group of direct isometries of real hyperbolic n-space. Since this group of isometries, SO^+(n,1), is simultaneously a subgroup of SL(n+1,R) and of SU(n,1), it is of interest to try to deform the holonomy representation into each of these larger Lie groups. Deformations into SL(n+1,R), when they exist, give rise to real projective structures on M, whereas deformations into SU(n,1) provide representations as groups of isometries of complex hyperbolic n-space, somewhat reminiscent of quasi-fuchsian deformations of a fuchsian group. In this talk we explain how to compute the corresponding representation varieties exactly, with particular emphasis on complex hyperbolic deformations of the closed 3-manifold Vol3, and of the 2-sphere with three cone points.
Tuesday Apr 24 3:00 pm	Tamas Kalman USC	Title: Legendrian knots bounding Lagrangian surfaces Abstract: We will consider Legendrian knots in standard contact three-space, as well as Lagrangians in its symplectization, bounding the Legendrians. I will show some constructions. These surfaces are closely related to augmentations and rulings of their boundary (viewed as a Legendrian object), as well as to its slice genus (which is a notion of the smooth category). This work (in progress) is joint with Tobias Ekholm and Ko Honda.
Tuesday May 1 3:00 pm	Danielle O'Donnol UCLA	Title: Intrinsically n-linked Complete Graphs Abstract: A graph G is intrinsically n-linked if every embedding of G into R^3 contains a non-split n-component link. I will discuss the problem of finding a function f(n) defined on the natural numbers where m=f(n) is the smallest natural number such that K_m is intrinsically n-linked.