Seminars and Colloquia by Series

Cube knots and a homology theory from cube diagrams

Series
Geometry Topology Seminar
Time
Monday, April 20, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Scott Baldridge – LSU
In this talk we will introduce the notion of a cube diagram---a surprisingly subtle, extremely powerful new way to represent a knot or link. One of the motivations for creating cube diagrams was to develop a 3-dimensional "Reidemeister's theorem''. Recall that many knot invariants can be easily be proven by showing that they are invariant under the three Reidemeister moves. On the other hand, simple, easy-to-check 3-dimensional moves (like triangle moves) are generally ineffective for defining and proving knot invariants: such moves are simply too flexible and/or uncontrollable to check whether a quantity is a knot invariant or not. Cube diagrams are our attempt to "split the difference" between the flexibility of ambient isotopy of R^3 and specific, controllable moves in a knot projection. The main goal in defining cube diagrams was to develop a data structure that describes an embedding of a knot in R^3 such that (1) every link is represented by a cube diagram, (2) the data structure is rigid enough to easily define invariants, yet (3) a limited number of 5 moves are all that are necessary to transform one cube diagram of a link into any other cube diagram of the same link. As an example of the usefulness of cube diagrams we present a homology theory constructed from cube diagrams and show that it is equivalent to knot Floer homology, one of the most powerful known knot invariants.

The Sutured Embedded Contact Homology of S^1\times D^2

Series
Geometry Topology Seminar
Time
Monday, April 13, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Roman Golovko – USC
We will define the sutured version of embedded contact homology for sutured contact 3-manifolds. After that, we will show that the sutured version of embedded contact homology of S^1\times D^2, equipped with 2n sutures of integral or infinite slope on the boundary, coincides with the sutured Floer homology.

Hypergeometric functions - the GKZ-perspective

Series
Geometry Topology Seminar
Time
Monday, April 13, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Uli Walther – Purdue University
Starting with some classical hypergeometric functions, we explain how to derive their classical univariate differential equations. A severe change of coordinates transforms this ODE into a system of PDE's that has nice geometric aspects. This type of system, called A-hypergeometric, was introduced by Gelfand, Graev, Kapranov and Zelevinsky in about 1985. We explain some basic questions regarding these systems. These are addressed through homology, combinatorics, and toric geometry.

Density of isoperimetric spectra

Series
Geometry Topology Seminar
Time
Monday, April 6, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Emory, W306 MSC (Math and Science Center)
Speaker
Noel Brady – University of Oklahoma

Please Note: Joint meeting at Emory

A k--dimensional Dehn function of a group gives bounds on the volumes of (k+1)-balls which fill k--spheres in a geometric model for the group. For example, the 1-dimensional Dehn function of the group Z^2 is quadratic. This corresponds to the fact that loops in the euclidean plane R^2 (which is a geometric model for Z^2) have quadratic area disk fillings. In this talk we will consider the countable sets IP^{(k)} of numbers a for which x^a is a k-dimensional Dehn function of some group. The situation k \geq 2 is very different from the case k=1.

Contact geometry, open books and monodromy

Series
Geometry Topology Seminar
Time
Monday, April 6, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Emory, W306 MSC (Math and Science Center)
Speaker
John Etnyre – School of Mathematics, Georgia Tech

Please Note: Joint meeting at Emory

Recall that an open book decomposition of a 3-manifold M is a link L in M whose complement fibers over the circle with fiber a Seifert surface for L. Giroux's correspondence relates open book decompositions of a manifold M to contact structures on M. This correspondence has been fundamental to our understanding of contact geometry. An intriguing question raised by this correspondence is how geometric properties of a contact structure are reflected in the monodromy map describing the open book decomposition. In this talk I will show that there are several interesting monoids in the mapping class group that are related to various properties of a contact structure (like being Stein fillable, weakly fillable, . . .). I will also show that there are open book decompositions of Stein fillable contact structures whose monodromy cannot be factored as a product of positive Dehn twists. This is joint work with Jeremy Van Horn-Morris and Ken Baker.

Hyperbolic volume and the complexity of Heegaard splittings of 3-manifolds

Series
Geometry Topology Seminar
Time
Monday, March 30, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Yo'av Rieck – University of Arkansas
Let M be a hyperbolic 3-manifold, that is, a 3-manifold admitting a complete, finite volume Riemannian metric of constant section curvature -1. Let S be a Heegaard surface in M, that is, M cut open along S consists of two handlebodies. Our goal is to prove that is the volume of M (denoted Vol(M)) if small than S is simple. To that end we define two complexities for Heegaard surfaces. The first is the genus of the surface (denoted g(S)) and the second is the distance of the surface, as defined by Hempel (denoted d(S)). We prove that there exists a constant K>0 so that for a generic manifold M, if g(S) \geq 76KVol(M) + 26, then d(S) \leq 2. Thus we see that for a generic manifold of small volume, either the genus of S is small or its distance is at most two. The term generic will be explained in the talk.

Invariants of Legendrian Knots from Open Book Decompositions

Series
Geometry Topology Seminar
Time
Monday, March 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sinem Onaran – School of Mathematics, Georgia Tech
Given any contact 3-manifold, Etnyre and Ozbagci defined new invariants of the contact structures in terms of open book decompositions supporting the contact structure. One of the invariants is the support genus of the contact structure which is defined as the minimal genus of a page of an open book that supports the contact structure. In a similar fashion, we define the support genus sg(L) of a Legendrian knot L in a contact manifold M as the minimal genus of a page of an open book of M supporting the contact structure such that L sits on a page and the framing given by the contact structure and by the page agree. In this talk, we will discuss the support genus of Legendrian knots in contact 3-manifolds. We will show any null-homologous loose knot has support genus zero. To prove this, we observe an interesting topological property of knots and links on the way. We observe any topological knot or link in a 3-manifold sits on a planar page (genus zero page) of an open book decomposition.

Annulus open book decompositions and the self linking number

Series
Geometry Topology Seminar
Time
Monday, March 2, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Keiko Kawamuro – IAS
We introduce a construction of an immersed surface for a null-homologous braid in an annulus open book decomposition. This is hinted by the so called Bennequin surface for a braid in R^3. By resolving the singularities of the immersed surface, we obtain an embedded Seifert surface for the braid. Then we compute a self-linking number formula using this embedded surface and observe that the Bennequin inequality is satisfied if and only the contact structure is tight. We also prove that our self-linking formula is invariant (changes by 2) under a positive (negative) braid stabilization which preserves (changes) the transverse knot class.

Cubic graphs and number fields

Series
Geometry Topology Seminar
Time
Monday, February 23, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Stavros Garoufalidis – School of Mathematics, Georgia Tech –
A cubic graph is a graph with all vertices of valency 3. We will show how to assign two numerical invariants to a cubic graph: its spectral radius, and a number field. These invariants appear in asymptotics of classical spin networks, and are notoriously hard to compute. They are known for the Theta graph, the Tetrahedron, but already unknown for the Cube and the K_{3,3} graph. This is joint work with Roland van der Veen: arXiv:0902.3113.

Transverse knots and contact structures

Series
Geometry Topology Seminar
Time
Monday, February 16, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John Etnyre – School of Mathematics, Georgia Tech
I will discuss a couple of applications of transverse knot theory to the classification of contact structures and braid theory. In particular I will make the statement "transverse knots classify contact structures" precise and then prove it (if we have time). I will also discuss how progress on two of Orevkov's questions concerning quasi-positive knots that have implications for Hilbert's 16th problem.

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