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Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

I will discuss a "duality" among the linearized contact homology groups of a Legendrian submanifold in certain contact manifolds (in particular in Euclidean (2n+1)-space). This duality is expressed in a long exact sequence relating the linearized contact homology, linearized contact cohomology and the ordinary homology of the Legendrian submanifold. One can use this structure to ease difficult computations of linearized contact homology in high dimensions and further illuminate the proof of cases of the Arnold Conjecture for the double points of an exact Lagrangian in complex n- space.

Series: Geometry Topology Seminar

A broken fibration is a map from a smooth 4-manifold to S^2 with isolated Lefschetz singularities and isolated fold singularities along circles. These structures provide a new framework for studying the topology of 4-manifolds and a new way of studying Floer theoretical invariants of low dimensional manifolds. In this talk, we will first talk about topological constructions of broken Lefschetz fibrations. Then, we will describe Perutz's 4-manifold invariants associated with broken fibrations and a TQFT-like structure corresponding to these invariants. The main goal of this talk is to sketch a program for relating these invariants to Ozsváth-Szabó invariants.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

Cannon: "A f.g. negatively curved group with boundary homeomorphic to the round two sphere is Kleinian". We shall outline a combinatorial (complex analysis motivated) approach to this interesting conjecture (following Cannon, Cannon-Floyd-Parry). If time allows we will hint on another approach (Bonk-Kleiner) (as well as ours). The talk should be accessible to graduate students with solid background in: complex analysis, group theory and basic topology.