# Seminars & Events for Topology Seminar

##### Transverse invariants in Heegaard Floer homology

Using the language of Heegaard Floer knot homology recently two invariants were defined for Legendrian knots. One in the standard contact 3-sphere defined by Ozsvath, Szabo and Thurston in the combinatorial settings of knot Floer homology, one by Lisca, Ozsvath, Stipsicz and Szabo in knot Floer homology for a general contact 3-manifold. Both of them naturally generalizes to transverse knots. In this talk I will give a characterization of the transverse invariant, similar to the one given by Ozsvath and Szabo for the contact invariant. Namely for transverse braids both transverse invariants are given as the bottommost elements with respect to the filtration of knot Floer homology given by the axis. The above characterization allows us to prove that the two invariants are the same in the standard contact 3-sphere.

##### Volume optimization on triangulated 3-manifolds

We propose a finite dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurston's equation and Haken's normal surface equation. The action functional is the volume. This is a generalization of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary

##### A rank inequality for the knot Floer homology of branched double covers

Given a knot $K$ in the three sphere, we compare the knot Floer homology of $(S3, K)$ with the knot Floer homology of $(\Sigma(K), K)$, where $\Sigma(K)$ is the double branched cover of the three-sphere over $K$. By studying an involution on the symmetric product of a Heegaard surface for $(\Sigma(K), K)$ whose fixed set is a symmetric product of a Heegaard surface for $(S3, K)$, and applying recent work of Seidel and Smith, we produce an analog of the classical Smith inequality for cohomology for knot Floer homology. To wit, we show that the rank of the knot Floer homology of $(S3,K)$ is less than or equal to the rank of the knot Floer homology of $(\Sigma(K), K)$

##### Combinatorial Heegaard Floer homology

After quickly reviewing the construction beyond the topological/combinatorial version of (stable) Heegaard Floer homology, we show how to define the theory over the integers, and how to get the $spin^c$ refined theory. This is a joint work with Peter Ozsvath and Zoltan Szabo.

##### A Kunneth formula in monopole Floer homology

We establish a Kunneth formula in monopole Floer homology, which describes how the groups associated to 3-manifolds behave under the operation of connected sum. The proof is based on a rigidity principal for Floer theories satisfying the surgery exact triangle. This is joint work with Tom Mrowka and Peter Ozsvath.

##### Toward a minus version of bordered Heegaard Floer homology

Bordered Heegaard Floer homology, in its current state, can recover the hat version of Heegaard Floer homology for closed manifolds. It is related to the sutured Floer homology, and gluing sutured manifolds along surfaces with boundary. One idea for expanding the bordered framework to obtain the stronger plus/minus versions of HF, is to look at sutured manifolds and gluing along closed boundary components. I will discuss progress along this lines, as well as connections to other possible approaches.

##### Towards cornered Floer homology

As part of the bordered Floer homology package, Lipshitz, Ozsvath and D. Thurston have associated to a parametrized oriented surface a certain differential graded algebra. I will describe a decomposition theorem for this algebra, corresponding to cutting the surface along a circle. Moreover, I will discuss a decomposition theorem for bordered modules associated to nice diagrams, corresponding to cutting a 3-manifold with boundary along a surface transverse to the boundary. This is joint work with Christopher Douglas.

##### The Jones polynomial and surfaces far from fibers

This talk explores relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We also show that certain coefficients of the Jones and colored Jones polynomials measure the "guts" of the surface—a measurement of how far this surface is from being a fiber in the knot complement.

This is joint work with Effie Kalfagianni and Jessica Purcell.

##### The Jones polynomial and surfaces far from fibers

This talk explores relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We also show that certain coefficients of the Jones and colored Jones polynomials measure the "guts" of the surface -- a measurement of how far this surface is from being a fiber in the knot complement. This is joint work with Effie Kalfagianni and Jessica Purcell.

##### Mapping class groups of Heegaard splittings

The mapping class group of a Heegaard splitting for a given 3-manifold is the group of automorphisms of the 3-manifold that take the Heegaard surface onto itself, modulo isotopies that preserve the surface setwise. This can be viewed as a subgroup of the mapping class group of the surface. I will discuss a number of equivalent definitions of this group and describe some recent examples that demonstrate an interesting relationship between the structure of the mapping class group and the topology of the ambient 3-manifold.

##### Representation Theory and Homological Stability

Homological stability is the remarkable phenomenon where for certain sequences $X_n$ of groups or spaces -- for example $SL(n,Z)$, the braid group $B_n$, or the moduli space $M_n$ of genus $n$ curves -- it turns out that the homology groups $H_i(X_n)$ do not depend on n once n is large enough. But for many natural analogous sequences, from pure braid groups to congruence matrix groups to Torelli groups, homological stability fails horribly. In these cases the rank of $H_i(X_n)$ blows up to infinity, and in the latter two cases almost nothing was known about $H_i(X_n)$; indeed it's possible there is no nice "closed form" for the answers.

##### Counting Connections and the Ehrenpreis Conjecture

Let $S$ be a closed hyperbolic surface. We review the theory of counting connections on $S$---between points, between horocycles, and between geodesic segments---and we explain how this relates to the proof of the Ehrenpreis conjecture.

##### A Khovanov Homotopy Type

We will start by describing Khovanov's categorification of the Jones polynomial from a cube of resolutions of a link diagram. We will then introduce the notion of a framed flow category, as defined by Cohen, Jones and Segal. We will see how a cube of resolutions produces a framed flow category for the Khovanov chain complex, and how the framed flow category produces a space whose reduced cohomology is the Khovanov homology. We will show that the stable homotopy type of the space is a link invariant. Time permitting, we will show that the space is often non-trivial, i.e., not a wedge sum of Moore spaces. This work is joint with Robert Lipshitz.

##### Invariants of Links in (contact) Circle Bundles

In 1992 Turaev introduced the notions of ``shadows" and ``gleams" to define quantum invariants of links in circle bundles over surfaces. We generalize his shadow link construction to links in circle bundles over orbifolds and then focus on the case where the ambient manifold is equipped with a compatible contact structure. We interpret the gleam of a Legendrian knot in terms of a defining contact form and use it to compute the rational Thurston-Bennequin and rotation numbers. This is joint work with Josh Sabloff.

##### Strong L-spaces

A strong L-space is a 3-manifold defined by a certain simple combinatorial condition. The definition of this family of spaces is motivated by Heegaard Floer homology, and I will discuss what is known about them, with a view towards their classification. In the spirit of left-orderability, I will give the talk with my left hand.

##### Structure and examples of pseudo-Anosov flows in graph manifolds and Seifert fibered pieces

We describe an interaction of a pseudo-Anosov flow with possible Seifert fibered pieces in the torus decomposition of the underlying manifold: if the fiber is associated to a periodic orbit of the flow, we produce a standard form for the flow in the piece which is a neighborhood of finitely many weakly embedded Birkhoff annuli. A Birkhoff annulus is an annulus so that each boundary component is a closed orbit of the flow and the flow is transverse to the interior of the annulus. Using collections of Birkhoff annuli as a skeleton for some flows, we then produce a very large class of new examples of pseudo-Anosov flows in graph manifolds. This is joint work with Thierry Barbot.

##### Heegaard Floer homology solid tori

It has been conjectured that L-spaces are equivalent to 3-manifolds with non-left-orderable fundamental group. Supposing that this conjecture is true, some interesting (perhaps even surprising) behaviour is suggested both for L-spaces and for left-orderable groups. This talk will outline some of the supporting evidence for the conjecture, and then discuss some calculations in bordered Heegaard Floer homology for studying a particular family of graph manifolds that do not admit taut foliations. In particular, we'll give examples of what might be termed 'Heegaard Floer homology solid tori'.

##### Nonorientable four-ball genus can be arbitrarily large

A classical problem in low-dimensional topology is to find a surface of minimal genus bounding a given knot K in the 3-sphere. Of course, the minimal genus will depend on the class of surface allowed: must it lie in S3 as well, or can it bend into B4? must the embedding be smooth, or only locally flat? must the surface admit an orientation, or can it be nonorientable? Our ability to bound or compute these genera varies dramatically between classes. Orientable surfaces form homology classes, so are amenable to algebraic topology (cf Alexander polynomial), and they admit complex structures, so can be understood using gauge theory (cf Ozsvath-Szabo's \tau). In contrast, the largest lower bound on the genus of a nonorientable surface smoothly embedded in B4 bounding any knot K was, until recently, the integer 3.