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Berkeley Topology Seminar
The Topology Seminar meets on Wednesdays from 4.10 to 5.00 in 3 Evans unless otherwise indicated.
Spring 2013
January 31
Liam Watson (UCLA)
Heegaard Floer homology solid tori
This talk is an attempt to treat and provide context for the following question: what are the simplest bordered Heegaard Floer invariants? We will focus on the case of manifolds with torus boundary and, by analogy with Heegaard Floer homology lens spaces-known as L-spaces-for closed three-manifolds, introduce the notion of a Heegaard Floer homology solid torus. These manifolds satisfy a type of Alexander trick at the level of Heegaard Floer homology; the goal of the talk will be to make this precise. We will give some explicit examples and discuss some applications and related questions.
February 6
Ronen Mukamel (Stanford)
Billiards in polygons, the square root of 11 and the first
non-arithmetic Teichmuller curve of genus one
An immersion f : H/G ---> M_g from a finite volume hyperbolic orbifold
into the moduli space of Riemann surfaces is called a Teichmuller
curve if it is both algebraic and isometric. The first example of
such an immersion is the modular curve f : H/SL(2,Z) ---> M_1; other
arithmetic examples (i.e. with G commensurable to SL(2,Z)) emerge from
the study of square tiled surfaces. In genus two, there is a
non-arithmetic example associated to each real quadratic ring O
without zero divisors. For most O, little is known about the
associated group G_O or the Riemann surface H/G_O.
We will present the first explicit example of a Fuchsian group G
uniformizing a non-arithmetic Teichmuller curve H/G of genus one as
well as an explicit algebraic model for H/G.
February 13
Ilya Grigoriev (Stanford)
Relations in cohomology of the classifying space of manifold bundles
The characteristic classes of surface bundles with fiber of genus g
coincide with the elements of the cohomology of the classifying space
of surface bundles, denoted BDiff \Sigma_g. The n-th cohomology of
this space is known in the "stable range" (n <= (2g-2)/3) by theorems
of Madsen-Weiss, Harer, and others. In this range, the map from a free
algebra generated by the so-called "kappa-classes" to H^*(BDiff
\Sigma_g) is an isomorphism. Recently, Soren Galatius and Oscar
Randal-Williams have obtained similar results for the case where the
surface \Sigma_g is replaced with a certain high-dimensional manifold,
namely the connect sum of g copies of the product of spheres S^k x
S^k.
Outside the stable range, the kernel of the above-mentioned map for
surfaces has been studied by Morita, Faber, Looijenga, Pandharipande
and many others. In this talk, I will describe a vast family of
elements in the kernel that also works in the the high-dimensional
case (for odd k >= 1). The kernel is large enough to imply that the
image of this map ("the tautological subring") is finitely-generated
for all odd k, rationally, even though there are infinitely many kappa
classes. It also implies upper bounds on the stable range of
cohomology for fixed g and k.
February 27
Rob Kirby (Berkeley)
Smale's proof of the immersion theorem, eversions of the 2-sphere,
and historical remarks.
The title says it all.
March 6
Russell Avdek (USC)
The Liouville connect sum and its applications
We introduce a new surgery operation for contact manifolds called the Liouville connect sum. This operation -- which includes Weinstein handle attachment as a special case -- is designed to study the relationship between contact topology and symplectomorphism groups established by work of Giroux and Thurston-Winkelnkemper. The Liouville connect sum is used to generalize results of Baker-Etnyre-Van Horn-Morris and Baldwin on the existence of "monodromy multiplication cobordisms" as well as results of Seidel regarding squares of symplectic Dehn twists.
March 13
Joseph Hirsh (CUNY)
Derived noncommutative deformation theory
In this talk I will describe the classical yoga of deformation theory---that commutative moduli problems are locally described by dgLie algebras in characteristic zero---and provide some examples. Then, importing tools from operad theory---such as Koszul Duality---I will describe a generalization that classifies the local structure of noncommutative moduli problems, and explain how this result can be understood in terms of structured moduli spaces. I will conclude by describing some ongoing work to apply these ideas to number theory and manifold topology.
March 20
Philip Hackney (UCR)
Actions on Operads
We study up-to-homotopy group actions on operads. We
describe an extension of the Moerdijk-Weiss dendroidal category Omega
which allows us to provide a combinatorial model for such actions and
prove a strictification result.
April 3
Moritz Groth (Utrecht)
Grothendieck derivators
The theory of derivators (going back to Grothendieck, Heller,
and others) provides an axiomatic approach to homotopy theory. It adresses
the problem that the rather crude passage from model categories to
homotopy categories results in a serious loss of information. In the
stable context, the typical defects of triangulated categories
(non-functoriality of cone construction, lack of homotopy colimits) can be
seen as a reminiscent of this fact.
The basic idea behind a derivator is that one should form homotopy
categories of diagram categories and also keep track of the calculus of
homotopy Kan extensions. In the stable context this calculus allows one to
canonically construct triangulations -- emphasizing the idea that stable
derivators provide an enhancement of triangulated categories. Morover, for
stable, closed symmetric monoidal derivators one can establish an
additivity result for traces -- a result which is known to be false at the
level of triangulated categories.
The aim of this talk is to give a short introduction to the theory and to
(hopefully) advertise derivators as a convenient, 'weakly terminal'
approach to axiomatic homotopy theory.
April 10
Kevin Walker (Station Q)
(n+ŠĆ)-dimensional TQFTs and a higher dimensional Deligne conjecture
The classical Deligne conjecture (now a theorem with several published proofs) says that chains on the little disks operad act on Hochschild cohomology. I'll describe a higher dimensional generalization of this result. In fact, even in the dimension of the original Deligne conjecture the generalization has something new to say: Hochschild chains and Hochschild cochains are the first two members of an infinite family of chain complexes associated to an arbitrary associative algebra, and there is a colored, higher genus operad which acts on these chain complexes.
April 17
Saul Schleimer (Warwick)
Curves in the Masur domain
Suppose V is a handlebody with boundary S. The Masur
domain M(V) in PML(S) is the maximal open subset where the mapping
class group of V acts properly discontinuously. We show if \alpha is
a curve in S with distance at least three from the disk set D(V)
then \alpha lies in the Masur domain. This answers a question asked
independently by Feng Luo and Juan Souto.
April 18 (special day and time)
Misha Kapovich (UC Davis)
Universality in 3-dimensional topology
We are all used to the fact that (real) 3-dimensional manifolds are very "special", e.g., their fundamental groups are very limited, word problem for fundamental groups and the homeomorphism
problem are algorithmically solvable, and so on. In the talk I will discuss "universality" for 3-dimensional manifolds ("Murphy's Law" in Ravi Vakil's terminology), where things could be "as
bad as possible". I will explain why singularities of SL(2)-character schemes of closed 3-dimensional
manifolds could be "arbitrary" (subject to some natural constraints), while SU(2)-character varieties can contain (up to stabilization by products of 3-spheres) arbitrary smooth compact manifolds
as connected components. The main sources of universality (in my talk) is a (relatively) recent
theorem of Panov and Petrunin about 3-dimensional hyperbolic orbifolds as well as my old work
with John Millson on representation theory of Artin groups.
May 1
Sebastian Thyssen (Bochum)
Higher q-expansions and TAF-character maps
In number theory, the q-expansion map takes modular forms to their Fourier expansions. This has a topological realization, as a map between cohomology theories from topological modular forms to Tate K-theory. This is an example of a "transchromatic" character map, a higher height generalization of the Chern character from complex K-theory to rational cohomology.
In this talk, I will give some background and then discuss work in progress to obtain a generalization of the q-expansion map for automorphic forms on complex hyperbolic spaces. The main conjecture is that there is an associated topological realization as a map between different versions of the cohomology theory of topological automorphic forms.
Previous Semesters
Fall 2012
*Special Room*
August 22, 740 Evans
Saul Schleimer (Warwick)
The graph of handlebodies
(Joint work with Joseph Maher.) We introduce the graph of
handlebodies and prove that it is quasi-isometric to an electrification
of the curve complex. We show that this graph is Gromov-hyperbolic
and of infinite diameter. If time permits we will also discuss the
construction of pseudo-Anosov maps acting on the graph with
positive translation distance.
*Special Seminar*
August 23,
740 Evans,
2-3pm
Michael Freedman (Microsoft Station Q)
Closed 3-manifolds smoothly imbedded in R4.
Ian Agol and I have been working to sharpen up some tools for studing existance and uniqueness problems for smooth imbeddings of closed 3-manifolds in R4. Potential target applications include great problems so it is worth getting the tools sharp. The mapping class group of a Heegaard surface for the 3-manifold appears to plays an important role. One result is that every imbedding of a closed 3-manifold in R4 is isotopic to one with a unique local minimum and a unique local maximum.
August 29
Soren Galatius (Stanford)
Homological stability for moduli spaces of high dimensional manifolds
I will discuss recent joint work with Oscar Randal-Williams concerning the manifolds W_g^{2n} obtained as the connected sum of g copies of S^n x S^n. For n=1 this is a genus g surface, and there is a moduli space M_g parametrizing smooth surface bundles with genus g fibers. For higher n there is an analogous moduli space M_g^n parametrizing smooth fiber bundles with fibers W_g (although for n > 1 it is no longer finite dimensional). We prove that for n > 2 the cohomology groups H^k(M_g^n) are independent of g as long as g >> k, generalizing a result of John Harer and others for n=1.
September 5
Jacob Lurie (Harvard)
Ambidexterity
Let M be a compact oriented manifold. Poincare duality establishes an isomorphism between the homology and cohomology of M. For compact orbifolds, there is a similar isomorphism if we take homology and cohomology with rational coefficients, but not if we use integer coefficients. In this lecture, I'll discuss some other examples of cohomology theories which satisfy Poincare duality for orbifolds, and related duality phenomena.
September 12
Owen Gwilliam (Berkeley)
Higher enveloping algebras of Lie algebras, topological field
theories, and factorization algebras
If we view an E_n algebra (an algebra over the little
n-disks operad) as living on the n-disk, it is natural to ask for a
local-to-global object on an n-manifold that is locally an E_n
algebra. The notion of a factorization algebra makes this idea
precise. The talk will revolve around a general construction taking as
input a Lie algebra g and smooth manifold M, and it will motivate the
definition of factorization algebra via this construction. Along the
way, we will interpret this construction from various perspectives,
such as deformation theory, quantization, and manifold calculus. At
the end, we hope to sketch a relationship with vertex algebras and
affine Kac-Moody algebras.
September 19
Mohammed Abouzaid (Columbia/Simons Center)
On embeddings and immersions of Lagrangians spheres
A conjecture of Arnold would imply that every exact Lagrangian in a cotangent bundle is Hamiltonian isotopic to the zero section. We now know that every such Lagrangian is diffeomorphic to the zero section. I will explain how, combining the h-principle with the spectrum-valued invariants introduced by Kragh, one can hope to show that such Lagrangians are in fact isotopic to the zero section through Lagrangian immersions. I will discuss the case of spheres, where we can at least find obstructions to the classes of Lagrangian immersions which admit embeddings. This is joint work with T. Kragh.
September 26
Dylan Thurston (MSRI/Berkeley/Indiana)
4-manifolds and the categorification of surfaces
Smooth 4-manifolds are the last frontier of low-dimensional topology.
For instance, four is the only dimension in which the smooth Poincaré
conjecture is still open. Prominent in this problem is the search for
non-trivial invariants of smooth 4-manifolds. There is essentially
only one such invariant known, the Donaldson invariant and various
other invariants conjectured to be equivalent to it. In order to get
a handle on these invariants of 4-manifolds, it makes sense to get a
handle on their dimensional reductions, which ought to assign some
sort of vector space to 3-manifolds and some sort of category to
surfaces.
In this talk, we will see how this program is starting to shape up for
Heegaard Floer homology (one of the family of invariants
conjecturally-equivalent to the Donaldson invariants) and some
glimmerings of how one might try to get another such invariant from
categorifying surface cluster algebras.
Portions of this talk are joint work with P. Ozsváth and R. Lipshitz.
October 3
David Nadler (Berkeley)
Lagrangian cobordisms
The notion of semi-infinite homology of path spaces in symplectic manifolds is fundamental but mysterious. We will discuss a proposal (joint with Hiro Tanaka) to approach semi-infinite homology via Lagrangian cobordisms. Understanding the simplest examples leads to concrete questions about E∞-ring spectra.
*Special Time and Room*
October 10, 939 Evans, 2-3pm
Julie Bergner (UC Riverside)
Comparing models for (∞, n)-categories
With many definitions being given for (∞, n)-categories, one criterion to check is whether they can be thought of as categories enriched in (∞, n-1)-categories. In joint work with Charles Rezk, we are establishing a chain of Quillen equivalences from the model structure for Θn-spaces and the model structure for categories enriched in Θn-1-spaces. This comparison also gives insight into how to compare to other known models.
October 17
Anatoly Preygel (Berkeley)
Thom-Sebastiani and duality for matrix factorizations (via DAG)
The Z/2-graded dg-category MF of matrix factorizations is a
categorical singularity invariant appearing on the algebro-geometric
side of mirror symmetry. Though it lives in commutative algebra, and
the first definitions were very explicit, it turns out to admit a few
descriptions very much in the spirit of homotopy theory. After
recalling a bit about sheaf theories in derived algebraic geometry,
we'll sketch at one of these definitions and show how it can be used
to give conceptual ('homotopy theoretic') proofs of some fundamental
properties of MF. Topology (mostly homotopy theory) will make regular
side appearances throughout.
*Special seminar joint with Representation Theory, Geometry and Combinatorics meets from 4-5.30 in 3 Evans*
October 24
Kevin Costello (Northwestern)
N=1 supersymmetric gauge theory and the Yangian
I'll explain a dictionary relating a deformation of the N=1 supersymmetric gauge theory and the Yangian. This dictionary allows one to perform exact calculations in the deformed N=1 gauge theory using the representation theory of the Yangian. The talk will start with a brief discussion of what I mean by a quantum field theory.
October 31
David Futer (Temple)
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 how far this surface is from being a fiber in the knot complement. This is joint work with Effie Kalfagianni and Jessica Purcell.
November 7
Ian Biringer (Boston College)
Growth of Betti numbers and a probabilistic take on geometric convergence
We will describe an asymptotic relationship between the volume and the Betti numbers of certain locally symmetric spaces. The proof uses a synthesis of Gromov-Hausdorff convergence of Riemannian manifolds and Benjamini-Schramm convergence from graph theory
November 14
Marc Culler (UIC)
Peripherally elliptic representations and character varieties
I am interested in SL(2,C) representations of knot groups
with the property that the peripheral elements are sent to elliptic
elements. I will describe some computations, known results and
questions about the structure of the set of these representations and
connections with character varieties and A-polynomials.
November 21
Jon Bloom (MIT)
A bordered monopole Floer theory
I will report on work-in-progress to develop a bordered monopole Floer theory. We associate an algebra to a surface, a module to 3-manifold with boundary, and a map of modules to a 4-manifold with corners (all in the A-infinity sense). These structures satisfy the natural gluing theorems inherent in a 4-dimensional TQFT with corners, and are closely related to Khovanov's invariant of tangles and Szabo's geometric spectral sequence. This is joint work with John Baldwin.
November 28
Ko Honda (USC)
A categorification of the quantum superalgebra U_q sl(1|1)
In the framework of Reshetikhin-Turaev invariants, the
quantum group U_q sl(1|1) of the super Lie algebra sl(1|1) gives rise
to the Alexander polynomial of links just as the quantum group U_q
sl(2) gives rise to the Jones and colored Jones polynomials. I will
describe the work of my student Yin Tian which gives a
categorification of U_q sl(1|1) which is motivated by contact geometry
and in particular by the contact category.
Spring 2012
January 18
Nathan Dunfield (UIUC)
Integer homology 3-spheres with large injectivity radius
Conjecturally, the amount of torsion in the first homology group of a hyperbolic 3-manifold must grow rapidly in any exhaustive tower of covers (see Bergeron-Venkatesh and F. Calegari-Venkatesh). In contrast, the first betti number can stay constant (and zero) in such covers. Here "exhaustive" means that the injectivity radius of the covers goes to infinity. In this talk, I will explain how to construct hyperbolic 3-manifolds with trivial first homology where the injectivity radius is big almost everywhere by using ideas from Kleinian groups. I will then relate this to the recent work of Abert, Bergeron, Biringer, et. al. In particular, these examples show a differing approximation behavior for L^2 torsion as compared to L^2 betti numbers. This is joint work with Jeff Brock.
January 25
Tom Church (Stanford)
Invariant invariants of surface bundles
The Morita-Mumford-Miller classes are the quintessential characteristic classes of surface bundles. These characteristic classes depend on a space E together with a fibering S_g -> E -> B as a surface bundle. Or do they? Well, the Hirzebruch signature formula implies that for a fibered 4-manifold, the first MMM class is just the signature -- so it doesn't depend on the fibering, only on the 4-manifold E.
It turns out that this is not a coincidence. Say that a characteristic class is "geometric" if it only depends on the total space of the surface bundle, and not on the fibering. I will explain the surprising result that all the odd MMM classes are geometric, joint with Benson Farb and Matthew Thibault. Time permitting, I will describe a complete classification of which characteristic classes are geometric w.r.t. cobordism, joint with Martin Crossley and Jeffrey Giansiracusa.
February 1
Dan Margalit (Georgia Tech)
Hyperelliptic curves, braid groups, and congruence subgroups
The hyperelliptic Torelli group is the subgroup of the mapping class group of a surface consisting of elements that act trivially on the homology of the surface and also commute with some fixed hyperelliptic involution. This group can also be characterized as the fundamental group of the branch locus of the period mapping, as well as the kernel of the (specialized) Burau representation of the pure braid group. Hain has conjectured that the hyperelliptic Torelli group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. His conjecture gives a meaningful description of the topology of the branch locus of the period mapping. We present some evidence for the conjecture, as well as progress towards its resolution. Some of the results rely on some new, intricate relations in the pure braid group. This is joint work with Tara Brendle.
February 8
Robert Lipshitz (Columbia)
A spectrification of a categorification of the Jones polynomial
In the early 1980's, Jones introduced a new knot invariant, now called the Jones polynomial. Roughly ten years ago, Khovanov gave a refinement -- or categorification -- of the Jones polynomial; this refinement is now called Khovanov homology. In this talk we will sketch definitions of the Jones polynomial and Khovanov homology, and mention some of their most spectacular applications. We will then discuss a recent space-level refinement of Khovanov homology; this refinement is joint work with S. Sarkar.
February 15
Gabriel C. Drummond-Cole (Northwestern)
Formal formality of low dimensional Calabi-Yau manifolds
Barannikov and Konsevich defined a "hypercommutative" algebra structure on the cohomology of a Calabi-Yau variety. It is natural to consider this hypercommutative algebra as the first piece of a homotopy hypercommutative algebra structure. This homotopy structure can be truncated to Barannikov-Kontsevich's. I'll talk a bit about how to see that this truncation loses no information for some cases, including all Calabi-Yau 3-folds.
February 22
John Pardon (Stanford)
Totally disconnected groups (not) acting on three-manifolds
Hilbert's Fifth Problem asks whether every topological group
which is a manifold is in fact a (smooth!) Lie group; this was solved in
the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture
is that a locally compact topological group which acts faithfully on a
manifold must be a Lie group. This is the Hilbert-Smith Conjecture,
which in full generality is still wide open. It is known, however (as a
corollary to the work of Gleason and Montgomery-Zippin) that it suffices
to rule out the case of the additive group of p-adic integers acting
faithfully on a manifold. I will present a solution in dimension three.
The proof uses tools from low-dimensional topology, for example
incompressible surfaces, minimal surfaces, and a property of the mapping
class group.
February 29
Nathaniel Rounds (Indiana)
Compactifying string topology
String topology studies the algebraic topology of the free loop space of a manifold. In this talk, we describe a compact space of graphs and show how this space gives algebraic operations on the singular chains of the free loop space. In particular, our chain level operations induce Cohen and Godin's "positive boundary TQFT" on the homology the free loop space. This project is joint work with Kate Poirier.
March 7
Curt McMullen (Harvard)
Cascades in the dynamics of foliations on surfaces
We will discuss harmonic foliations on Riemann surfaces
and the notion of "zero flux".
In genus two we will see such foliations go through
`period-doubling' in self-similar cascades that can be
explained through homological invariants.
March 14
Peter Teichner (Berkeley/Max Planck-Bonn)
Whitney towers for classical links
We'll give a survey of recent joint work with Jim Conant and Rob Schneiderman on a quantitative way of measuring the failure of the Whitney trick in dimension four.
March 12
Motohico Mulase (UC Davis)
The mirror dual of the Catalan numbers and the
Eynard-Orantin topological recursion
What is the mirror dual to the Catalan numbers?
By answering this question, I will explain the mathematical
framework of the Eynard-Orantin topological recursion
theory. This theory is a proposal for the universal
B-model that is mirror to "integrable" A-model
(or counting) problems.
In this talk I will be focused on presenting the simplest
mathematical example of the theory for which we
can actually calculate everything. This example
illustrates the general theory behind the scene.
April 4
Ina Petkova (Columbia)
Bordered Heegaard Floer homology - recent developments and applications
Bordered Heegaard Floer homology is a Floer theory over Z/2 for manifolds with boundary. After gluing, it recovers Heegaard Floer homology for closed manifolds. I will briefly describe the bordered Floer package and some applications to knot theory. Then I will discuss its decategorification, and some progress towards extending the theory over the integers.
April 11
Shelly Harvey (Rice)
Filtering smooth concordance classes of topologically slice knots
The n-solvable filtration of the smooth knot concordance
group, suggested by work of Cochran-Orr-Teichner, is slightly flawed
in the sense that any topologically slice knot lies in every term of
the filtration. To repair this we investigate a new filtration, {B_n},
that is simultaneously a refinement of the n-solvable filtration and a
generalization of notions of positivity studied by Gompf and Cochran.
As is the case for the n-solvable filtration, each B_n/B_{n+1} has
infinite rank. Our primary interest is in the induced filtration,
{T_n}, on the subgroup, T, of knots that are topologically slice. We
prove that T/T_0 is large, detected by gauge-theoretic invariants and
the tau, s, and epsilon-invariants; while the non-triviality of
T_0/T_1 can be detected by certain Heegaard Floer d-invariants. All of
these concordance obstructions vanish for knots in T_1. Nonetheless,
going beyond this, our main result is that T_1/T_2 has positive rank.
Moreover under a "weak homotopy-ribbon" condition, we show that each
T_n/T_{n+1} has positive rank.
April 18
Ralph Cohen (Stanford)
Topological Field Theories, Koszul Duality, and String Topology
In this talk I will describe the role of generalized Fukaya categories in the Costello-Lurie classification of two dimensional topological field theories. Then I will switch gears and describe a variety of properties of Koszul dual differential graded algebras. The archetypical examples are the algebra of cochains, C^*(X), and the Pontrjagin algebra C_*(\Omega X), when X is simply connected. I will emphasize the effect of Koszul duality when one of the algebras satisfies Poincare duality up to homotopy. Finally I will connect these topics by discussing the effect of Koszul duality in the classification of topological field theories, while continuing to emphasize the example of the string topology of a manifold.
April 25
Douglas LaFountain (Aarhus/Berkeley)
Decorated Teichmuller theory and compactifications of moduli space
For a genus g surface with s>0 punctures and 2g+s>2, decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over the usual Teichmuller space, where the fiber corresponds to families of horocycles peripheral to each puncture. As proved by R. Penner, DTeich admits a mapping class group-invariant cell decomposition, which then descends to a cell decomposition of Riemann's moduli space. In this talk we introduce a new cellular bordification of DTeich which is also MCG-invariant, namely the space of filtered screens. After an appropriate quotient, we obtain a cell decomposition for a new compactification of moduli space, which is shown to be homotopy equivalent to the Deligne-Mumford compactification. This work is joint with R. Penner.
May 2
Shea Vela-Vick (Columbia)
The equivalence of transverse link invariants in knot Floer homology
The Heegaard Floer package provides a robust tool for
studying contact 3-manifolds and their subspaces. Within the sphere
of Heegaard Floer homology, several invariants of Legendrian and
transverse knots have been defined. The first such invariant,
constructed by Ozsvath, Szabo and Thurston, was defined
combinatorially using grid diagrams. The second invariant was
obtained by geometric means using open book decompositions by Lisca,
Ozsvath, Stipsicz and Szabo. We show that these two previously
defined invariant agree. Along the way, we define a third, equivalent
Legendrian/transverse invariant which arises naturally when studying
transverse knots which are braided with respect to an open book
decomposition.
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