: The Joint Los Angeles Topology Seminar


Year 2015-16


Monday 3/21
4:30-6:30pm
KAP 414

@ USC

Nicolas Tholozan (Univ. Luxembourg): Compact quotients of pseudo-Riemannian hyperbolic spaces
A pseudo-Riemannian manifold is a manifold where each tangent space is endowed with a quadratic form that is non-degenerate, but not necessarily positive definite. A typical example is the hyperbolic space H(p,q), which is a pseudo-Riemannian manifold of signature (p,q) and constant negative sectional curvature. It is homogeneous, as it admits a transitive isometric action of the Lie group SO(p,q+1). A long standing question is to determine for which values of (p,q) one can find a discrete subgroup of SO(p,q+1) acting properly discontinuously and cocompactly on H(p,q). In this talk I will show that there is no such action when p is odd and q >0. The proof relies on a computation of the volume of the corresponding quotient manifold. The proof also implies that, when p is even, this volume is essentially rational. I will discuss in more details the case of H(2,1) (the 3-dimensional anti-de Sitter space), for which compact quotients exist and have been described by work of Kulkarni-Raymond and Kassel.

Peter Samuelson (University of Iowa):
The Homfly skein and elliptic Hall algebras
The Homfly skein relations from knot theory can be used to associate an algebra to each (topological) surface. The Hall algebra construction associates an algebra to each smooth (algebraic) curve over a finite field. Using work of Burban and Schiffmann, we show that the skein algebra of the torus is isomorphic to the Hall algebra of an elliptic curve. If time permits we discuss a third (categorical) construction of the same algebra. (Joint with Morton and Licata.)

Monday 2/29
4:15-6:30pm
MS 5127

@ UCLA

Eugene Gorsky (UC Davis): Heegaard Floer homology of some L-space links
A link is called an L-space link if all sufficiently large surgeries along it are L-spaces. It is well known that the Heegaard Floer homology of L-space knots have rank 0 or 1 at each Alexander grading. However, for L-space links with many components the homology usually has bigger ranks and a rich structure. I will describe the homology for algebraic and cable links, following joint works with Jen Hom and Andras Nemethi. In particular, for algebraic links I will construct explicit topological spaces with homology isomorphic to link Floer homology.

Sheel Ganatra (Stanford University):
Automatically generating Fukaya categories and computing quantum cohomology
Suppose one has determined the Floer theory algebra of a finite non-empty collection of Lagrangians in a Calabi-Yau manifold. I will explain that, if the resulting algebra satisfies a finiteness condition called homological smoothness, then the collection automatically split-generates the Fukaya category. In addition, the Hochschild invariants of the algebra (and hence of the whole Fukaya category) are automatically isomorphic to the quantum cohomology ring. This result immediately extends to the setting of monotone/non-Calabi-Yau symplectic manifolds, under an additional hypothesis on the rank of the algebra???s 0th Hochschild cohomology. The proofs make large use of joint work with Perutz and Sheridan, which in turn is part of a further story about recovering Gromov-Witten invariants from the Fukaya category.

Monday 2/8
4-6pm
Sloan 153

@ Caltech

Anna Wienhard (University of Heidelberg): Maximal representations and projective structures on iterated sphere bundles
The Toledo number is a numerical invariant associated to representations of fundamental groups of surfaces into Lie groups of Hermitian type. Maximal representations are those representations for which the Toledo number is maximal. They form connected components of the representation variety. In the case when the Lie group is SL(2,R)= Sp(2,R) they correspond precisely to holonomy representations of hyperbolic structures. Maximal representations into the symplectic group Sp(2n,R) generalize this situation with a lot of new features appearing. I will describe some of these new features and explain how maximal representations arise as homonym representations of projective structures on iterated sphere bundles over surfaces.

Shicheng Wang (Peking University):
Chern--Simons theory, surface separability, representation volumes, and dominations of 3-manifolds
The talk will start with mapping degree sets and simplicial volumes. We then discuss recent results on virtual representation volumes and on virtual dominations of 3-manifolds, as well as their relations. Time permitted, we may end with the high dimensional applications of representation volumes. This is joint work with P. Derbez, Y. Liu and H. Sun.

Monday 11/30
4-6pm
MS 6229

@ UCLA

Ailsa Keating (Columbia University): Higher-dimensional Dehn twists and symplectic mapping class groups
Given a Lagrangian sphere S in a symplectic manifold M of any dimension, one can associate to it a symplectomorphism of M, the Dehn twist about S. This generalises the classical two-dimensional notion. These higher-dimensional Dehn twists naturally give elements of the symplectic mapping class group of M, i.e. $\pi_0 (Symp (M))$. The goal of the talk is to present parallels between properties of Dehn twists in dimension 2 and in higher dimensions, with an emphasis on relations in the mapping class group.

Hiro Lee Tanaka (Harvard University):
Factorization homology and topological field theories
This is joint work with David Ayala and John Francis. Factorization homology is a way to construct invariants of manifolds out of some algebraic data. Examples so far include singular homology, intersection homology, Bartlett's spin net formalism for Turaev-Viro invariants, Reshetikhin-Turaev invariants for framed knots, and Salvatore's non-Abelian Poincare Duality. It has also been used by Ayala-Francis to prove the cobordism hypothesis. In this talk we'll give some basic examples and prove some classification results akin to Brown Representability.

Monday 11/16
4-6pm
Sloan 151

@ Caltech

Mike Hill (UCLA): A higher-height lift of Rohlin's Theorem: on \eta^3
Rohlin's theorem on the signature of Spin 4-manifolds can be restated in terms of the connection between real and complex K-theory given by homotopy fixed points. This comes from a bordism result about Real manifolds versus unoriented manifolds, which in turn, comes from a C_2-equivariant story . I'll describe a surprising analogue of this for larger cyclic 2 groups, showing that the element eta cubed is never detected! In particular, for any bordism theory orienting these generalizations of Real manifolds, the three torus is always a boundary.

Joshua Greene (Boston College):
Definite surfaces and alternating links
I will describe a characterization of alternating links in terms intrinsic to the link complement and derive some consequences of it, including new proofs of some of Tait's conjectures.

Monday 10/19
4:30-6:30pm
KAP 245

@ USC

Jeff Danciger (UT Austin): Convex projective structures on non-hyperbolic three-manifolds
We discuss a program underway to determine which closed three-manifolds admit convex real projective structures and its implications in the search for low-dimensional matrix representations of three-manifold groups. While every hyperbolic structure is a convex projective structure, examples of convex projective structures on non-hyperbolic three-manifolds were found only recently by Benoist. We produce a large source of new examples, including the doubles of many hyperbolic knot and link complements. The strategy is to suitably deform cusped hyperbolic three-manifolds and then (convexly) glue them together. Joint work with Sam Ballas and Gye-Seon Lee.

Faramarz Vafaee (Caltech):
L-spaces and rationally fibered knots
The main focus of the talk will be on proving fiberedness results for knots in L-spaces with either L-space or S1 x S2 surgeries. Recall that an L-space is defined to be a rational homology three-sphere with the same Heegaard Floer homology as a lens space. We prove that knots in L-spaces with S1 x S2 surgeries are Floer simple and fibered. Moreover, the induced contact structure on the ambient manifold is tight. We also prove that a knot K in an L-space Y with a non-trivial L-space surgery is fibered provided that the orthogonal complement of K with respect to the linking form of Y vanishes. This generalizes the result of Boileau-Boyer-Cebanu-Walsh, in which they assume the knot is primitive. This work is joint with Yi Ni.

Year 2014-15


Monday 4/6
4-6pm
MS 6627

@ UCLA

Steven Sivek (Princeton University): Augmentations of Legendrian knots and constructible sheaves
Given a Legendrian knot in R^3, Shende-Treumann-Zaslow defined a category of constructible sheaves on the plane with singular support controlled by the front projection of the knot. They conjectured that this is equivalent to a category determined by the Legendrian contact homology of the knot, namely Bourgeois-Chantraine's augmentation category. Although this conjecture is false, it does hold if one replaces the augmentation category with a closely related variant. In this talk, I will describe this category and some of its properties and outline the proof of equivalence. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Zaslow.

Hirofumi Sasahira (Nagoya University):
Spin structures on Seiberg-Witten moduli spaces
We will prove that under a certain condition the moduli space of solutions to the Seiberg-Witten equations on a 4-manifold has a canonical spin structure. The spin bordism class of the moduli space is a differential topological invariant of the 4-manifold. We will show that this invariant is nontrivial for the connected sum of some symplectic 4-manifolds.

Monday 12/1
4-6pm
MS 5127

@ UCLA

David Rose (USC): Annular Khovanov homology via trace decategorification
We'll review work of the speaker, joint with Lauda and Queffelec, relating Khovanov(-Rozansky) homology to categorified quantum sl_m via categorical skew Howe duality. We'll then discuss work in progress (joint with Queffelec) showing how to obtain annular Khovanov homology from this "skew Howe 2-functor" via trace decategorification. This provides a conceptual basis for this invariant, and in particular explains the recent discovery of Grigsby-Licata-Wehrli that the annular Khovanov homology of a link carries an action of sl_2. Our framework extends to give the first construction of sl_n annular Khovanov-Rozansky homology (which carries an action of sl_n), and should lead to a proof of a conjecture of Auroux-Grigsby-Wehrli relating annular Khovanov homology to the Hochschild homology of endomorphism algebras in category O.

Liam Watson (University of Glasgow):
A categorified view of the Alexander invariant
Alexander invariants are classical objects in low-dimensional topology stemming from a natural module structure on the homology of the universal abelian cover. This is the natural setting in which to define the Alexander polynomial of a knot, for example, and given that this polynomial arises as graded Euler characteristic in knot Floer homology, it is natural to ask if there is a Floer-theoretic counterpart to the Alexander invariant. There is: This talk will describe a TQFT due to Donaldson, explain how it is categorified by bordered Heegaard Floer homology, and from this place the Alexander invariant in a Heegaard Floer setting. This is joint work with Jen Hom and Tye Lidman.

Monday 11/17
4-6pm
Sloan 151

@ Caltech

Boris Coskunuzer (Koc University and MIT): Minimal Surfaces with Arbitrary Topology in H^2xR
In this talk, we show that any open orientable surface can be embedded in H^2xR as a complete area minimizing surface. Furthermore, we will discuss the asymptotic Plateau problem in H^2xR, and give a fairly complete solution.

Ina Petkova (Rice University):
Combinatorial tangle Floer homology
In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss the construction of this combinatorial invariant for tangles in S^3, D^3, and I x S^2. The special case of S^3 gives back a stabilized version of knot Floer homology.

Monday 11/3
4-6pm
KAP 414

@ USC

Anna Wienhard (Heidelberg and Caltech): Anosov representations and proper actions
When M is a Riemannian manifold, a discrete subgroup of isometries acts properly on M. This is not true for semi-Riemannian manifolds. For a homogeneous space there is criterion, due to Benoist and Kobayashi, which describes when the action of a discrete subgroup of isometries is proper. In this talk I will explain a connection between Anosov representations and proper actions on homogeneous spaces, which relies on a new characterization of Anosov representations. As an application, for a fixed convex cocompact subgroup G' of a Lie group G of rank one, one gets a precise description of the set of proper actions of G' on the group G by left and right multiplication. This is joint work with Francois Gueritaud, Olivier Guichard, and Fanny Kassel.

Jeremy Toulisse (University of Luxembourg):
Minimal maps between hyperbolic surfaces, and anti-de Sitter geometry
Around 1990, Geoff Mess discovered deep connections between 3-dimensional anti-de Sitter (AdS) geometry and the theory of hyperbolic surfaces. These ideas were further expanded by Schoen, Labourie, Schlenker, Krasnov and others to establish an equivalence between minimal Lagrangian diffeomorphisms between hyperbolic surfaces and maximal surfaces in AdS space-time. We will explain this connection, and extend it to manifolds with conical singularities.

Year 2013-14


Monday 5/5
4-6pm
KAP 245

@ USC

Thang Le (Georgia Tech): The Habiro ring and invariants of 3-manifolds
The Habiro ring, first appeared in Habiro's work on sl_2 quantum invariants, is a completion of the ring of polynomials with integer coefficients. The Habiro ring has attracted attentions of number theorists (Manin, Marcolli...) who suggest that the Habiro ring can be considered as the ring of regular functions on the elusive "field with 1 element". In the talk we will explain the Habiro ring and show that for any simple Lie agebra, there is a unified invariant of integral homology 3-spheres with values in the Habiro ring which reduces to the Witten-Reshetikhin-Turaev invariant at roots of unity. This is joint work with Habiro.

Joshua Sussan (CUNY Medgar Evers College):
Braid group actions and Heisenberg categorification
Associated to a simply laced Dynkin diagram there is a corresponding affine Lie algebra and a Heisenberg sub algebra. On the Fock space representation of this (quantized) Heisenberg algebra we construct an action of a braid group. We also category this action by constructing functors on a certain category of modules and show that these functors satisfy the braid group relations.

Monday 2/24
4-6pm
Sloan 159

@ Caltech

Yong Hou (Zanty Electronics): Dimensions and complexity of Kleinian groups
I will talk about complexity of Kleinian groups $\Gamma$ with limit sets $\Lambda(\Gamma)$ that are of small Hausdorff dimension $D_\Gamma$, and in addition address the classical retrosection conjecture for Riemann surfaces. It is well-known theorem of Doyle, and Phillips & Sarnak (for higher dimensions), that finitely generated classical Schottky groups have Hausdorff dimension bounded strictly away from dimension of $\partial\mathbb{H}^n$. However it is rather difficult to obtain structures of Kleinian groups with given Hausdorff dimensions spectrum in general. In this talk we will discuss my recent result which establish the existence of a universal constant $\lambda>0$ such that ALL finitely generated Kleinian groups $\Gamma$ with $D_\Gamma$<\Lambda$ are classical Schottky groups. This in fact generalizes my earlier result which is proved for 2-generated $\Gamma$. The result can also be viewed as converse to Doyle's theorem. The universality of $\Lambda$ have many important implications including, complexity consequence such as finite index subgroups of non-classical Schottky groups and finitely generated subgroup of classical Schottky group are complexity invariant. Finally I'll also mention implications to Marden's classical retrosection conjecture for Riemann surfaces.

Jonathan Bloom (MIT):
The monopole category and invariants of bordered 3-manifolds
I'll discuss work-in-progress with John Baldwin toward constructing a gauge-theoretic analogue of the Fukaya category and monopole Floer theoretic invariants of bordered 3-manifolds. Our construction associates an $A_\infty$ category to a surface, an $A_\infty$ functor to a bordered 3-manifold, and an $A_\infty$ natural transformation to a 4-dimensional cobordism of bordered 3-manifolds. I'll also describe how surgery provides a finite set of bordered handlebodies which generate our category. Our approach is strongly motivated by Khovanov's H^n algebras and functor-valued invariant of tangles, which embed in our construction on the level of homology via branched double cover.

Monday 1/27
4-6pm
MS 6627

@ UCLA

Tirasan Khandhawit (Kavli IPMU Tokyo): Stable homotopy type for monopole Floer homology
In this talk, I will describe an attempt to extend Manolescu's construction of stable Floer homotopy type. The construction associates a stable homotopy object to a 3-manifold and we expect to recover the Floer groups from appropriate homology groups of this stable object. The main ingredients are finite dimensional approximation technique and Conley index theory. In addition, I will demonstrate the construction for certain 3-manifolds such as the 3-torus.

Jennifer Hom (Columbia):
An infinite rank summand of topologically slice knots
Let C_{TS} be the subgroup of the smooth knot concordance group generated by topologically slice knots. Endo showed that C_{TS} contains an infinite rank subgroup, and Livingston and Manolescu-Owens showed that C_{TS} contains a Z^3 summand. We show that in fact C_{TS} contains a Z^\infty summand. The proof relies on the knot Floer homology package of Ozsvath-Szabo and the concordance invariant epsilon.

Monday 11/4
4-6pm
KAP 156

@ USC

Matt Hogancamp (Indiana University): A quasi-local approach to link homology
There exist many categorifications of quantum link invariants, but as yet none of their ``colored" versions are functorial under 4-dimensional link cobordisms. In this talk I propose a new categorification of the colored Jones polynomial and provide evidence that it is functorial (up to sign). The construction uses a new, quasi-idempotent chain complex which categorifies a multiple of the Jones-Wenzl projector.

Vincent Colin (University of Nantes):
Higher-dimensional Heegaard Floer homology
In a work in progress with Ko Honda, we extend the definition of the hat version of Heegaard Floer homology to contact manifolds of arbitrary odd dimension using higher-dimensional open book decompositions and the theory of Weinstein domains. This also suggests a reformulation and an extension of symplectic Khovanov homology to links in arbitrary 3-manifolds.

Monday 10/20
4-6pm
MS 6229

@ UCLA

Lukas Lewark (Durham University): The Khovanov-Rozansky concordance invariants
The Khovanov-Rozansky homologies induce a family of knot concordance invariants (among them the Rasmussen invariant) which give strong lower bounds to the slice genus. We will see why some of those concordance invariants are distinct from the rest, using amongst others various spectral sequences that relate the different Khovanov-Rozansky homologies.

Mohammed Abouzaid (Columbia):
Formality and Symplectic Khovanov Homology
I will describe one aspect of the proof that Khovanov homology agrees with the symplectic analogue, focusing on the formality of a subcategory of the Fukaya category of the nilpotent slice studied by Seidel-Smith. They key new ingredient is an abstract criterion for formality due to Seidel, and its implementation using counts of holomorphic curves in a partial compactification of these spaces. This is joint work with I. Smith.

Monday 10/7
4-6pm
Sloan 159

@ Caltech

Lawrence Roberts (University of Alabama): "Bordered" Khovanov homology and its decategorification
Khovanov homology is an invariant of a link in S^3 which refines the Jones polynomial of the link. Recently I defined a version of Khovanov homology for tangles with interesting locality and gluing properties, currently called bordered Khovanov homology, which follows the algebraic pattern of bordered Floer homology. I will describe the ideas behind bordered Khovanov homology, and (time permitting) describe what appears to be the Jones polynomial-like structure which bordered Khovanov homology refines.

Gang Liu (UC Berkeley):
On 3-manifolds with nonnegative Ricci curvature
For a noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to R^3 or the universal cover splits. As a corollary, it confirms a conjecture of Milnor in dimension 3.

Year 2012-13


Monday 4/15
4-6pm
MS 6229

@ UCLA

John Pardon (Stanford University): Obstruction bundles and counting holomorphic disks in Heegaard Floer homology
We discuss an approach to count the holomorphic disks which give the boundary operator in Heegaard Floer homology. Fix some Heegaard diagram and a domain D from a generator x to a generator y. We consider the ``Hurwitz space'' B of ramified maps from S to Sigma_g which represent D. We will discuss how counting the number of points in the moduli space of holomorphic disks from x to y is equivalent to calculating the relative Euler class of a certain ``obstruction bundle'' over B. For indecomposable domains D, this gives a purely topological definition of the contribution to the boundary operator.

David Nadler (UC Berkeley):
Stable Legendrian singularities and combinatorial quantization
We will describe stable singularities of Legendrian subvarieties and how to deform arbitrary singularities to stable ones. Similar patterns appear in Waldhausen's S-construction and have close connections with ribbon graphs. As an application, we will construct an elementary combinatorial model of Fukaya categories realizing an expectation of Kontsevich.

Monday 2/25
4-6pm
257 Sloan

@ Caltech

Rachel Roberts (Washington University): On the interplay between foliations, laminations and contact structures
Let M be a 3-manifold. I will discuss some ways in which information about codimension-one foliations and laminations in M yields information about contact structures in M, and vice versa. I will discuss work joint with Will Kazez and work joint with Tejas Kalelkar and Will Kazez.

Yi Liu (Caltech):
Virtual positivity of representation volumes
In this talk, we discuss hyperbolic volume and Seifert volume of closed mixed 3-manifolds. In particular, we show that these volumes are virtually positive if a corresponding geometric piece presents. We construct virtual representations using ingredients from recent work of Przytycki and Wise. This is joint work with Pierre Derbez and Shicheng Wang.

Monday 11/26
4-6pm
MS 6229

@ UCLA

Hans Boden (McMaster University): An SU(n) Casson-Lin invariant for links
This talk will describe some recent joint work with Eric Harper defining a family of invariants of links in S^3 in terms of irreducible projective SU(N) representations of the link group. We will explain the compactness and irreducibility results needed to construct the invariants and outline how they can be computed in terms of braid representatives of the link.

Cameron Gordon (University of Texas at Austin):
Bridge number, Heegaard genus, and Dehn surgery
Several well-known examples suggest that if $M$ is obtained by non-trivial Dehn surgery on a knot in $S^3$ then the bridge number of the dual knot in $M$, say with respect to a minimal genus Heegaard splitting, is small. We will show that this is true for non-integral surgeries, and deduce results about the relationship between the Heegaard genera of $M$ and the knot exterior under such surgeries. On the other hand, we will show that for integral surgeries, for a given $M$ the bridge number of the dual knot can be arbitrarily large. This is joint work with Ken Baker and John Luecke.

Monday 11/5
4-6pm
151 Sloan

@ Caltech

Jesse Johnson (Oklahoma State): 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.

Yanki Lekili (University of Cambridge / Simons Center):
An arithmetic refinement of homological mirror symmetry for the 2-torus
We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes on the Tate curve over Z[[q]]. It specializes to an equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the nodal Weierstrass curve y^2+xy=x^3, and, over the punctured disc Z((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. This is joint work with Tim Perutz.

Monday 10/22
4-6pm
KAP 113

@ USC

Stephen Bigelow (UC Santa Barbara): Diagrammatic invariants of tangles
I will mainly talk about the Alexander polynomial. There is a way to compute the Alexander polynomial of a knot diagram by resolving crossings into linear combinations of diagrams that have no crossings, but perhaps some "dead-ends". This works just as well to give an invariant that sends a tangle to a linear combination of simple diagrams that are easy to work with. I will explain this, the colored Alexander polynomial, and prospects for the Jones and HOMFLY polynomials.

Martin Scharlemann (UC Santa Barbara):
Proposed Property 2R counterexamples classified
Earlier work with Robert Gompf and Abigail Thompson classified, via a natural slope indexed by the rationals, all two-component links which contain the square knot and from which (S^1 \times S^2) # (S^1 \times S^2) can be obtained by surgery. It was argued there that each of a certain family L_n of such links probably contradicts the Generalized Property R Conjecture. Left unresolved was how the family L_n fits into the classification scheme. This question is resolved here, in part by giving varied perspectives and more detail on the construction of the L_n. The interest in these examples comes from their mathematical location: at the nexus of three old problems on which progress has been very difficult: the Schoenflies Conjecture, the Generalized Property R Conjecture, and the Andrews-Curtis Conjecture.

Year 2011-12


Friday 4/13
4-6pm
257 Sloan

@ Caltech

Juan Souto (University of British Columbia): Metrics on the sphere with large volume and spectral gap
I will explain how to construct, for d>2, metrics on the d-dimensional sphere with bounded geometry, arbitrarily large volume, and spectral gap bounded from below away from 0. As a consequence we obtain that there are hyperbolic knot complements M_i whose volume tends to infinity and whose Cheeger constant is larger than some epsilon>0 for all i. This is joint work with Marc Lackenby.

Nick Sheridan (MIT):
Homological Mirror Symmetry for a Calabi-Yau hypersurface in projective space
We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3 quartic surface, and our result reproduces that of Seidel; and in the three-dimensional case, it is the quintic three-fold. After stating the result carefully, we will describe some of the techniques used in its proof, and draw lots of pictures in the one-dimensional case.

Friday 3/2
4-6pm
257 Sloan

@ Caltech

David Futer (Temple University): 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.

Sucharit Sarkar (Clay Math Institute / Columbia):
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.

Monday 2/13
4-6pm
MS 6229

@ UCLA

Selman Akbulut (Michigan State): Exotic smooth structures on 4-manifolds
I will discuss corks and plugs (and possibly anti-corks) which are useful tools for understanding exotic smooth manifolds. A natural puzzle is to find the corks and plugs of a given small exotic manifold, such as the Dolgachev surface and the Akhmedov-Park's exotic CP^2 # 2(-CP^2), whose handle-body pictures I will describe.

Vladimir Markovic (Caltech):
Virtual geometry of Riemann surfaces and 3-manifolds
I will discuss my work with J. Kahn about the Ehrenpreis conjecture and the surface subgroup theorem for hyperbolic 3-manifolds.

Monday 1/23
3:30-5:30pm
KAP 148

@ USC

Nathan Dunfield (UIUC): Twisted Alexander polynomials of hyperbolic knots
I will discuss a twisted Alexander polynomial naturally associated to a hyperbolic knot in the 3-sphere via a lift of its holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in a number field coming from the hyperbolic geometry. The polynomial can be defined as the Reidmeister torsion of a certain acyclic chain complex, namely the first homology of the knot exterior with coefficients twisted by the holonomy representation tensored with the abelianization map. This polynomial contains much topological information, for instance about the simplest surface bounded by the knot. I will present computations showing that for all 313,209 hyperbolic knots in S^3 with at most 15 crossings it in fact gives perfect such information, in contrast with a related polynomial coming from the adjoint representation of SL(2, C) on it's Lie algebra. This is joint work with Stefan Friedl and Nicholas Jackson.

Sergei Gukov (Caltech):
Mirror symmetry for colored knot homology
It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzburg models. One important application to knot homologies is the existence of "colored differentials" that relate homological invariants of knots colored by different representations. Based on this structure, we formulate a list of properties of the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By calculating the colored HOMFLY homology for symmetric and anti-symmetric representations, we find a remarkable "mirror symmetry" between these triply-graded theories. This talk is based on http://arxiv.org/abs/1112.0030.

Monday 11/28
3:30-5:30pm
KAP 245

@ USC

Aaron Lauda (USC): Towards odd Khovanov homology via odd categorified quantum groups
Khovanov homology is a categorification of the Jones polynomial that paved the way for other categorifications of quantum link invariants. The theory of categorified quantum groups provides a representation theoretic explanation of these homological link invariants via the work of Webster and others. Surprisingly, the categorification of the Jones polynomial is not unique. Ozsvath, Rasmussen, and Szabo introduced an "odd" analog of Khovanov homology that also categorifies the Jones polynomial, and the even and odd categorification are not equivalent. In this talk I will explain joint work with Alexander Ellis and Mikhail Khovanov that aims to develop odd analogs of categorified quantum groups to give a representation theoretic explanation of odd Khovanov homology.

Vera Vertesi (MIT):
Transverse invariants in Heegaard Floer homology
Using the language of knot Floer homology recently two invariants were defined for Legendrian knots. One in the standard contact 3-sphere defined by Ozsvath, Szabo and Thurston in thecombinatorial 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. This is a joint work with J. Baldwin and D.S. Vela-Vick.

Monday 10/24
4-6pm
MS 6229

@ UCLA

Kristen Hendricks (Columbia): 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 (S^3, 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 (S^3, 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 (S^3,K) is less than or equal to the rank of the knot Floer homology of (Sigma(K), K).

Matt Hedden (Michigan State):
The Khovanov module and unlink detection
I'll discuss a module structure on Khovanov homology and prove that it detects unlinks. The prove uses Kronheimer and Mrowka's result that Khovanov homology detects unknots, a refinement of Ozsvath and Szabo's spectral sequence from Khovanov homology to the Heegaard Floer homology of the branched double cover of a link, and a theorem which shows that Heegaard Floer homology detects S^1xS^2 summands in the prime decomposition of a 3-manifold. This is joint work with Yi Ni.

Friday 9/30
4-6pm
257 Sloan

@ Caltech

Steven Sivek (Harvard): Monopole Floer homology and Legendrian knots
I will define invariants of Legendrian knots using Kronheimer and Mrowka's construction of monopole Floer homology for sutured manifolds. These invariants have several interesting properties: their behavior under stabilization and contact surgery suggests that they are closely related to the Lisca-Ozsvath-Stipsicz-Szabo invariant in knot Floer homology, and they are functorial with respect to Lagrangian concordance. As an application, I will construct many examples of non-loose knots in overtwisted contact manifolds.

Qian Yin (University of Michigan / UCLA):
Lattes Maps and Combinatorial Expansion
A Lattes map is a rational map that is obtained from a finite quotient of a conformal torus endomorphism.Thurston maps are branched covering maps over the 2-sphere with finite post-critical sets. We characterize Lattes maps by their combinatorial expansion behavior, and deduce new necessary and sufficient conditions for a Thurston map to be topologically conjugate to a Lattes map. In the Sullivan dictionary, this characterization corresponds to Hamenstadt's entropy rigidity theorem.


Year 2010-11


Monday 2/28
4-6pm
MS 6229

@ UCLA

Ko Honda (USC): HF=ECH via open book decompositions
The goal of this talk is to sketch a proof of the equivalence of Heegaard Floer homology (due to Ozsvath-Szabo) and embedded contact homology (due to Hutchings). This is joint work with Vincent Colin and Paolo Ghiggini.

Thomas Kragh (MIT):
Nearby Lagrangians and Fibered Spectra
I will start by giving basic definitions, state the nearby Lagrangian conjecture and describe some previous results on this. I will then discuss fibered (ring)-spectra and how to use these on the nearby Lagrangian problem. I will sketch how to prove that any nearby Lagrangian is up to a finite covering space a homology equivalence. Combining this with results by Abouzaid we arrive at the conclusion that any nearby Lagrangian is a homotopy equivalence.

Monday 2/7
4-6pm
KAP 156

@ USC

Rumen Zarev (Columbia): Bordered Floer homology and the contact category
Bordered Heegaard Floer homology is a verison of Heegaard Floer homology for 3-manifolds with boundary, developed by Lipshitz, Ozsvath, and Thurston. A key component of the theory is a DG-algebra A(F) associated to a parametrized surface F. After describing the basics of bordered Floer homology, I will discuss how the homology of A(F) can be naturally identified with a full subcategory of the category of contact structures on Fx[0,1], with convex boundary conditions.

Friday 1/14
4-6pm
257 Sloan

@ Caltech

Tian-Jun Li (Minnesota): Symplectic 4-manifolds with Kodaira dimension zero
Symplectic 4-manifolds with negative Kodaira dimension have been classified up to symplectomorphisms. In the next case, Kodaira dimension zero, there is a speculation that such a manifold is diffeomorphic to K3, Enriques surface or a torus bundle over torus. I will discuss what is known towards this conjectured smooth classification.

Matthew Day (Caltech):
A Birman exact sequence for automorphism groups of free groups
The classical Birman exact sequence relates the mapping class group of a surface with punctures to the mapping class group of the same surface with some of the punctures patched over. We describe a similar sequence for automorphism groups of free groups. Instead of mapping class groups of surfaces with punctures, we consider groups of automorphisms of a free group that fix the conjugacy classes of some of the basis elements. Instead of filling in punctures, we induce automorphisms on a free group of lower rank by deleting basis elements. Our results concern the kernel of this map: the kernel is finitely generated and we give an explicit recursive presentation for it. Further, the kernel is not finitely presentable (aside from a few trivial cases). We show this by building explicit infinite-rank subgroups of its second homology and cohomology. This is joint work with Andrew Putman.

Friday 12/3
4-6pm
257 Sloan

@ Caltech

Joshua Greene (Columbia): Mutation and alternating links
I will discuss the proof and consequences of the following result. Suppose that D_1 and D_2 are reduced alternating diagrams for a pair of links whose branched double-covers have isomorphic Heegaard Floer homology groups. Then the spaces are diffeomorphic, and moreover D_1 and D_2 are related by a sequence of Conway mutations.

Andrew Putman (Rice):
Equivariant homological stability for congruence subgroups
An important structural feature of the kth homology group of SL_n(Z) is that it is independent of n once n is sufficiently large. This property is called "homological stability" for SL_n(Z). Congruence subgroups of SL_n(Z) do not satisfy homological stability; however, I will discuss a theorem that says that they do satisfy a certain equivariant version of homological stability.

Monday 11/1
4-6pm
MS 6229

@ UCLA

Allison Gilmore (Columbia): An algebraic proof of invariance for knot Floer homology
We investigate the algebraic structure of knot Floer homology in the context of categorification. Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. Starting with a braid diagram for a knot, one singularizes or smooths each crossing, then associates an algebra to each resulting singular braid. These can be arranged into a chain complex that computes knot Floer homology. Using this construction, we give a fully algebraic proof of invariance for knot Floer homology that avoids any mention of holomorphic disks or grid diagrams. We close with an alternative description of knot Floer homology in terms of Soergel bimodules that suggests a close relationship with HOMFLY-PT homology.

John Baldwin (Princeton):
A combinatorial spanning tree model for knot Floer homology
I'll describe an ongoing project with Adam Levine to iterate Manolescu's exact triangle in knot Floer homology. This iteration results in a spectral sequence which converges to (a stabilized version of) knot Floer homology. With coefficients in a Novikov ring, the E_2 term of this spectral sequence is roughly generated by Kauffman states of the knot or, equivalently, by spanning trees of its black graph. One can describe the d_2 differential combinatorially and can prove that this spectral sequence collapses at E_3. Therefore (E_2,d_2) provides a combinatorial chain complex for delta-graded knot Floer homology.

Monday 9/27
3:30-5:30pm
KAP 245

@ USC

Anthony Licata (Stanford): Heisenberg algebras, Hilbert schemes, and categorification
Representations of Heisenberg algebras appear naturally in many mathematical contexts. In geometry, one notable appearance is in the work of Nakajima and Grojnowski relating Heisenberg algebras and Hilbert schemes of points on a complex surface. We will describe a categorification of the Nakajima-Grojnowski construction. This is joint work with Sabin Cautis.

Joan Licata (Stanford):
Legendrian contact homology for Seifert fibered spaces
In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form. A Legendrian knot in such a manifold is described by a specially labeled Lagrangian diagram, and from this data one can compute both the ``classical" invariants for Legendrian knots in rational homology three-spheres and also a new invariant which takes the form of a differential graded algebra. This work is joint with J. Sabloff.



Year 2009-10


Monday 5/3
3:30-5:30pm
KAP 245

@ USC

Vera Vertesi (MSRI): Legendrian and transverse classification of twist knots.
In 1997 Chekanov gave the first example of a knot type whose Legendrian representations are not distinguishable using only the classical invariants: the 52 knot. Epstein, Fuchs and Meyer extended his result by showing that there are at least n different Legendrian representations of the (2n+1)2 knot with maximal Thurston-Bennequin number. The aim of this talk to give a complete classification of Legendrian representations of twist knots. In particular the (2n+1)2 knot has exactly ceil(n2/2) Legendrian representations with  maximal Thurston-Bennequin number. This is a joint work (in progress) with John Etnyre and Lenhard Ng.

Daniel Krasner (Columbia and MSRI):
Graphical calculus of Soergel bimodules in Khovanov-Rozansky link homology
I will outline a graphical calculus of Soergel bimodules, developed by B. Elias and M. Khovanov, and describe how it can be used to construct an integral version of sl(n) and HOMLFYPT link homology, as well as prove functoriality of the latter.

Friday 4/23
4-6pm
257 Sloan

@ Caltech

Yanki Lekili (MSRI): Quilted Floer homology of 3-manifolds
We introduce quilted Floer homology (QFH), a new invariant of 3-manifolds equipped with a circle valued Morse function. This is a natural extension of Perutz's 4-manifold invariants, making it a (3+1) theory. We relate Perutz's theory to Heegaard Floer theory by showing an isomorphism between QFH and HF^+ for extremal spin^c structures with respect to the fibre of the Morse function. As applications, we give new calculations of Heegaard Floer theory and a characterization of sutured Floer homology.
Tao Li (Boston College):
Rank and genus of amalgamated 3-manifolds
A fundamental question in 3-manifold topology is whether the rank of the fundamental group a 3-manifold is equal to its Heegaard genus. We use hyperbolic JSJ pieces to construct closed 3-manifolds with rank smaller than genus.

Monday 3/1
4-5pm
MS 6627

@ UCLA

Christopher Douglas (Berkeley): 2-dimensional algebra and 3-dimensional local field theory
Witten showed that the Jones polynomial invariants of knots can be computed in terms of partition functions of a (2+1)-dimensional topological field theory, namely the SU(2) Chern-Simons theory.  Reshetikhin and Turaev showed that this theory extends to a (1+1+1)-dimensional topological field theory---that is, there is a Chern-Simons-type invariant associated to 3-manifolds, 3-manifolds with boundary, and 3-manifolds with codimension-2 corners. I will explain the notion of a local or (0+1+1+1)-dimensional topological field theory,  which has, in addition to the structure of a (1+1+1)-dimensional theory, invariants associated to 3-manifolds with codimension-3 corners.  I will describe a notion of 2-dimensional algebra that allows us to construct and investigate such local field theories. Along the way I will discuss the geometric classification of local field theories, and explain the dichotomy between categorification and algebraification.

Friday 12/4
4-6pm
257 Sloan

@ Caltech

Yi Liu (UC Berkeley): Tiny groups and the simplicial volume
A group is called tiny if it cannot map onto the fundamental group of any aspherical 3-manifold of negative Euler characteristic. For example, knot groups are tiny. In this talk we show that if a finitely presented tiny group G maps onto the fundamental group of a compact aspherical 3-manifold N, then the simplicial volume of N is bounded above in terms of G. This is joint work with Ian Agol.

Shelly Harvey (Rice University): Filtrations of the Knot Concordance Group
For each sequence of polynomials P={p1(t), ... }, we define a characteristic series of groups, called the derived series localized at P. Given a knot K, such a sequence of polynomials arises naturally as the sequence of orders of the higher-order Alexander modules of K. These new series yield filtrations of the smooth knot concordance group that refine the (n)-solvable filtration. We show that each of the successive quotients of this refined filtration contains 2-torsion and elements of infinite order.  These results generalize the p(t)-primary decomposition of the algebraic knot concordance group due to Milnor, Kervaire and Levine. This is joint work with Tim Cochran and Constance Leidy.

Monday 11/2
4-6pm
KAP 145

@ USC

Kefeng Liu (UCLA): Recent results on moduli spaces

Paul Melvin (Bryn Mawr College):
Degree formulas for higher order linking
The linking number of a pair of closed curves in 3-space can be expressed as the degree of a map from the 2-torus to the 2-sphere, by means of the linking integral Gauss wrote down in 1833.  In the early 1950's, John Milnor introduced a family of higher order linking numbers (the "mu-bar invariants").  In this talk I will describe a formula for Milnor's triple linking number as the "degree" of a map from the 3-torus to the 2-sphere.  This is joint work with DeTurck, Gluck, Komendarczyk, Shonkwiler and Vela-Vick.

Friday 10/9
4-6pm
MS 6229

@ UCLA

Scott Morrison (UC Berkeley): Blob homology
Blob homology is a new gadget that takes an n-manifold and an n-category with duals, and produces a graded vector space. It's a simultaneous generalization of two important constructions: the 0-th graded piece recovers the usual "skein module" invariant, and in the special case of n=1, where the manifold is the circle, blob homology reduces to Hochschild homology. I'll begin by reviewing these ideas, then give the definition of blob homology. Finally, I'll describe some of its nice formal properties, including an action of chains of diffeomorphisms generalizing the action of diffeomorphisms on a skein module, and a nice gluing formula in terms of A_\infty modules.

Yi Ni (CalTech):     Some applications of Heegaard Floer homology to Dehn surgery
In recent years, Heegaard Floer homology has become a very powerful tool for studying Dehn surgery. In this talk, we will discuss two kinds of such applications. One is to exploit the relationship between the sutured structure of the knot complement and longitudinal surgery, the other consists of some results about cosmetic surgeries.



Year 2008-09

Monday 4/13
4-6pm
DRB 337

@ USC

Stefan Friedl (University of Warwick): Fibered 3-Manifolds and Symplectic 4-Manifolds


Monday 3/30
4-5pm
MS 3915H

@ UCLA

Samuel Lisi (Stanford University): Homoclinic Orbits and Pseudohomoclinic Curves


Monday 2/23
4-5:30pm
MS3915A

@ UCLA

Brett Parker (UC Berkeley): Orthogonally Intersecting Symplectic Divisors


Monday 2/9
4-6pm
KAP 427

@ USC

Andrew Blumberg (Stanford University): Algebraic K-Theory, Higher Homotopy Theory of Derived Categories and Localization

Chris Judge (Indiana University): Small Eigenvalues and Maximal Laminations


Monday 12/1
4-6pm
MS 5147

@ UCLA

Tim Perutz (Columbia): A hypercube for fixed-point Floer homology
The fixed-point Floer homology of a symplectomorphism gives a lower bound for the number of fixed points of non-degenerate representatives of its Hamiltonian isotopy class. It's usually hard to compute. Seidel (2001) conjectured a description of the Floer homology of a composite of Dehn twists along Lagrangian spheres. I'll give a concrete formulation of the conjecture,and a status-report on my efforts to prove it. This formulation involves a "hypercube" chain complex, similar (and perhaps in some cases isomorphic) to structures appearing in Heegaard Floer homology and in Khovanov homology.

Andrew Cotton-Clay (UC Berkeley): Fixed point bounds for symplectic mapping classes
We show that the rank of a certain twisted version of symplectic Floer homology gives a bound on the number of fixed points of any map with nondegenerate fixed points in a given symplectic mapping class on a monotone symplectic manifold. We apply this to the case of area-preserving surface diffeomorphisms to give a sharp lower bound on the number of fixed points in a given mapping class which frequently exceeds the classical sharp bound on non-area-preserving maps given by Nielsen theory. This generalizes the Poincaré-Birkhoff fixed point theorem for area-preserving twist maps on the cylinder to arbitrary surfaces and mapping classes.


Monday 11/17
4:00-6:00

@ USC

Yanki Lekili (MIT): Broken Lefschetz fibration and Floer theoretical invariants
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 describe Perutz's 4-manifold invariants associated with broken Lefschetz 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 Ozsvath-Szabo invariants.

Mohammed Abouzaid (Clay / MIT): Genus 2 Lagrangians in the 4-Torus Via Mirror Symmetry
I will explain how the pseudo-holomorphic quilts of Wehrheim-Woodward can be used to understand the Fukaya category of a product of symplectic manifolds (under the usual assumptions which prevent bubbling). Applying this to the product of the 2-torus with itself, we obtain a correspondence between Lagrangians in the 4-torus of vanishing Maslov index and coherent sheaves on a certain abelian variety. In the case of genus 2 surfaces, we use some sheaf theory on the complex side to obtain numerical restrictions, in the spirit of the Arnol'd conjecture, on the number of intersections of such a surface with the fibres of a Lagrangians torus fibration. This is work in progress, joint with Ivan Smith.

Monday 11/3
4-6pm
MS 5147

@ UCLA

Robert Lipshitz (Columbia): Heegaard Floer Theory for Three-Manifolds with Boundary
Heegaard Floer homology is a (3+1)-dimensional topological field theory (conjecturally) coming from Seiberg-Witten theory. We will discuss an extension of Heegaard Floer homology to three-manifolds with boundary, focusing on the geometric setting and algebraic properties of the extension. This is joint work with P. Ozsvath and D. Thurston.

Sucharit Sarkar (Princeton): Constructing CW Complexes from Grid Diagrams
Given a knot presented in a grid diagram, we can associate to it a chain complex such that the homology of the chain complex is a powerful invariant called the Heegaard Floer homology. We try to construct a CW complex which has one cell for each generator of the chain complex and whose attaching maps correspond to the chain complex boundary map. Time permitting, we will show that the stable homotopy type of this CW complex is actually a knot invariant.

Monday 10/20
4-6pm
KAP 427

@ USC

David Shea Vela-Vick (University of Pennsylvania): Transverse Invariants and Bindings of Open Books
Let T be a transverse knot in $(Y, \xi)$ which is the binding of some open book, $(T, \pi)$, for the ambient contact manifold $(Y, \xi )$. In this talk, we show that the transverse invariant, defined by Lisca, Ozsváth, Stipsicz, and Szabó (LOSS), is nonvanishing for such transverse knots. We will also discuss a vanishing theorem for the invariants defined by LOSS. As a corollary, we will see that if $(T, \pi)$ is an open book with connected binding, then the complement of $T$ has no Giroux torsion. Time permitting, we will also talk about a generalization of this theorem which removes the connected binding condition.

Ben Webster (MIT): Knot Homology and Geometry
I'll try to describe how Khovanov's HOMFLY homology for knots, while initially a bit daunting, is really a very natural construction, if one defines the HOMFLY polynomial correctly. Along the way, we'll discuss what Hecke thought the Hecke algebra was, why the structure of the flag variety is important in knot invariants, and a little bit of philosophy about why geometrizations and categorifications tend to come in pairs.

Monday 10/6
4-5pm
MS 5147

@ UCLA

Cagatay Kutluhan (University of Michigan): Seiberg-Witten Floer homology and symplectic forms on S1 X M3
Let M be a closed, oriented 3-manifold. Suppose S1 X M admits a symplectic form. Then, does M fiber over S1? I will talk about joint work with Clifford H. Taubes on how one can go about proving an affirmative answer to this question using Seiberg-Witten Floer homology. In particular, I will show how to prove that the answer to this question is ``Yes'' when M has first Betti number equal to 1 and S1 X M has non-torsion canonical class.

Monday 9/22
4:30-6pm
KAP 113

@ USC

Ciprian Manolescu (UCLA): Link Homology Theories from Symplectic Geometry
I will start with an introduction to quiver varieties, then explain how one can use Lagrangian Floer homology on them to define link invariants, one for each positive integer n. When n=2, this construction is due to Seidel and Smith and the resulting invariant is conjectured to equal Khovanov homology.
 
This page is maintained by Ciprian Manolescu.