University of Virginia Geometry Seminar 2014-15

An L-space link is a link on which all large surgeries are L-spaces. These links turn out to be rich in geometry and simple in algebra. I will present some properties and examples of L-space links in contrast to L-space knots, give bounds on the ranks of their Floer homology and on the coefficients in the multi-variable Alexander polynomials, and show that how to use Manolescu-Ozsvath link surgery formula to give a free resolution of the Floer homology of surgeries on any L-space links. As an application, we compute the whole package of Heegaard Floer homology of surgeries on 2-component L-space links only in terms of Alexander polynomial and the surgery framing.

This talk will introduce the Khovanov Flow Category which gives rise to the construction of the Lipshitz-Sarkar stable homotopy type of a knot (ie a suspension spectrum X(K)). The cohomology H*(X(K)) coincides with the Khovanov homology of the knot, but the spectrum X(K) carries more information, including stable cohomology operations.

The cabling conjecture asserts that surgery on a knot in S^3 can only be a nontrivial connected sum if the knot is a (p,q)-cable of another knot and the surgery slope is pq. In this talk, I will use contact geometry to put strong restrictions on the possible slopes of reducible surgeries below the Thurston-Bennequin number of a knot. In particular, if n-surgery on a positive hyperbolic knot K is irreducible, then we will see that n=2g(K)-1 and g(K) > 2. This is joint work with Tye Lidman.

Convex cocompact groups originate in the study of the geometry of low-dimensional manifolds. An analogue of convex cocompactness in the theory of mapping class groups was developed by B. Farb and L. Mosher in 2001. Since then, convex cocompact subgroups have been the subject of intense study, because they give insight into the geometry of mapping class groups as well as into the structure of Gromov-hyperbolic surface group extensions. In this talk, I will discuss the geometry of convex cocompact subgroups of mapping class groups which arise from representations of right-angled Artin groups, as well as an analogue of convex cocompactness for subgroups of right-angled Artin groups. This represents joint work with J. Mangahas and S. Taylor.

A Legendrian knot in R^3 with the standard contact structure is a knot for which dz-ydx=0. Given a Legendrian knot, one can associate the Chekanov-Eliashberg differential graded algebra (DGA) over Z/2. Fuchs and Sabloff showed there is a correspondence between augmentations to Z/2 of the DGA and rulings of the knot diagram. Etnyre, Ng, and Sabloff showed that one can define a lift of the Chekanov-Eliashberg DGA over Z/2 to a DGA over Z[t,t^{-1}]. This talk will give an extension of the relationship between rulings and augmentations to Z/2 of the DGA over Z/2, to a relationship between rulings and augmentations to a field of the DGA over Z[t,t^{-1}]. No knowledge of the Chekanov-Eliashberg DGA will be assumed.

We initiate an extensive study of multisections of Lefschetz
fibrations via positive factorizations in framed mapping class groups.
Using our techniques, we can reformulate and tackle various interesting conjectures and problems related to the topology of symplectic
4-manifolds. In this talk we will focus on the symplectic Kodaira dimension, and the conjectural smooth classification of symplectic Calabi-Yau surfaces and their fundamental groups in real dimension 4. If time permits, we will discuss some other applications as well; such as Stipsicz's conjectures on minimality and fiber sum decompositions, constructions of inequivalent Lefschetz fibrations and exotic pencils. Joint work with K. Hayano.

Consider the double cover of the 3-sphere branched over a link. We will discuss how the instanton homology of this 3-manifold is related to the Khovanov homology of the link. When signs (integer coefficients) are taken into account, the relevant link homology is odd Khovanov homology, which is genuinely distinct from Khovanov homology. This material is motivated by work of Ozsváth and Szabó in the Heegaard-Floer setting.

In a series of surprising conjectures, Gorsky-Oblomkov-Rasmussen-Shende relate Khovanov-Rozansky homologies of the torus knot T(p,q) with the representation theory of the rational Cherednik algebra for SL_p. One such conjecture states that the triply graded Khovanov-Rozansky homology has a well-defined limit as $q\to \infty$, and that this homology is very simple: it is a (super) polynomial ring. In this talk we discuss and outline a proof of this limiting version of their conjecture. Along the way we discuss a categorified Young symmetrizer, which plays an essential role in our proof.

The HOMFLY-PT polynomial colored by one-column partitions, P_{1^n}(L) can be constructed by cabling a braid by objects known as Young antisymmetrizers in the Hecke algebra of S_n . In this talk, after reviewing the basic construction of HOMFLY-PT homology, we will show how to categorify these objects in the homotopy category of Soergel bimodules. These categorical antisymmetrizers can also be twisted by permutations. Each of these twisted objects can be used to define a link homology theory H_{1^n,w}(L) categorifying P_{1^n}(L). As an application we will give a method of computing the limit of the HOMFLY-PT homology of the (p,q)-torus knot as q approaches infinity. This is joint work with Matt Hogancamp.

The 2-Engel relation, classically studied in group theory, will be used to analyze the topological surgery conjecture, a long-standing problem in 4-manifold topology.

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.

While many open 4-manifolds are known to admit uncountably many
diffeomorphism classes of smooth structures, it is still unknown if this
behavior can be expected in general. The goal of this talk is to introduce
a family of "large" R^4's with the property that each is contained in some
compact Stein surface. We will discuss why these new R^4's are relevant to
understanding smoothing theory on more general open 4-manifolds and
outline their construction. Along the way, we will define Casson handles
and describe their use in cut-and-paste arguments.

The existence problem of a tight contact structure on a fixed three manifold is hard and still widely open. The classification problem is even less well-understood. In this talk, we will focus on the classification of tight contact structures on (certain) Seifert fibered manifolds on which the existence problem was settled recently by Lisca and Stipsicz.

Abstract: Knots and links are modeled as ribbons immersed in the plane.
This is a 2-dimensional analogue of thick knots and the ropelength problem. This
talk will introduce the idea of a flat ribbon link and give examples of
'tight' ribbonlength knots. It turns out there are some surprising
technicalities involved - moving from 3 to 2 dimensions does not
necessarily simplify the mathematics.

For $C$ a smooth projective curve, and $G$ a simple, simply connected
complex group, let $M_C(G)$ be the moduli space of semistable
$G-$principal bundles on $C$. As the curve $C$ moves in the moduli
$\mathcal{M}_g$ of smooth curves, the spaces $M_C(G)$ are known to
define a flat family of schemes, and this family can be extended to
the Deligne-Mumford compactification $\bar{\mathcal{M}}_g$. We
describe the geometry of the fibers of this family which appear at the
stable boundary, in particular we discuss a recent result which shows
that the fibers over maximally singular curves contain an important
and ubiquitous moduli space, the free group character variety
$\mathcal{X}(F_g, G),$ as a dense, open subspace. The latter is a
moduli space of representations of the free group $F_g$ in $G$, and
naturally appears as an object of interest in Teichm\"uller theory,
the theory of geometric structures, and the theory of Higgs bundles.
For $G = SL_2(\C)$ and $SL_3(\C)$ we describe maximal rank valuations
on the coordinate rings of these spaces, and how the associated
Newton-Okounkov polyhedra can be used to study the geometry of both
$\mathcal{X}(F_g, G)$ and $M_C(G).$

Most work on surgeries in contact manifolds has focused upon determining the situations where tightness is preserved. We will discuss an approach to this problem from the reverse angle: when negative surgery on a fibred knot in an overtwisted contact manifold produces a tight one. We will examine the various phenomena that occur, and discuss an approach to characterising them via Heegaard Floer homology.

Symplectic homology is a version of Floer homology defined for symplectic manifolds with contact-type boundaries. This invariant detects a number of interesting properties of Weinstein domains, and is very rich in many examples; for instance, the symplectic homology of a cotangent bundle is isomorphic to the homology of the loop space (by Viterbo; Abbondandolo and Schwarz).
I will provide some background and motivation for studying symplectic homology, and will discuss the computation of symplectic homology for certain affine algebraic varieties in terms of relative Gromov-Witten invariants. This is joint work with Luis Diogo.

I will report on progress in defining an invariant of 3-manifolds with boundary in the spirit of C. Manolescu's stable homotopy type for rational homology 3-spheres.

A Casson tower is constructed by thickening a 2-complex built from layers of immersed discs in a 4-manifold. A Casson tower has a height and an attaching circle, which is the boundary of its base disc. A higher tower, namely a tower with more layers of immersed discs, is a better approximation to an embedded disc. Casson towers featured prominently in Freedman's original proof of the 4-dimensional topological Poincaré conjecture. In particular Freedman showed that an infinite Casson tower, which is called a Casson handle, contains an embedded disc. The ability to embed discs is the key to being able to apply surgery and h-cobordism techniques, originally from high dimensional topology, to 4-manifolds.
I will explain what a Casson tower is and present embedding results, from work with Jae Choon Cha, on Casson towers of height 4, 3 and 2. The height 2 result in particular can be used to find a new family of topologically slice knots. I will explain why we think these slice knots are interesting.

Generalized geometry is the study of the geometry of the generalized tangent bundle (direct sum of the tangent and cotangent bundle) of a manifold. Partly because of the connection with physics, most of the literature on generalized geometry specializes to even-dimensional manifolds. The odd-dimensional case, pioneered by Vaisman, Wade and Poon, is much less understood. In this talk we report on an on-going project (joint with D. Grandini) aimed at developing a unified framework that provides generalized complex geometry concepts (such as type, pure spinors, T-duality etc.) with natural odd-dimensional counterparts.

We analyze the algebraic structure of Khovanov homology and related chromatic homology
theories, with a focus on understanding torsion. Although computations hint at the abundance of torsion, describing and understanding it is no easy task. We will discuss current results on 2-torsion, which we obtain using relations to chromatic graph homology theories and spectral sequence arguments. We also state non-existence results for odd torsion in certain gradings of Khovanov homology of semi-adequate knots and links.