Friday talks take place in Willard 010, Saturday and Sunday talks take place in Math Sciences 101. See the map on the main conference page for these locations.
Friday (Willard 010)
2:30 - Stephan Burton, Geometrically Maximal Knots
The volume density of a hyperbolic knot is the ratio of its volume to its crossing number. It is known that the volume density of a knot is bounded above by the volume of a regular ideal octahedron. Champanerkar, Kofman, and Purcell constructed sequences of alternating knots with volume density approaching this upper bound. Such a sequence is called geometrically maximal. This construction requires that the knots of the sequence do not contain a cycle of tangles. In this talk I will present geometrically maximal sequences of knots that have a cycle of tangles and which are not alternating.
2:55 - Christian Millichap, Mutations of hyperbolic knots
In this talk, we shall examine the topology and geometry of hyperbolic knots that differ by mutations. A mutation is a topological cut and paste operation along a 4-punctured sphere in the knot complement. This action is often violent enough to create a different knot, yet subtle enough that many topological and geometric invariants of the knot are preserved. In particular, mutant hyperbolic knot complements will all have the same volume and (sufficiently) shortest geodesic lengths, yet such knot complements can often be constructed to be pairwise incommensurable, i.e., they do not share a common finite-sheeted cover. Here, we will construct a large class of pairwise incommensurable mutant knots and get our hands on their geometry by realizing them as long Dehn fillings of the same augmented link.
3:20 - Christine Lee, The Slope Conjecture and 3-string pretzel knots
The Slope Conjecture relates the degrees of the colored Jones polynomial to the boundary slopes of essential surfaces in the knot complement. We will discuss how this is proven for a class of pretzel knots $P(r,s,t)$ using the Hatcher-Oertel algorithm. This is joint work with R. van der Veen.
5:00 - Katherine Vance, Tau invariants for balanced spatial graphs
Recently Harvey and O'Donnol defined a combinatorial Heegaard Floer homology theory $\widehat{HFG}$ for spatial graphs. Their theory is relatively bigraded, with an integer-valued Maslov grading and a relative Alexander grading, which takes values in the first homology of the spatial graph exterior. We define a $\mathbb{Z}$-filtered chain complex $\widehat{CG}$ for balanced spatial graphs whose associated graded chain complex has homology determined by $\widehat{HFG}$. One step in showing this filtered chain complex exists is to lift the relative Alexander filtration to an absolute grading. We then use the filtered chain complex $\widehat{CG}$ to show that there is a well-defined $\tau$ invariant for balanced spatial graphs which generalizes the $\tau$ knot concordance homomorphism defined by Ozsvath-Szabo and Rasmussen.
5:25 - Julia Bennett, Large $R^4$'s in Stein Surfaces
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 outline the construction of these $R^4$'s and discuss how they can be used as building blocks to produce new smooth structures on open 4-manifolds. Along the way, we will describe the basic invariants that are available to distinguish exotic smoothings in this setting.
Saturday (Math Sciences 101)
9:30am - John Etnyre, Embeddings of contact 3-manifolds
In this talk I will discuss some preliminary results and observations about embedding contact 3-manifolds in Euclidean 5-space with the standard contact structure, in addition to other contact 5-manifolds. These results come from discussions with Yanki Lekili and Ryo Furukawa and will joint results between various combinations of them and myself.
10:30 - Sinem Onaran, Invariants of knots in overtwisted contact 3-manifolds
In this talk, we will discuss recently defined depth, tension and order invariants for knots in overtwisted contact 3-manifolds. We will discuss applications of these invariants and list several open problems related to the invariants. This work is joint with K. Baker.
11:45 - Laura Starkston, Symplectic embeddings and mapping class monoid relations
For certain contact 3-manifolds supported by a planar open book decomposition, there are two ways of constructing and classifying symplectic fillings whose boundary is that contact 3-manifold. One way involves understanding factorizations of the monodromy of the open book into positive Dehn twists. The other way is to look at embeddings of concave neighborhoods of a collection of symplectic surfaces into a well understood (rational) symplectic 4-manifold. Depending on the bounding contact 3-manifold, each of these methods has different strengths and data from one method can provide information about the other. Therefore it is very useful to understand how to translate the information coming from symplectic surface embeddings to mapping class group relations or vice versa. The goal of this talk will be to discuss some correspondences between these two methods in large classes of examples, where the boundary 3-manifold is Seifert fibered over the 2-sphere.
2:30 - Andrew Wand, Tightness and open book decompositions
We will discuss a characterization of tightness of a contact 3-manifold in terms of supporting open book decompostions, some applications, and generalizations.
3:30 - Matthew Hedden, Khovanov-Floer theories are functorial
Khovanov homology is an easily defined homological invariant of links in the 3-sphere, which generalizes the Jones polynomial. Motivation for this definition is provided by the additional "functoriality" gained by associating groups to links (as opposed to polynomials). This allows for the definition of maps on Khovanov homology associated to cobordisms between links. There are several other, much less easily defined homological invariants of links e.g. singular instanton link homology, Heegaard, Monopole, or Instanton Floer homologies of the branched double cover of the link. These theories are also functorial with respect to link cobordisms. Surprisingly, Khovanov homology is connected to each of these theories (and several others) through a spectral sequence, and a natural question asks whether this connection is functorial. In this talk I'll define an abstract algebraic notion of a "Khovanov-Floer" theory, and prove that such theories are functorial with respect to link cobordisms. I'll then show that all the aforementioned invariants satisfy our definition, thus proving that the spectral sequences connnecting them to Khovanov homology are functorial with respect to link cobordisms. This is joint work with John Baldwin and Andrew Lobb.
4:45 - Efstratia Kalfagianni, Topology in the degrees of knot polynomials
I plan to discuss results and questions relating the degree(s) of the colored Jones knot polynomial(s) to the topology of essential surfaces in knot complements.
Sunday (Math Sciences 101)
9:00am - Dave Futer, Geometrically similar knots
There are several known ways to produce hyperbolic 3-manifolds that isospectral (i.e. have the same spectrum of geodesic lengths) but not isometric. All known constructions of of this sort involve finite covers of the same base manifold, leading Reid to ask whether this is a necessary feature. That is, are isospectral manifolds necessarily commensurable? I will describe a way to build pairs of knot complements that are incommensurable but have the same closed geodesics up to length L, where L is as large as one likes. This is joint work with Christian Millichap.
10:00 - Jessica Purcell, Geometrically maximal knots
In this talk, we consider the ratio of volumes of hyperbolic knots to their crossing numbers. This ratio is known to have maximum value less than the volume of a regular ideal octahedron. This motivates several questions, such as, for which knots is the ratio very near the maximum? For fixed crossing number, what links maximize this ratio? We say that a sequence of hyperbolic knots is geometrically maximal if these ratios limit to the maximum value. In this talk, we describe several sequences of geometrically maximal knots, and present several conjectures. We discuss weaving knots, which are alternating knots with the same projection as a torus knot, and which were conjectured by Lin to be among the maximum volume knots for fixed crossing number. We prove weaving knots are geometrically maximal. We discuss a method, using ideas known to Agol, for constructing other sequences of geometrically maximal knots. This is joint with Abhijit Champanerkar and Ilya Kofman.
11:15 - Emmy Murphy, Dissolving finite index covers of 4-manifolds and connections to coarse geometry
Supposing $M$ and $N$ are s-cobordant oriented 4-manifolds, it is known that M and N become diffeomorphic after connect summing with sufficiently many copies of $S^2\times S^2$. Certainly $M$ and $N$ need not be diffeomorphic, however it is an open problem whether a single copy of $S^2\times S^2$ always suffices, noting that this operation destroys all known gauge theory invariants. We discuss a much weaker result: if $π_1(M) = π_1(N) = \mathbb{Z}$ and $M_i$, $N_i$ are the covering spaces corresponding to the subgroup $i\mathbb{Z}$, then there is a constant $C$ so that $M_i$ is diffeomorphic to $N_i$ after summing $C$ copies of $S^2\times S^2$. Moving to more general fundamental groups we are lead to study groups whose coset spaces have slender waists, and we show how this is tied to amenability. This project is joint work with M. Freedman and L. Guth.
12:15 - Ken Baker, New surgeries between the Poincare Sphere and lens spaces
We exhibit an infinite family of hyperbolic knots in the Poincare Homology Sphere with tunnel number 2 and a lens space surgery and discuss the implications. Notably, this is in contrast to the previously known examples due to Hedden and Tange which are all doubly primitive.
"Cercis Canadensis Leaf" by Klikini - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.
