Low Dimensional Topology and Circle-valued Morse Functions

Luis Valdez, The University of Texas at El Paso: The Kakimizu Complex for Genus One Hyperbolic Knots in the 3-Sphere

Kakimizu complex $MS(K)$ for a knot $K$ in the 3-sphere $\mathbb{S}^3$ is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the knot exterior, where a collection of vertices forms a simplex if they have mutually disjoint representatives in $X_K$. The Kakimizu complex of a hyperbolic knot is a full, finite and contractible simplicial complex. In this talk we discuss the structure of the Kakimizu complex for genus one hyperbolic knots in detail. For instance, if $d$ is the dimension of $MS(K)$ then $0\leq d\leq 4$ and there are infinitely many knots realizing the upper bound $d=4$. For $d=2,3$ the complex $MS(K)$ consists of one or two $d$-simplices sharing a common $d-1$-face, while for $d=0,4$ the complex is a single $d$-simplex. We also present examples in each dimension that produce the distinct types of the complex $MS(K)$.  Video

Alex Zupan, University of Nebraska-Lincoln: Circular Morse functions and Handle-Ribbon Knots

A wonderful theorem of Casson and Gordon asserts that for a fibered knot K in the 3-sphere, K is homotopy-ribbon (a condition in between ribbonness and sliceness) if and only if a specific surface diffeomorphism, the "closed monodromy" of K, can be extended over a handlebody.  In this talk, we extend this result to non-fibered knots K, proving that K is handle-ribbon (another condition in between ribbonness and sliceness, slightly more restrictive than homotopy-ribbonness) if and only if the exterior of K has a singular fibration (determined by a circular Morse function) that extends over handlebodies. This is joint work with Maggie Miller.  Video

Mario Eudave-Muñoz, Universidad Nacional Autónoma de México: On Non Almost-Fibered Knots

Let K be a knot in the 3-sphere. A regular circle-valued Morse function on the knot complement induces a handle decomposition on the knot exterior. Starting with a Seifert surface F, we can add 1-handles, 2-handles, then again 1-handles, 2-handles and so on. After adding 1-handles and before adding 2-handles we get a surface Si, called a thick surface, and after adding 2-handles and before adding 1-handles we get a surface Fi, called a thin surface. The complexity of the decomposition measures the complexity of the thick surfaces Si. A circular thin position of the knot exterior is one with minimal complexity, and by results of Fabiola Manjarrez-Gutierrez, this gives a sequence of Seifert surfaces that are alternately incompressible (the thin surfaces) and weakly incompressible (the thick surfaces). A fibered knot is then a knot whose exterior has a circular thin position with one and only one incompressible level surface and no weakly incompressible level surface. An almost-fibered knot is a knot whose complement possesses a circular thin position in which there is exactly one thick surface and one thin surface. There are many known families of almost-fibered knots.
We consider families of hyperbolic genus one knots which have 3, 4 or 5 disjoint Seifert surfaces of genus 1, and show that the knots are not almost-fibered, that is, they have a circular thin position which contains more than one thick surface and thin surface. This is a joint work with Araceli Guzman-Tristan, Fabiola Manjarrez-Gutierrez and Enrique Ramirez-Losada.  Video

Zhenkun Li, Stanford University: A Surgery Formula in Instanton Floer Theory

Instanton Floer homology is introduced by Floer in 1980s. It is a powerful invariant for 3-manifolds and knots and links inside them. There have been many important applications of Instanton Floer homology, such as the approval of Property P conjecture. It has been conjectured that the instanton Floer homology is isomorphic to other versions of Floer theory. Though this conjecture is still widely open, one could ask whether some important properties that have been known to be true in other Floer theory also hold for instanton theory. One such property is the surgery formula, which relates the instanton Floer homology of a 3-manifold coming from Dehn surgeries with the instanton Floer homology of the knot. In this talk, we will present a surgery formula for instanton theory, and describe how this formula can be applied in computing instanton Floer homology and study the SU(2)-representations of fundamental groups of 3-manifolds.  Video

Christine Lee, Texas State University: Characterizing the Khovanov Complex for Infinite Torus Braids.

The Khovanov homology of torus braids on n strands is a central object in quantum topology. By Rozansky, the direct limit of the complex of infinitely many full twists on n strands recovers the categorified Jones-Wenzl projector, and they feature in conjectures by Gorsky-Oblomkov-Rasmussen-Shende, which relate the Khovanov-Rozansky homology of an algebraic link to the Hilbert scheme of points on its defining complex curve. Despite substantial progress on these conjectures, the differentials of the conjectured forms of the complex remain difficult to determine. In this talk, we discuss recent work, joint with Carmen Caprau, Nicolle Gonzalez, Radmila Sazdanovic, and Melissa Zhang, that allows us to characterize a complex simplified from the Khovanov complex of torus braids on n strands. As an application, we give a formula for the Khovanov homology of torus braids on 3-strands and show that Plamenevskaya’s invariant for closed braids with sufficiently large fractional Dehn twist coefficient does not vanish.  Video

Gage Martin, Massachusetts Institute of Technology: Annular Links, Double Branched Covers, and Annular Khovanov Homology

Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.  Video

Ian Agol, UC Berkeley: Guts in Sutured Decompositions and the Seifert Genus

Given a knot complement, one may decompose along a maximal collection of disjoint minimal genus Seifert surfaces, then remove product bundles by cutting along product annuli. We prove that the resulting sutured manifold is unique up to isotopy, independent of the choice of Seifert surfaces. This shows that the dimension of maximal simplices of the Kakimuzu complex is constant. More generally, one has a similar result for Thurston norm-minimizing surfaces realizing a second homology class in certain 3-manifolds.  This is joint work with Yue Zhang.  Video

Yue Zhang, UC Berkeley: Applications of Guts on Minimal Volumes of Hyperbolic Manifolds and Sufficiently Long Dehn Fillings

This talk consists of two applications of guts in 3-manifolds. 
The minimal volume of orientable hyperbolic manifolds with a given number of cusps has been found for 0,1,2,4 cusps, while the minimal volume of 3-cusped orientable hyperbolic manifolds remains unknown. By using guts in sutured manifolds and pared manifolds, we are able to show that for an orientable hyperbolic 3-manifold with 3 cusps such that every second homology class is libroid, its volume is at least 6 Catalan’s constant. 
By using guts, we also develop a method for controlling the upper bound of the Euler characteristic of surfaces in sufficiently long Dehn fillings which has many applications for links and 3-manifolds. This is joint work with Ian Agol.  Video

Jennifer Schultens, UC Davis: Flipping Heegaard Splittings

Heegaard splittings are usually studied up to isotopy of their splitting surfaces.  In several contexts, it is of interest to study them up to oriented isotopy of their splitting surfaces.  A natural question that then arises is whether or not the two sides of a Heegaard splitting can be exchanged or ``flipped" via such an isotopy.   We will consider the context in which this question was first raised and discuss examples where it can be answered.  Video

Fabiola Manjarrez-Gutiérrez, Universidad Nacional Autónoma de México: Bounds on the Handle Number of Sutured Manifolds

Developed from geometric arguments for bounding the Morse-Novikov number of a link in terms of its tunnel number, we obtain upper and lower bounds on the handle number of a Heegaard splitting of a sutured manifold $(M, \lambda)$ in terms of the handle number of its decompositions along a surface representing a given 2nd homology class. Fixing the sutured structure $ (M, \lambda)$, this leads us to develop the handle number function $h: H_2(M, \partial M; \mathbb{R}) \rightarrow \mathbb{N}$ which is bounded, constant on rays from the origin, and locally maximal. This is joint work with Kenneth L. Baker.  Video

Dave Futer, Temple University: In Search of the Margulis Constant

The (3-dimensional) Margulis constant is the largest universal constant epsilon with the following property. If two isometries of hyperbolic space H^3 move a common point by distance less than epsilon, and generate a discrete group, then the isometries must commute. That such a universal constant exists is a beautiful and important theorem of Margulis from over 50 years ago. This result and the closely related thick-thin decomposition is used for many structural theorems in hyperbolic geometry. However, its value is still unknown. I will describe a massively computer-assisted project, which seeks to pin down the value of the Margulis constant by searching through a parameter space of isometries. While we do not yet have a complete answer, we do have a theorem if one adds a symmetry hypothesis. This symmetric result can likely be used in a bootstrapping argument for the general case of this problem. Joint work with David Gabai and Andrew Yarmola.  Video

Tao Li, Boston College: Taut Foliations of 3-Manifolds with Heegaard Genus Two

Let M be a closed, orientable, and irreducible 3-manifold with Heegaard genus two. We prove that if the fundamental group of M is left-orderable then M admits a co-orientable taut foliation.  Video

Steve Boyer, Université du Québec à Montréal: Homeo_+(S^1) Representations and the L-Space Conjecture

The asymptotic behaviour of foliations and flows on 3-manifolds often lead to representations of their fundamental groups with values in Homeo_+(S^1). Motivated by the L-space conjecture, we discuss how such representations can be used to verify the left-orderability of 3-manifold groups in a number of interesting situations (e.g. toroidal integer homology spheres, surgeries on knots, cyclic branched covers of hyperbolic links). We also discuss the extent to which the analogous results for the existence of taut foliations hold, as predicted by the L-space conjecture.  This is joint work with Cameron Gordon and Ying Hu.  Video

Allison Moore, Virginia Commonwealth University: Tangle Decompositions and Immersed Curves

A tangle decomposition along a Conway sphere breaks a knot or link into simpler pieces, each of which is a two-string tangle. We will discuss some of the ways in which Khovanov homology can be approached and calculated using tangle decompositions. In particular, the algebraic invariants associated with tangles can be translated into sets of immersed curves on the four-punctured sphere. This strategy turns out to be quite useful for investigating two classic open problems: the cosmetic surgery conjecture and the cosmetic crossing conjecture. This is joint with Kotelskiy, Lidman, Watson and Zibrowius.  Video

Makoto Ozawa, Komazawa University: Forbidden Minor Multibranched Surfaces and Critical Complexes

We show that all $K_5 \times S^1$ and $K_{3,3} \times S^1$ families are forbidden minor for multibranched surfaces embeddable in the 3-sphere $S^3$. Unlike previously known examples of forbidden minors, those examples except for two are not critical for $S^3$, where we say that a complex is {\em critical} for $S^3$ if it cannot be embedded in $S^3$ but after removing any one point, it can be embedded. We exhibit all critical complexes which are contained in $K_5 \times S^1$ and $K_{3,3} \times S^1$ families. In general, we show that any forbidden minor multibranched surface contains a critical complex which is a union of a multibranchedsurface and a (possibly empty) graph. On the other hand, we show that the cone over $K_5$ cannot be embedded in $S^3$, but it does not contain any critical complex. From those examples, a partially ordered set of complexes naturally arises, and some properties are studied. Finally, we will classify all critical complexes which have a form $(G\times S^1)\cup H$, where $G$ and $H$ are graphs.This is a joint work with Mario Eudave-Muñoz.  Video

Rachael Roberts, Washington University in St. Louis: Morse Normal Form for Branched Surfaces

I will discuss Morse normal form for branched surfaces and its use in constructing taut foliations. The results build on earlier work of Charles Delman and this work is joint with Delman.  Video

Scott Taylor, Colby College, University of Central Florida: Handle Structures and Additivity

We’ll discuss our work over the past decade, developing a thin position theory that works for knots, links, spatial graphs, and orbifolds. A key property of the theory is that it produces “cut”-incompressible surfaces, allowing us to apply it to questions regarding the additivity and non-additivity of certain knot invariants. After giving an overview of the theory, we’ll describe applications to some of width, tunnel number, and bridge number.  Video

Problem Session