ENG 
ESP Session S28 - Knots, Surfaces, 3-manifolds
Talks
Wednesday, July 14, 16:00 ~ 16:30 UTC-3
Cyclic quadrilaterals and smooth Jordan curves
Joshua Greene
Boston College, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss the context and proof of the following result: for every smooth Jordan curve and for every four points on a circle, there exists an orientation-preserving similarity taking the four points onto the curve. The proof involves symplectic geometry in a surprising way.
Joint work with Andrew Lobb (Durham University).
Wednesday, July 14, 16:40 ~ 17:10 UTC-3
Taut foliations from double-diamond replacements
Rachel Roberts
Washington University in St Louis, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Suppose $M$ is an oriented 3-manifold with connected boundary a torus, and suppose $M$ contains a properly embedded, compact, oriented, surface $R$ with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)$. We define a readily recognizable type of sutured manifold decomposition, which for notational reasons we call double-diamond taut, and show that if $R$ admits a double-diamond taut sutured manifold decomposition, then for every boundary slope except one, there is a co-oriented taut foliation of $M$ that intersects $\partial M$ transversely in a foliation by curves of that slope. In the case that $M$ is the complement of a knot $\kappa$ in $S^3$, the exceptional filling is the meridional one; in particular, restricting attention to rational slopes, it follows that every manifold obtained by non-trivial Dehn surgery along $\kappa$ admits a co-oriented taut foliation. As an application, we show that if $R$ is a Murasugi sum of surfaces $R_1$ and $R_2$, where $R_2$ is an unknotted band with an even number $2m\ge 4$ of half-twists, then every manifold obtained by non-trivial surgery on $\kappa= \partial R$ admits a co-oriented taut foliation.
Joint work with Charles Delman (Eastern Illinois University).
Wednesday, July 14, 17:20 ~ 17:50 UTC-3
The Gordon-Litherland pairing for links in thickened surfaces
Hans Boden
McMaster University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce the Gordon-Litherland pairing for knots and links in thickened surfaces that bound unoriented spanning surfaces. Using the GL pairing, we define signature and determinant invariants and relate them to invariants derived from the Tait graph and Goeritz matrices. These invariants depend only on the $S^*$ equivalence class of the spanning surface, and the determinants give a simple criterion to check if the knot or link has minimal genus. The GL pairing is isometric to the relative intersection pairing on a 4-manifold obtained as the 2-fold cover along the surface. Time permitting, we will explain how to use the GL pairing to give a topological characterization of alternating links in thickened surfaces, extending the results of Josh Greene and Josh Howie.
Joint work with Micah Chrisman (Ohio State University, Marion) and Homayun Karimi (McMaster University).
Wednesday, July 14, 18:00 ~ 18:30 UTC-3
On classification of genus $g$ knots which admit a $(1,1)$-decomposition
Fabiola Manjarrez-Gutiérrez
UNAM, México - This email address is being protected from spambots. You need JavaScript enabled to view it.
Given an oriented minimal genus Seifert surface $F'$ for a $(1,1)$-knot $K$ it is possible to surger $F'$ along annuli to obtain a simple minimal Seifert surface $F$. Such a surface can be put in a very nice position with respect to the $(1,1)$-position of the knot $K$. Using this kind of surfaces we give a description of a $(1,1)$-knot of genus $g$ as a vertical banding of $(1,1)$-knots of genus smaller than $g$. In addition, we show that any rational knot of genus $g$ is obtained as a vertical banding of $g$ genus one rational knots.
Joint work with Mario Eudave-Muñoz (UNAM) and Enrique Ramírez-Losada (CIMAT).
Wednesday, July 14, 18:40 ~ 19:10 UTC-3
Meridionally essential one-sided spanning surfaces
Joshua Howie
University of California, Davis, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
The geography problem for spanning surfaces asks for a classification of all pairs of Euler characteristic and slope which can be realised by a spanning surface for a given knot in the 3-sphere. It is enough to understand the meridionally essential one-sided spanning surfaces, a somewhat larger class of surfaces than the geometrically essential spanning surfaces. We will discuss the existence of such one-sided surfaces, and give an algorithmic solution to the geography problem.
Wednesday, July 14, 19:20 ~ 19:50 UTC-3
Satellites and Lorenz knots
Jessica Purcell
Monash University, Australia - This email address is being protected from spambots. You need JavaScript enabled to view it.
We construct infinitely many families of Lorenz knots that are satellites but not cables, giving counterexamples to a conjecture attributed to Morton. We amend the conjecture to state that Lorenz knots that are satellite have companion a Lorenz knot, and pattern equivalent to a Lorenz knot. We show this amended conjecture holds very broadly: it is true for all Lorenz knots obtained by high Dehn filling on a parent link, and other examples.
Joint work with Thiago de Paiva (Monash University, Australia).
Thursday, July 15, 16:00 ~ 16:30 UTC-3
Prime quasi-alternating links are atoroidal
Cameron Gordon
University of Texas at Austin, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
A classical result of Menasco is that a prime non-split alternating link is either hyperbolic or a (2,q)-torus link. In 2005 Ozsvath and Szabo introduced the class of quasi-alternating links, which (properly) contains the non-split alternating links. We prove that Menasco's result holds for this more general class: a prime quasi-alternating link is either hyperbolic or a (2,q)-torus link.
Joint work with Steve Boyer (University of Quebec at Montreal) and Ying Hu (University of Nebraska Omaha).
Thursday, July 15, 16:40 ~ 17:10 UTC-3
The L-space conjecture and toroidal 3-manifolds
Steven Boyer
UQAM, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Hanselman, Rasmussen and Watson have characterised closed, toroidal, non-L-space 3-manifolds expressed as the union of manifolds with incompressible torus boundaries in terms of the gluing map. We discuss analogous results on the left-orderabilty of their fundamental groups suggested by the L-space conjecture together with some applications.
Joint work with Cameron McA. Gordon (University of Texas at Austin) and Ying Hu (University of Nebraska at Omaha).
Thursday, July 15, 17:20 ~ 17:50 UTC-3
Dynamical systems on hyperbolic groups
Yo'av Rieck
University of Arkansas , USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $G$ be an infinite hyperbolic group.
By a \em dynamical system \em on $G$ we mean an action of $G$ on a compact space $X$. The most commonly studied (and best understood) type of dynamical system, called SFT (subshift of finite type), is given by a closed, $G$-invariant subspace $X \subset A^G$, where $A$ is any finite set. We will explain these terms in the talk and show why an SFT on $G$ is essentially a ``tiling'' of $G$.
Gromov studied SFT's on $G$ in his original paper about hyperbolic groups, and much work on the subject was done by Coornaert and Papadopoulos. In particular, $G$ admits an SFT that can be used to study its action on its boundary.
A non-empty SFT is called \em strongly aperiodic \em if the stabilizer of every point is trivial. The question of which finitely generated, infinite groups admits a strongly aperiodic SFT has a long history, dating back to the foundational work of Wang and Berger in the 60's. Few groups are known not to admit one, and many are known to admit one; however, until the current work, the only hyperbolic groups that were known to admit a strongly aperiodic SFT were surface groups (Cohen and Goodman-Strauss).
In this talk we will describe the construction of a strongly aperiodic SFT when $G$ is one-ended, which is the key for the following result:
{\bf Theorem.} An infinite hyperbolic group admits a strongly aperiodic SFT if and only if it is one-ended.
Time permitting we will discuss further development related to this construction.
Joint work with David Bruce Cohen and Chaim Goodman--Strauss.
Thursday, July 15, 18:00 ~ 18:30 UTC-3
On non almost-fibered knots
Araceli Guzmán Tristán
CIMAT Guanajuato, México - This email address is being protected from spambots. You need JavaScript enabled to view it.
An almost-fibered knot is a knot whose complement possesses a circular thin position in which there is one and only one weakly incompressible Seifert surface and one incompressible Seifert surface. Infinite examples of almost-fibered knots are known. In this talk, we will show the existence of infinitely many hyperbolic genus one knots that are not almost-fibered.
Joint work with Mario Eudave Muñoz (Instituto de Matemáticas, UNAM) and Enrique Ramírez Losada (CIMAT, Guanajuato).
Thursday, July 15, 18:40 ~ 19:10 UTC-3
Berge Conjecture for tunnel number one knots
Tao Li
Boston College, U.S.A. - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $K$ be a tunnel number one knot in $M$, where $M$ is either $S^3$, $S^2\times S^1$, or a connected sum of $S^2\times S^1$ with a lens space. We prove that if a Dehn surgery on $K$ yields a lens space, then $K$ is a doubly primitive knot in $M$. For $M = S^3$ this resolves the tunnel number one Berge Conjecture. For $M = S^2\times S^1$ this resolves a conjecture of Greene and Baker-Buck-Lecuona for tunnel number one knots.
Joint work with Yoav Moriah (Technion, Israel) and Tali Pinsky (Technion, Israel).
Thursday, July 15, 19:20 ~ 19:50 UTC-3
Embeddability in $\mathbb R^3$ is NP-hard
Eric Sedgwick
DePaul University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
We prove that the problem of deciding whether a 2–or 3–dimensional simplicial complex embeds into $\mathbb R^3$ is NP-hard. This stands in contrast with the lower dimensional cases which can be solved in linear time, and a variety of computational problems in $\mathbb R^3$ like unknot or 3–sphere recognition which are in NP ∩ co-NP (assuming the generalized Riemann hypothesis). Our reduction encodes a satisfiability instance into the embeddability problem of a 3–manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings on link complements.
Joint work with Arnaud de Mesmay (CNRS, GIPSA-Lab, France), Yo'av Rieck (University of Arkansas, USA) and Martin Tancer (Charles University, Czech Republic)..
Friday, July 23, 16:00 ~ 16:30 UTC-3
Generalizing classical knot invariants
Maggy Tomova
The University of Iowa, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Abstract: Tunnel number and knot width are well-known and very useful invariants. They are however not additive. In this talk, I will present joint work with Scott Taylor of generalizations of these invariants that are additive.
Joint work with Scott Taylor (Colby College).
Friday, July 23, 16:40 ~ 17:10 UTC-3
Instantons and Knot Concordance
Juanita Pinzón Caicedo
University of Notre Dame, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots $K_0$ and $K_1$ are said to be smoothly concordant if there is a smooth embedding of the annulus $S^1 \times [0, 1]$ into the ``cylinder'' $S^3 \times [0, 1]$ that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set $\mathcal{C}$ of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. The algebraic structure of $\mathcal{C}$, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In particular, the study of anti-self dual connections on 4-manifolds can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to $\mathbb{Z}^\infty$, and (2) satellite operations that are similar to cables are not homomorphisms on $\mathcal{C}$.
Joint work with Matt Hedden (Michigan State University, USA) and Tye Lidman (North Carolina State University, USA).
Friday, July 23, 17:20 ~ 17:50 UTC-3
Flows, growth rates and veering triangulations
Yair Minsky
Yale University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The work of Thurston, Fried, McMullen, Mosher, Fenley and others weaves together a rich picture of fibrations and flows in 3-manifolds, linking growth rates of orbits, dilatations of pseudo-Anosov maps, and Thurston's norm on homology. Agol and Gueritaud introduced veering triangulations, which are ideal triangulations associated with (certain) pseudo-Anosov flows. We use these triangulations to construct a polynomial invariant that extends McMullen's Teichmuller polynomial from suspension flows to the more general setting. We develop a combinatorial model for the flow which, with the polynomial, permits explicit computations of growth rates of orbits in naturally defined subsets of the flow. As an application we obtain, for a fibered 3-manifold, a description of the limit set of the dilatations of fibrations belonging to a fibered face of Thurston's norm. This is joint work with Michael Landry and Sam Taylor.
Friday, July 23, 18:00 ~ 18:30 UTC-3
The Strong Slope Conjecture for Mazur pattern satellite knots
Kimihiko Motegi
Nihon University, Japan - This email address is being protected from spambots. You need JavaScript enabled to view it.
The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that a Mazur pattern satellite knots satisfy the Strong Slope Conjecture if the original knot does. Consequently, combining with previous results, any knot obtained by a finite sequence of cabling, connected sums, Whitehead doubling and taking Mazur pattern satellites of adequate knots (including alternating knots) or torus knots satisfies the Strong Slope Conjecture.
Joint work with Kenneth L. Baker (University of Miami) and Toshie Takata (Kyushu University).
Friday, July 23, 18:40 ~ 19:10 UTC-3
Guaranteed-quality triangular meshes
Joel Hass
UC Davis, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
All surfaces can be triangulated, but in applications one seeks triangulations that have nice regularity properties. For computer graphics, finite elements, morphing, image recognition and other uses, one would like to have triangles whose angles are bounded away from zero degrees, and even better as close to 60 degrees as possible. I will talk here about a new algorithm developed jointly with Maria Trnkova that improves previously obtained bounds. It produces a triangulation, or mesh, with all angles in the interval $[35.2^o, 101.5^o$].
Friday, July 23, 19:20 ~ 19:50 UTC-3
Trisections and link surgeries
Abigail Thompson
University of California, Davis, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I’ll discuss some natural conjectures that arise about integral surgeries on the links generated by trisected 4-manifolds.
Joint work with Robion Kirby, University of California, Berkeley.