RESÚMENES

1. Clasificación de nudos

Luis Celso Chan Palomo, Universidad Autónoma de Yucatán

 

El curso consistirá en ilustrar algunas técnicas combinatorias, algebraicas y geométricas que han sido útiles para estudiar la clasificación de los nudos. Algunos de los temas a cubrir serán: Historia de la clasificación de nudos, Conjeturas de Tait, Nudos primos, Los primeros 1,701,936 nudos, Álgebra Lineal y Nudos, Grupos y Nudos y finalmente Superficies y Nudos.


2. Introducción a la Homología de Khovanov

Ma. de los Angeles Guevara, Instituto Potosino de Investigación Científica y Tecnológica, IPICYT

 

Mikhail Khovanov, en busca de un invariante de enlaces más fuerte que el polinomio de Jones, introdujo la ahora llamada homología de Khovanov. Dicha homología determina si un nudo es el trivial, mientras que aún no es conocido si el polinomio de Jones puede hacer lo mismo.
El curso empezará con una breve introducción al polinomio bracket y al polinomio de Jones. Después, se definirá de manera elemental la homología de Khovanov y nos enfocaremos en dar ejemplos de cómo calcular la homología. Finalmente, si el tiempo lo permite, usaremos la homología de Khovanov como herramienta para calcular otros invariantes que miden que tan “lejos” están los enlaces de ser alternantes.



3. Thin position for knots and 3-manifolds

Alex Zupan, University of Nebraska-Lincoln

 

Lecture 1:  Thin position for knots in the 3-sphere

Abstract:  In a groundbreaking discovery in the late 1980s, David Gabai proved the Property R Conjecture for knots in the 3-sphere.  Along the way, he introduced a powerful new combinatorial tool:  thin position for knots.  Every knot K can have infinitely many different knot diagrams, and the crossing number of K is determined by finding a diagram with the smallest possible crossing number.  Similarly, a knot K has infinitely many distinct embeddings in 3-space, and thin position minimizes a different measure of complexity, called knot width, over all possible embeddings.  We will define width and go over the basics of this fundamental idea.

 

Lecture 2: Width complexes for knots

Abstract:  While thin position concerns only those embeddings of a knot that have the smallest possible knot width, there are many interesting questions related to the set of all embeddings.  For example, given any embedding of the unknot, is it possible to simplify the embedding to the standard one without ever increasing the width?  For the purpose of better understanding these types of structures, Jennifer Schultens introduced the width complex in 2008, an abstract structure relating all embeddings of a single knot.  We will talk about some interesting and surprising properties of this complex.

 

Lecture 3: Thin position and essential surfaces

Abstract:  One useful way to study 3-dimensional spaces is to to understand various surfaces embedded within them, and an interesting class of embedded surfaces are given by those surfaces known as essential, which have been “cut up and capped off” as much as possible.  One example of an essential surface is a minimal Seifert surface.  In 2005, Ying-Qing Wu proved that there is a strong connection between certain essential surfaces and thin position.  We will discuss this result, as well as the curious relationship between thin position and connected sums of knots.

 

Lecture 4: Generalizing thin position to 3-manifolds and beyond

Abstract: In the mid-1990s, Martin Scharlemann and Abigail Thompson synthesized Gabai’s thin position with the idea of weak reduction of Andrew Casson and Cameron Gordon to produce a powerful concept called thin position of 3-manifolds, spaces that locally resemble 3-dimensional real space.  We will define thin position of 3-manifolds and discuss some of the applications of this version of thin position.  Following the work of Scharlemann and Thompson, many other researchers developed new and interesting ideas of thin position in other contexts.  We will also discuss key aspects of these generalizations.