# Low Dimensional Topology Seminar

## Fico González-Acuña

This seminar is intended to promote collaboration among the LDT community. The seminar is held twice a month and it is broadcasted to remote participants via BlueJeans.

Please subscribe to our mailing list to get a BlueJeans email invitation to future talks and notifications about related events.

We have an associated Google Calendar to this seminar, you may add it to your Google Calendar and consult the dates and times in which talks are scheduled. They are usually on Thursdays at 11:00 AM (Mexico City Time).

If you use a different calendar program, you can download the ics file.

# Upcoming Talks

• Jan 22
10:00
##### Low-dimensional topology and non-Euclidean geometry in nature

In the talk I demonstrate on specific examples the emergence of a new actively developing field, the “statistical topology”, which unifies topology, noncommutative geometry, probability theory and random walks. In particular, I plan to discuss the following interlinked questions: (i) statistics of random walks on hyperbolic manifolds and graphs in connection with the topology and fractal structure of unknotted long polymer chain confined in a bounding box and hierarchical DNA folding, and (ii) optimal embedding in the three-dimensional space of exponentially growing tissues, like, for example, the salad leaf, and how the hierarchical ultrametric geometry emerges in that case.

# Past Talks

• Dec 17
11:00
##### Isotopy classes of relatively trisected 4-manifolds with boundary

A relative trisection of a smooth, compact, oriented 4-manifold with boundary X is a decomposition of X into three diffeomorphic pieces which have “nice” intersection properties. The trisection induces an open book decomposition on the boundary, which is a surface bundle over $S^1$ in the compliment of a link in $\partial X$. It is known that every such 4-manifold admits a trisection and that any two trisections can be made isotopic after suitable “stabilization” operations. In this talk, I will show that any two diffeomorphic relative trisections of the 4-ball which induce isotopic open books on the boundary 3-sphere are in fact isotopic trisections. An interesting feature of the argument is that we do not show that the original diffeomorphism is isotopic to the identity! I will give a good deal of background on trisections, trisection diagrams, and open books. If time permits, I will discuss some practical features of relative trisections which allow us to classify low-“complexity” relative trisections. This work is joint with Patrick Naylor.

• Dec 3
10:00
##### Genus one hyperbolic knots in the 3-sphere and the Kakimizu complex

The Kakimizu complex $MS(K)$ for a knot $K\subset\mathbb{S}^3$ is the simplicial complex with simplices the collections of isotopy classes of minimal genus Seifert surfaces in the knot exterior that can be realized by mutually disjoint and non-parallel such surfaces.

For genus one hyperbolic knots $K\subset\mathbb{S}^3$ the optimal bounds for the dimension and diameter of the complex $MS(K)$ are known to be $4$ and $2$, respectively. We refine these results by showing that, in the top dimension, the complex ${MS}(K)$ consists of at most $2$ simplices, and exactly one $4$-simplex in the $4$-dimensional case. We also provide infinitely many examples of such knots for which $MS(K)$ consists of exactly one or exactly two top-dimensional simplices.

• Nov 20
10:00
##### A landscape of knots: TDA and dimension reduction techniques in theoretical mathematics

A multitude of knot invariants, including quantum invariants and their categorifications, have been introduced to aid with characterizing and classifying knots and their topological properties. Relations between knot invariants and their relative strengths at distinguishing knots are still mostly elusive. In addition to Ball Mapper, and machine learning techniques, we develop a new approach using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. Although of different origins, these methods confirm and illuminate similar substructures in knot data created for 10 million knots up to 17 crossings

• Nov 19
11:00
##### Comparación de 4-géneros topológicos y suaves de nudos satélites.

(Talk in spanish) El estudio de objetos de dimensión 4 es especial: una variedad de dimensión 4 puede admitir infinitas estructuras suaves no equivalentes, y variedades de dimensión 4 pueden ser homeomorfas pero no difeomorfas. Esta diferencia entre estructuras topológicas y suaves se puede abordar en términos del estudio de los nudos como fronteras de superficies embebidas en la 4-bola. En esta charla, me centraré en operadores satelitales de nudos y mostraré que los satélites pueden ser frontera de superficies muy diferentes en la categoría suave y en la topológica. Este es un trabajo conjunto con Allison Miller y Peter Feller.

• Nov 5
11:00
##### Rigidez topológica y geométrica de espacios de Alexandrov

Los espacios de Alexandrov son espacios métricos, no necesariamente suaves, que admiten curvatura (seccional) acotada por debajo. En esta plática hablaré de la topología y geometría de espacios de Alexandrov dimensión 3 que están “suficientemente colapsados respecto a su diámetro”. Más precisamente mostraré que, dependiendo del factor de colapso, esta familia de espacios satisface la conjetura de Borel o estos espacios están modelados en alguna de las geometrías de Thurston. Estos resultados son conjuntos con Noé Bárcenas, Fernando Galaz-García y Luis Guijarro.

• Oct 22
11:00
##### Generalizing classical knot invariants

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.

• Oct 8
11:00
##### Un polinomio tipo de Alexander para doodles.

Un doodle es una inmersión de un número finito de círculos en la 2-esfera sin intersecciones triples. En esta charla platicaremos un poco acerca de estos objetos anudados planos, además de una versión plana del grupo de trenzas, el llamado twin group. Describiremos la construcción de un invariante polinomial para doodles a través de una representación del twin group y los polinomios de Chebyshev de segundo tipo. Por su construcción y comportamiento, este invariante es un análogo al polinomio de Alexander. Este es un trabajo realizado en colaboración con Bruno Cisneros, Marcelo Flores y Jesús Juyumaya.

See all past talks

The mailing list for this seminar is a Google Group.