# 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

• Apr 30
11:00
##### Colored knots and 3-manifolds

I will explain what colored knots are and how every colored knot gives rise to a 3-manifold. Also give an idea of the proof, via Lickorish twists, of how every 3-manifold is obtained this way.

# Past Talks

• Apr 15
11:00
##### "String representation" of trivalent 2-stratifolds

(Talk in Spanish)

Two dimensional stratifolds are topological objects that have been of particular interest in recent years, due to its possible applications to TDA and other fields of knowledge. However, we really don't know much about them yet. In this talk, we focus only on simply connected 2-stratifolds.

Inspired by the surface classification theorem and based on the algorithm already developed to build this type of stratifolds, we designed an algorithm that not only constructs them but also classifies and counts them. The heart of this algorithm is the invariant called string representation which allows us to identify each trivalent 2-stratifold in a unique way by means of a character string. In this talk, we will explain this algorithm and the invariant of interest.

This is joint work with Jesus Rodriguez-Viorato.

• Mar 25
17:00
##### Multibranched surfaces in 3-manifolds

A multibranched manifold is a second countable Hausdorff space that is locally homeomorphic to multibranched Euclidean space. In this talk, we concentrate compact 2-dimensional multibranched manifolds (multibranched surfaces) embedded in 3-manifolds. We give a necessary and sufficient condition for a multibranched surface to be embedded in some closed orientable 3-manifold. Then we can define the genus of a multibranched surface in virtue of the Heegaard genera of 3-manifolds, and show an inequality between the genus, the number of branch loci and regions. We determine whether two multibranched surfaces have the same neighborhood by means of local moves. Similarly to the graph minor, we also introduce a minor on multibranched surfaces, and consider the obstruction set for the set of multibranched surfaces embedded in the 3-sphere. This talk is a survey including recent joint works with Kazufumi Eto, Shosaku Matsuzaki, Mario Eudave-Munoz, Kai Ishihara, Yuya Koda, Koya Shimokawa.

• Mar 11
11:00
##### Knots, Bands, Integrals, and a little bit of DNA

(Talk in Spanish) In this talk we will talk about a topological invariant, the link number, and its relationship with the Gaussian integral. Then we will talk about two related geometric invariants, one we will call “the total twist of a band” and the other “average Tait number of the knot”. These two geometric invariants are computed by integrals, and we will explore some examples. Also, very briefly, we will consider two ways of modeling a circular DNA molecule, one as a knot, the other as a band, and their relationship to the previous invariants. If time allow us, I will say something about the total curvature of a knot.

• Feb 26
11:00
##### The splitting genus of Alexander split links

We say 2-component link L in $\mathbb{S}^3$ is “Alexander split” if its Alexander polynomial is zero. It turns out that L is Alexander split exactly when the maximal abelian cover of its exterior has non-zero $H_2$. In fact, with its natural module structure, this $H_2$ has rank 1. We define the splitting genus of L to be the minimal genus of surfaces representing a generator. I’ll discuss the development of this invariant, fundamental results, and potential applications. Parts are joint work with Chris Anderson.

• Feb 11
11:00
##### On alternating closed braids

(Talk in Spanish)

Due to Alexander’s theorem, it is well known that any link can be represented as a closed braid. Besides, a link is alternating or non-alternating depending on whether it possesses an alternating diagram or not. However, there are alternating links that cannot be represented as alternating closed braids. In this talk, we shall discuss the set of links that can be represented as alternating closed braids, and we shall introduce invariants that measure how far the links are from this set. We will show the relations of these invariants with others as the unknotting number and the alternation number. Furthermore, we will give the value of these invariants for some knot families. This work is partially joint with A. Kawauchi.

• 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.

• 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.

See all past talks

The mailing list for this seminar is a Google Group.