Low Dimensional Topology Seminar

Fico González-Acuña

Welcome to Fico González-Acuña Low Dimensional Topology Seminar home page.

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

  • Jun 8
    11:30
    (U. of Saskatchewan)
    Surfaces in four-manifolds as viewed from curve complexes

    A knotted surface in a 4-manifold can be represented as a path in a graph whose vertices are pants decompositions. Understanding the lengths of these paths allows us to measure how complicated the knotted surface is. In this talk, we compute the minimum length of a path one needs to represent some surfaces from the knot table. This complexity measure is inspired by the work of Blair, Campisi, Taylor, and Tomova.

Past Talks

  • May 25
    11:00
    (San José State University)
    Kirby-Thompson Distance for Trisections of Knotted Surfaces

    We adapt the work of Kirby-Thompson and Zupan to define an integer invariant $L(T)$ of a bridge trisection $T$ of a smooth surface $K$ in $S^4$ or $B^4$ . We show that when $L(T) = 0$, then the surface $K$ is unknotted. We also show that for a trisection $T$ of an irreducible surface, the bridge number produces a lower bound for $L(T)$. Consequently, $L$ can be arbitrarily large.

  • May 18
    11:00
    (UC Davis)
    On Newton polytopes of Legendrian knots.

    Legendrian submanifolds are an important object of study in contact geometry. Wavefronts are an example of a Legendrian submanifold. A Legendrian knot in $(\mathbb{R}^3, \xi=ker(dz-ydx))$ is a smooth knot whose tangent vectors must lie in the contact hyperplanes $\xi$. As with smooth knots, Legendrian knots can bound surfaces in the four-ball. Such a surface is called a filling of the knot. For Legendrian knots we require the surfaces to be exact and Lagrangian. An important problem in contact topology is the classification of exact Lagrangian fillings of Legendrian knots in the standard contact 3-sphere up to Hamiltonian isotopy. In joint work with Roger Casals we show that Newton polytopes can be used to distinguish infinitely many exact Lagrangian fillings of Legendrian links in the standard contact $3$-sphere and higher dimensional Legendrian spheres in the standard contact $(2n+1)$ sphere up to Hamiltonian isotopy. We also show that there exist Legendrian links with infinitely many exact non-orientable Lagrangian fillings.

  • May 11
    11:00
    (California State University)
    The Meridional Rank Conjecture via Wirtinger number and Coxeter quotients.

    The meridional rank conjecture asks if the minimal number of meridians needed to generate the knot group of a knot K is equal to the bridge number of K. The question was originally posed by Cappell and Shaneson in the 1970s. In this talk, we will introduce an alternative definition of bridge number known as Wirtinger number. By combining Wirtinger number and Coxeter quotients of knot groups, we will establish the meridional rank conjecture for several classes of knots. We will also discuss computational results in which we apply these techniques to all tabulated prime knots with at most 16 crossings

  • Apr 27
    11:00
    (UCDavis)
    Multisection diagrams with divides for Weinstein 4-manifolds

    Trisections and multisections are a way of decomposing a 4-manifold into simple pieces that are glued together along simple pieces. Such a decomposition yields a way to encode a 4-manifold diagrammatically. When our 4-manifold has a symplectic structure, we would like to encode that diagrammatically as well. I will explain joint work with Gabriel Islambouli which defines a new compatibility between symplectic structures and multisections, and how we can encode such compatible decompositions diagrammatically.

  • Apr 13
    11:00
    (CINC-UAEM)
    Imagine non-compact surfaces gridded

    In this talk we will study non-compact topological surfaces (orientable and non-orientable) as polyhedra whose faces are square. Discrete models embedded in regular tessellations by cubes and hypercubes in Euclidean and hyperbolic spaces of dimension 3 and 4.

  • Mar 23
    11:00
    (FC-UNAM)
    On the Khovanov homology of alternating knots

    The relationships of the Khovanov homology of a knot with the spanning trees of the Tait graph associated with a knot diagram have been studied (for example, Champanerkar and Kofman) and these relationships are simpler in the case of alternating knots. In this talk I will talk about a work in progress with the thesis student Danae Castillo.

  • Mar 9
    11:00
    (UG)
    Characteristic numbers of a representation.

    Given a compact and oriented 3-manifold and a representation of its fundamental group in the complex general linear group, we can define the characteristic class associated with the representation by taking the Chern classes of the complex vector bundle associated with the representation. Such a class is an element in the de Rham cohomology of the manifold. Cheeger and Simon gave a similar definition for the so-called secondary characteristic classes, but, this time, the classes are defined on the cohomology of the manifold with coefficients on the complexes modulo the integers. It is possible to identify this cohomology with the homomorphisms of the respective homology to the complexes modulo the integers. Characteristic numbers are defined by evaluating the characteristic class of the representation in the fundamental class of the manifold. In this talk we will give a way to calculate characteristic numbers by means of the determinant of representation and Atiyah Patodi Singer’s index theorem. We will give, as an example, the first characteristic numbers of representations of the fundamental group of spherical 3-manifolds. On the other hand, the representation induces a homomorphism between the homology of the manifold and the homology of the general linear group. Again, evaluating the fundamental class of the manifold defines an invariant of it, for the case of rational 3-spheres of homology, the invariant is defined as an element in the third algebraic K-group of the complexes. Furthermore, we will give a construction that recovers the spectrum of the spherical 3-manifolds already mentioned.

  • Feb 23
    11:00
    (University of Warsaw)
    Khovanov and sl(N)-homology for periodic links

    I will describe the construction of a group action on Khovanov and sl(N)-homology of a periodic link. As an application, I will show a periodicity criterion from these theories. I will also discuss a relation of Khovanov homology of a periodic link and its quotient. This is an account of a joint work with Politarczyk, Silvero and Yozgyur

  • Feb 2
    11:00
    (UCDavis)
    Knots, Graphs and Surfaces

    Any connected, finite graph can be imbedded in the 3-dimensional sphere so that its complement has free fundamental group. Some graphs can be imbedded so that the complement of every subgraph also has free fundamental group. Planar graphs have such imbeddings, and so do some non-planar graphs, such as the complete graph on 5 vertices. However some graphs, such as the complete graph on 7 vertices, $K_7$, are intrinsically knotted; i.e., no matter how $K_7$ is imbedded in the 3-sphere, it contains a knotted cycle. I’ll talk about an integer invariant that measures how intrinsically knotted a graph is, and relate this invariant to a long-standing conjecture in graph theory, called the orientable cycle double cover conjecture.

See all past talks

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