| Speaker: Ryan Budney (U.Vic) |
| Title: The type-2 invariant |
| Abstract: I will give a brief discussion of Vassiliev's work on finite-type invariants, Birman's reformulation and a few geometric descriptions of the type-2 invariant of knots. |
| Location: COR B129, 4pm--5pm |
| Speaker: Ryan Budney (U.Vic) |
| Title: A variation on the type-2 invariant |
| Abstract: The idea behind the type-2 invariant, when extrapolated to higher-dimensional manifolds gives rise to an interesting invariant of mapping-class groups of manifolds. This presentation will survey such results. |
| Location: COR B129, 4pm--5pm |
| Speaker: Andrea Marino |
| Title: Embedding Calculus - Taylor Approximations for 'Geometric' Functors |
| Abstract: The talk aims to give an overview of the Embedded Calculus introduced by Goodwillie and Weiss. It is a powerful technique that can be used, among others, to approximate spaces of knots with configuration spaces. The framework is the following. Suppose X is a presheaf of spaces over a manifold M, and assume the restrictions X(U) → X(V) are homotopy equivalences whenever the inclusion V→U is a (sort of) homotopy equivalence. Can we determine the homotopy type of X(M) from the homotopy type of X on small disks? The answer is yes, whenever its “Taylor Tower” converges! We will see how cubical formalism helps us define the n-th derivatives, polynomial degree, and the Taylor Tower of presheaves. If time permits, we will apply the machinery to the space of knots in dimension 4 and higher. With some (deep) formality results, such a technique yields the rational cohomology of knot spaces. |
| Location: COR B129, 4pm--5pm |
| Speaker: Andrea Marino |
| Title: A combinatorial description of the Goodwillie-Sinha Spectral Sequence in terms of Fox-Neuwirth Trees |
| Abstract: Building on the previous seminar, we will present the Sinha’s description of the Goodwillie-Weiss Tower for Knot Spaces in terms of (Compactified) Configuration Spaces. The (co)homology of the latter admit a combinatorial presentation in terms of the so-called Fox-Neuwirth Trees. It turns out that the Sinha Spectral Sequence in (co)homology can be defined itself in terms of a degree-wise finite-dimensional bicomplex built from Fox-Neuwirth Trees. This equivalence stems from a geometric connection between Configuration Spaces and their Compactifications, through a series of Spaces based on "Weighted Fox-Neuwirth Trees". If time permits, we will show the connections of Sinha Spectral Sequence with Vassiliev Invariants and Formality of Little Disk Operads. This is joint work with Paolo Salvatore. |
| Location: COR B129, 4pm--5pm |
| Speaker: Andrea Marino |
| Title: Non-collapse of Sinha Spectral Sequence mod 2 |
| Abstract: The Sinha Spectral Sequence is a central object in Knot Theory, approximating the cohomology of the space of knots modulo immersions and providing insights into finite-type invariants. Exploiting the Fox-Neuwirth stratification of configuration spaces, we construct a combinatorial basis for this spectral sequence. However, the resulting "Barycentric Fox-Neuwirth" structure is prohibitively large for explicit computations. Using homotopy transfer ideas, we deform the bicomplex into a smaller multicomplex over characteristic 2 and ambient dimensions 2, 3. While the resulting differentials become more intricate, they remain tractable enough to demonstrate that the spectral sequence does not collapse at page 2. This result shows that formality techniques used in the rational case cannot be extended to finite-type invariants mod 2. This is joint work with Paolo Salvatore. |
| Location: COR B129, 4pm--5pm |
| Speaker: Nicolas Freches (Aachen) |
| Title: The Palais-Smale condition in geometric knot theory |
| Abstract: In geometric knot theory, we investigate questions from classical (i.e. topological) knot theory using methods from analysis. We study self-repulsive potentials on spaces of closed, injective and regular curves. By understanding the energy landscape of such potentials, we aim to gain information about the topology of the underlying knot classes. We construct the manifold of closed, embedded curves parametrized by arc length and show, that the famous Palais-Smale condition holds on this manifold for various families of knot energies. An application is given by a longtime existence and strong subconvergence result for the respective Sobolev gradient flow, as well as existence of energy minimizers in every knot class. |
| Location: DTB A203, 1pm--2pm |
| Speaker: Ryan Budney |
| Title: A 2-torsion invariant of 2-knots. |
| Abstract: I will describe an invariant of knots that makes sense in all dimensions. For embeddings of the 2-sphere in S^4, it takes its values in the group of order 2. Perhaps the most interesting feature of this invariant is it can be defined from the "double-point diagram" of a knot. In dimension 3, any invariant that satisfies a Skein relation can be defined from its double-point diagram (chord diagram), but as far as I know this is the first invariant of 2-knots definable from the double-point diagram. |
| Location: DTB 425a 1pm--2pm |
| Speaker: Josh Howie (Monash) |
| Title: Essential Checkerboard Surfaces Of Some m-Almost Alternating Knots |
| Abstract: Essential surfaces can help us understand the geometry and topology of 3-manifolds. It is a classical result that the checkerboard surfaces associated to an alternating knot are essential. We will prove that certain generalisations of alternating knots also have essential checkerboard surfaces. The existence of these pairs of surfaces will allow us to show that these knots satisfy a conjecture about the lengths of meridian curves in hyperbolic knot complements. There is also a connection to the Neuwirth conjecture which aims to explain topologically how a knot group splits as an amalgamated free product. |
| Location: DTB 203 2pm--3pm |