The Speakers

Title: Exponential Networks: The Resolved Conifold

Abstract: Building on J. Walcher’s introduction to exponential networks, I will discuss the resolved conifold, the example most relevant for applications to knot invariants. I will explain how the BPS spectrum is realized in terms of finite webs and conclude with some ideas on extending these methods to general knots.

Title: Revisiting 3d Modularity

Abstract: A few years ago, new topological invariants of closed three-manifolds, based on 3d SCFT obtained by compactifying on these manifolds, have been proposed. Subsequently, we have observed an interesting relation between quantum modular forms and vertex operator algebras and these so-called \hat Z -invariants. In this talk I will revisit a few key aspects of this relation between \hat Z -invariants and modular-type objects, including the role of the SL_2(Z) representation, the mock modular invariants, and the connection to vertex operator algebras. This talk is based on joint work with Ioana Coman, Davide Passaro, Piotr Kucharski and Gabriele Sgroi.

Title:  Introduction to colored knot homology

Abstract: I will give a pedagogical review of colored knot homology theories. These categorifications of colored knot polynomials admit several constructions (not all equivalent!), have interesting behaviour under framing changes, and feature in the computation of skein invariants of smooth 4-manifolds.

Title: Essential surfaces in link exteriors and link Floer homology

Abstract:  Knot/link Floer homology is a link invariant package, introduced independently by Ozsvath-Szabo and Rasmussen, has been shown to be quite useful to solve a number of questions in low dimensional topology in the last two decades. Although it is not a complete invariant of knots/links, a number of knots and links have been shown to be detected by this toolbox. The center of most of these results have been careful examination of certain essential surfaces in the knot/link exteriors and observing that operations on those surfaces can be kept track by the link/knot Floer homology of those knots/links. In this talk, we will be talking about those results and some new ones. This is based on joint work with Fraser Binns, some of which is ongoing.

Title: Twisted links, stability, and quivers

Abstract: Using the language of Knots-Quivers correspondence, we will discuss how HOMFLY-PT homologies behave under full twist insertion in a twist region of a knot or a link. Along the way, we will discover unexpected

relations between quivers for knots and links which differ by crossing resolution.

Title: 3d quantum trace map

Abstract: Let Y be an ideally triangulated 3-manifold. In this talk, I will describe the 3d quantum trace map, a homomorphism from the SL_2 skein module (i.e. quantization of SL_2 character variety) of Y to its quantum gluing module (i.e. quantization of Thurston's gluing variety), thereby giving a precise relationship between the two quantizations. This map, whose existence was conjectured earlier by Agarwal, Gang, Lee, and Romo, is a natural 3-dimensional analog of the 2d quantum trace map of Bonahon and Wong. This talk is based on arXiv:2403.12850 (joint work with Sam Panitch). 

Title: Black swans in complex Chern-Simons theory

Abstract: TBA

Title: Khovanov homology for knots in RP^3

Abstract: The beauty of the theory of knots is the multitude of interpretations it has. A knot in a three-manifold can be seen from several different perspectives starting from, the singularity for a foliation of the space around it, to. a Wilson line in a 2+1 spacetime. Each three-manifold has a unique knot theory. In this talk we will try to understand the knot theory of RP^3 using the theory of virtual knots. We will construct a Jones polynomial and a Khovanov homology theory for the knots in RP^3. Thus we show that virtual knot theory can be used as a powerful tool to study knot theory of other three manifolds. We will also discuss some features of knots in RP^3 which are quite challenging to understand.

Title: Lasagna s-invariant detects exotic $4$-manifolds

Abstract: We introduce a generalization of Rasmussen's $s$-invariant, called the lasagna $s$-invariant, which assigns either an integer or $-\infty$ to each second homology class of a smooth $4$-manifold. The construction is based on the construction of skein lasagna modules by Morrison-Walker-Wedrich. We present a few properties enjoyed by lasagna $s$-invariants, and we show that they detect the exotic pair of knot traces $X_{-1}(-5_2)$ and $X_{-1}(P(3,-3,-8))$, an example first discovered by Akbulut. This gives the first gauge/Floer-theory-free proof of the existence of compact orientable exotic $4$-manifolds. This is joint work with Michael Willis.

Title: HL Cone, Foams, and Graph Coloring

Abstract: We begin with a review of modern perspective on graph coloring which appeared in the work of Kronheimer-Mrowka and Khovanov-Robert. Next, we outline how the work of Treuman-Zaslow and Caslas-Zaslow lead to seeing graph coloring as topological defects labelled by the elements of Klein-Four Group. This highlights the quantum nature of graph coloring, namely, it satisfies the sum over all the possible intermediate state properties of a path integral. In our case, the topological field theory (TFT) with defects gives meaning to it. This TFT has the property that when evaluated on a planar trivalent graph, it provides the number of Tait-Coloring of it. Defects can be considered as a generalization of groups. With the Klein-four group as a 1-defect condition, we reinterpret graph coloring as sections of a certain cover, distinguishing a coloring (global-sections) from a coloring process (local-sections), and give a new formulation of some of Tait's work.

Title: Quantum trace and length conjecture

Abstract: I will review the quantum trace for hyperbolic knots and its relation with the 'length' conjecture and I will comment on some new developments.