2026 Lectures
Plenary Lectures
From Seifert Forms to Branched Covers of \(B^4\)
Alexandra Kjuchukova
University of Notre Dame
Given a knot \(K \subset S^3\), when does a surjective homeomorphism \(\rho: \pi_1(S^3\backslash K) \twoheadrightarrow G\) extend over the exterior of a surface \(F \subset B^4\) with \(\partial F = K\)? I’ll present a general obstruction theory answering this question. The focus of the lectures is on the dihedral case, \(G = D_n\), where the obstruction reduces to a Seifert-matrix computation, and, when it vanishes, the surface \(F\) can be constructed explicitly. In the first lecture I will develop the needed tools: Seifert forms, branched covers, the linking form on \(\Sigma_2 K\), and Cappell--Shaneson characteristic knots, with twist knots as a running example. In the second lecture I’ll introduce the obstruction space, specialize to dihedral groups, and give an explicit construction of \(F\) when the obstruction vanishes. On the side, I’ll establish the ribbonness status of all 3-colorable twist knots using the \(\Xi_3\) invariant of dihedral covers.
Interesting problems about the Jones polynomial
Christine Ruey Shan Lee
Texas State University
The Jones polynomial is the link invariant that started it all for the field of quantum topology. Despite numerous applications such as the resolution of the Tait conjectures, the construction of Khovanov homology leading to applications to problems in \(3\)-and \(4\)-manifold topology, we still cannot answer many fundamental questions about the Jones polynomial. In this talk, I will give an overview on quantum invariants for knots and links, and \(3\)-manifolds. I plan to discuss outstanding conjectures about quantum link invariants relevant to my research. This includes the Jones unknotting conjecture, the volume conjecture, the AJ conjecture, and the slope conjecture.
A whittled complex for the Khovanov homology of torus braids
Christine Ruey Shan Lee
Texas State University
We give an algorithm to reduce the number of generators of the Khovanov chain complex of torus braids \((\sigma_1 \sigma_2 \cdots \sigma_{n-1})^k\) on \(n\) strands. I will begin the talk with context on the stable Khovanov homology of torus links leading to the open question of the structure of their homology theory, as well as potential applications to open questions concerning the colored Jones polynomial. Next I will discuss our work, joint with Carmen Caprau, Nicolle Gonzalez, and Radial Sazdanovic, using Bar-Natan Gaussian elimination, that gives our whittled complex \(\mathcal{F}\mathcal{T}_n^k\). The whittled complex is homotopy-equivalent to the original Khovanov chain complex but with a reduced number of generators. After sketching the proof, I will end the talk discussing related future projects.
Topology through the lens of computer science
Eric Samperton
Purdue University
In these two talks, I will discuss how the theory of computational complexity can help us sharpen both our understanding and our goals when we study problems in topology. The first talk will focus mostly on the fundamental decision problems of geometric topology, such as the homeomorphism problem and sphere recognition. Tracing the history of results here serves as a natural way to introduce the basic notions of theoretical computer science (including computability, reduction, NP-hard, P vs NP, etc.) to our topology audience. In the second talk, I will pivot to focusing on the computational complexity of 3-manifold invariants associated to (2+1)-dimensional topological quantum field theories (TQFTs). I will explain how this line of inquiry is motivated by an approach to fault tolerant quantum computing by braiding anyons—particle-like excitations that are expected to arise in certain 2-dimensional quantum materials. I will outline both a conjectural program to characterize exactly which TQFTs host anyon capable of encoding universal quantum computation, as well as a parallel program to characterize exactly which 3-d TQFTs support #P-hard 3-manifold invariants.
Participant Lectures
The Skein Category TFT
Thomas Carlson
Montana State University
A result of Walker states that there is a Bimod-valued TFT defined by assigning to a surface \(\Sigma\), its skein category, \(\mathnormal{SkCat}_\mathcal{A}(\Sigma)\), and to a 3D bordism \(M : \Sigma_{in} \to \Sigma_{out}\), the functor \(\mathnormal{SkBimod}_\mathcal{A}(M, -, -) : \mathnormal{SkCat}_\mathcal{A}(\Sigma_{in})^{op} \times \mathnormal{SkCat}_\mathcal{A}(\Sigma_{out}) \to \mathnormal{Vect}\), which takes labelings of the incoming and outgoing surfaces to the skein module of \(M\) relative to those labelings. In particular, the functoriality of this assignment provides a formula for the skein module of a manifold resulting from gluing two manifolds along a common boundary. We will explore what these constructions look like, and briefly discuss an outline of a proof of this result.
Distinguishing exotic \(\mathbb{R}^4\)s with Heegaard Floer homology
Sean Eli
Georgia Tech
We give the first explicit infinite family of exotic \(\mathbb{R}^4\)s, distinguished with end Floer homology. We also construct exotica with various properties and re-prove Bizaca and Etnyre’s result on smoothings of \(M \times \mathbb{R}\), in the case \(M\) is closed. I will also talk about further directions and the noncompact classification problem.
Non-semisimple Topological Quantum Computation
Sung Kim
University of Southern California
There is an active paradigm shift within quantum topology from semisimple to non-semisimple phenomena. The primary technology that catalyzed this shift is called the modified trace. Mathematicians have shown that non-semisimple topological quantum field theories (TQFTs) can distinguish certain topological features that their semisimple counterparts cannot. One well-known application of quantum topology is topological quantum computation. Given this paradigm shift, it is natural to ask: what does it mean to do topological quantum computation via modified traces? In this talk, we explore this question through non-semisimple topological quantum computation accompanied with a concrete case study of the non-semisimple Ising model.
Thompson’s Groups and Link Homology
Louisa Liles
The Ohio State University
In 2014 Vaughan Jones showed how to build links in the \(3\)-sphere from elements of Thompson’s group F. We will explore extensions of this construction to build \((n,n)\)-tangles, annular tangles, and virtual links, and their connections with Khovanov homology. This talk includes joint work with Slava Krushkal, Yangxiao Luo, Micah Chrisman, and Melody Molander.
A Well-Defined Jellyfish Algorithm for Affine \(E\) Planar Algebras
Melody Molander
The Ohio State University
A planar algebra is an infinite collection of vector spaces equipped with an action of planar tangle diagrams. A first example is when the vector spaces are chosen to be the Temperley–Lieb algebras. Vaughan Jones first introduced planar algebras in the study of subfactors. Bigelow, Morrison, Peters, and Snyder introduced a powerful diagrammatic evaluation method on planar algebras called the jellyfish algorithm. In this talk, I will give a generators-and-relations presentation for planar algebras corresponding to subfactors with principal graph affine \(E_7\). Then I will define a jellyfish algorithm on this planar algebra and show that its nice braided structure gives rise to a well-defined map onto the complex numbers.
Extensions of Singular Fibrations and the Homology Cobordism Group
Trevor Oliveira-Smith
University of California, Davis
A homology sphere is said to be trivial in the homology cobordism group when it bounds a smooth homology ball. A homology ball is said to be ribbon if it can be built from only 4-dimensional 0,1, and 2-handles. It is an open question whether any homology sphere which is trivial in the homology cobordism group must be trivial through a ribbon homology ball. In this talk, we investigate this question by extending work of Miller and Zupan on singular fibrations to show that a homology sphere bounds a ribbon homology ball if and only if there is a certain singular fibration which “nicely” extends. We tie this work in with other open questions in homology cobordism, as well as the slice-ribbon conjecture.
\(\mathfrak{sl}(3)\) homology for strongly invertible links
Max Throm
Michigan State University
In Borodzik-Dai-Mallick-Stoffregen [arXiv:2507.13642], they introduce a refinement of Bar-Natan homology for involutive links and use it to give lower bounds on the equivariant 4-genus. Here we attempt to define a similar refinement for \(\mathfrak{sl}(3)\) homology. The ultimate goal of this work-in-progress is to use this refined \(\mathfrak{sl}(3)\) homology to obtain similar results to [arXiv:2507.13642], i.e., to gain further information regarding the equivariant slice genus and isotopy-equivariant slice genus of such links.
Odd Khovanov homology and \(2\)-knots
Rithwik Susheel Vidyarthi
Michigan State University
We prove a conjecture of Migdail and Wehrli regarding the maps on Odd Khovanov homology associated to knotted spheres. The key tool we use is Daemi’s plane Floer homology, which we use in place of Lee deformation. Continuing the analogy with Lee homology, we see this work as a potential first step toward a genuinely functorial model for odd Khovanov homology. This is joint work with Dean Spyropoulos and Chen Zhang.