The Topology Seminar
2024-2025 School Year Schedule:
(All talks 3:30-4:30 in SH 6635 unless otherwise stated)
- Tuesday, 11/19: Vijay Higgins (UCLA) "Annular webs, central elements of skein algebras, and miraculous cancellations"
Abstract
The skein algebra of a surface is spanned by links in the thickened surface, subject to skein relations which diagrammatically encode the data of a quantum group. The multiplication in the algebra is induced by stacking links in the thickened surface. This product is generally noncommutative. When the quantum parameter q is generic, the center of the skein algebra is essentially trivial. However, when q is a root of unity, interesting central elements arise. When the quantum group is quantum SL(2), the work of Bonahon-Wong shows that these central elements can be obtained by a topological operation of threading Chebyshev polynomials along knots. In this talk, I will discuss joint work with Francis Bonahon describing how the threading operation extends to SL(n) skein algebras, including a method for verifying the centrality of the elements by studying webs in the annulus. Then in the case of SL(3), I will discuss how 'stated skein algebras' can be used to show that the threading operation yields a well-defined algebra homomorphism, analogous to the Frobenius homomorphism for quantum groups.
- Monday, 12/02, 4-5 PM: Qiuyu Ren (UC Berkeley) "Lasagna s-invariant detects exotic 4-manifolds"
Abstract
We introduce a lasagna version of Rasmussen's s-invariant coming from the study of Khovanov/Lee skein lasagna modules, which assigns either an integer or \(-\infty\) to each second homology class of a given smooth 4-manifold. After presenting some properties of the lasagna s-invariant, we show that it detects the exotic pair of knot traces \( X_{-1}(-5_2) \) and \(X_{-1}(P(3,-3,-8))\). This gives the first gauge/Floer-theory-free proof of the existence of exotic compact orientable 4-manifolds. Time permitting, we mention some other applications of lasagna s-invariants. This is joint work with Michael Willis.
- Tuesday, 01/14: Jennifer Schultens (UC Davis) "Flipping Heegaard splittings"
Abstract
Heegaard splittings provide natural decompositions of 3-manifolds into two symmetric pieces. We provide examples of such decompositions and delve into the question of when the two pieces can be interchanged, or flipped, via an isotopy.
- Friday, 01/31, 12-1 PM: Aaron Lauda (USC) TBA
- Monday, 02/10, 4-5 PM: Peter Samuelson (UC Riverside) TBA
- Tuesday, 02/11: Anup Poudel (The Ohio State University) TBA
- Tuesday, 02/18: Agustina Czenky (USC) TBA
- Tuesday, 02/25: Christine Ruey Shan Lee (Texas State) TBA
- Tuesday, 03/04: Thang Lê (Georgia Tech) TBA
- Tuesday, 03/11: Corey Jones (NC State) TBA
- Monday, 03/17, 4-5 PM: Nathan Geer (Utah State) TBA
- Tuesday, 03/18: Colleen Delaney (Purdue University) TBA
- Tuesday, 04/01: David Penneys (The Ohio State University) TBA
- Tuesday, 04/15: Józef Przytycki (The George Washington University) TBA
- Tuesday, 04/29: Sujoy Mukherjee (University of Denver) TBA
- Tuesday, 05/06: Sunghyuk Park (Harvard) TBA