• Speaker: Amitesh Datta, Princeton University
• Title: Does the Jones polynomial of a knot detect the unknot? A novel approach via braid group representations and class numbers of number fields.
• Date: 02/07/2023
• Time: 3:30 PM - 4:30 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Peter Kilgore Johnson (john8251@msu.edu)

How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation. In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s and is equivalent to whether B_4 is a group of 3 x 3 matrices). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones. I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.

• Speaker: Ian Montague , Brandeis University
• Title: Seiberg-Witten Floer K-Theory and Cyclic Group Actions
• Date: 02/21/2023
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Peter Kilgore Johnson (john8251@msu.edu)

Given a spin rational homology sphere equipped with a cyclic group action, I will introduce equivariant refinements of Manolescu's kappa invariant, derived from the equivariant K-theory of the Seiberg--Witten Floer spectrum. These invariants give rise to equivariant relative 10/8-ths type inequalities for equivariant spin cobordisms between rational homology spheres. I will explain how these inequalities provide applications to knot concordance, obstruct cyclic group actions on spin fillings, and give genus bounds for knots in punctured 4-manifolds. If time permits I will explain how these invariants are related to equivariant eta-invariants of the Dirac operator, and describe work-in-progress which provides explicit formulas for the $S^1$-equivariant eta-invariants on Seifert-fibered spaces.

• Speaker: Shunyu Wan , University of Virginia
• Title: Naturality of Legendrian LOSS invariant under positive contact surgery and application
• Date: 03/14/2023
• Time: 11:30 AM - 12:30 PM
• Place: C304 Wells Hall
• Contact: Peter Kilgore Johnson (john8251@msu.edu)

Given a Legendrian Knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L which lives in the knot Floer homology group. We prove that the LOSS invariant is natural under the positive contact surgery. In this talk I will review some background and definition, get the idea of the proof and try to focus on the application which is about new examples of non-simple knots.

• Speaker: Siddhi Krishna, Columbia University
• Title: RTG Seminar: Fibered knots: what, why and how
• Date: 03/20/2023
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Peter Kilgore Johnson (john8251@msu.edu)

Fibered knots show up all over low-dimensional topology, as they provide a robust way to investigate interactions between phenomena of different dimensions. In this talk, I'll survey what they are, why you should care, and how to identify them. Then, as time permits, I'll also sketch a proof that positive braid knots are fibered. I will assume very little background for this talk -- all are welcome!

• Speaker: Anup Poudel, Ohio State
• Title: A comparison between $SL_n$ spider categories.
• Date: 03/21/2023
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Vijay B Higgins (higgi231@msu.edu)

In this talk, we will explore and make comparisons between various models that exist for spherical tensor categories associated to the category of representations of the quantum group $U_q(sl_n).$ In particular, we will discuss the combinatorial model of Murakami-Ohtsuki-Yamada (MOY), the n-valent ribbon model of Sikora and the trivalent spider category of Cautis-Kamnitzer-Morrison (CKM). We conclude by showing that the full subcategory of the spider category from CKM, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This proves a conjecture of Le and Sikora and also answers a question from Morrison's Ph.D. thesis.

• Speaker: Sam Gunningham, Montana State
• Title: RTG Seminar: TBA
• Date: 04/10/2023
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Vijay B Higgins (higgi231@msu.edu)

TBA

• Speaker: Pierre-Louis Blayac, University of Michigan
• Title: TBA
• Date: 04/18/2023
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Vijay B Higgins (higgi231@msu.edu)

TBA

• Speaker: Sarah Petersen, University of Colorado, Boulder
• Title: TBA
• Date: 04/25/2023
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Peter Kilgore Johnson (john8251@msu.edu)

No abstract available.