• Speaker: Matt Stoffregen, MSU
• Title: Surgery Exact Triangles in Involutive Floer homology
• Date: 01/26/2021
• Time: 2:50 PM - 3:40 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Honghao Gao (gaohongh@msu.edu)

We'll sketch the definition of the involutive Heegaard Floer homology constructed by Hendricks-Manolescu, and then explain how this homology theory behaves under surgery. As a consequence, we can use the surgery formula to construct three-manifolds which are not homology cobordant to any combination of Seifert fiber spaces. This is joint work Kristen Hendricks, Jen Hom and Ian Zemke.

• Speaker: Irving Dai, MIT
• Title: Lattice homology and instantons
• Date: 02/09/2021
• Time: 2:50 PM - 3:40 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Honghao Gao (gaohongh@msu.edu)

We show that if Y is the boundary of an almost-rational plumbing, then the framed instanton Floer homology of Y is isomorphic to its Heegaard Floer homology. This class of 3-manifolds includes all Seifert fibered rational homology spheres. Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps established by John Baldwin and Steven Sivek. This is joint work with Antonio Alfieri, John Baldwin, and Steven Sivek.

• Speaker: Thang Le, Georgia Tech
• Title: Quantum trace map for $SL_n$ skein algebras of surfaces
• Date: 02/23/2021
• Time: 2:50 PM - 3:40 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Honghao Gao (gaohongh@msu.edu)

For a punctured surface there are two quantizations of the $SL_n$ character variety. The first quantization is the $SL_n$ skein algebra, and the second one is the quantization of the higher Teichmuller space. When $n=2$ Bonahon and Wong showed that there is an algebra homomorphism, called the quantum trace, from the first quantized algebra to the second one. We show for general n a similar quantum trace map exists. The construction of the $SL_n$ quantum trace is based on the theory of stated $SL_n$ skein algebra, developed in a joint work with A. Sikora, and a work of Chekhov and Shapiro.

• Speaker: Roger Casals, UC Davis
• Title: Lagrangian Fillings of Legendrian links: Two Constructions in Floer Theory
• Date: 03/09/2021
• Time: 2:50 PM - 3:40 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Honghao Gao (gaohongh@msu.edu)

In this talk I will review our current understanding in the classification of Lagrangian fillings for Legendrian links in the standard contact 3-sphere. The talk will present two illustrative constructions: one explaining how to build and detect infinitely many Lagrangian fillings using the Legendrian Contact DGA, and the other explaining how to classify fillings for the Hopf link using pseudo-holomorphic foliations. First, I will present the basic objects of interest and survey the recent developments in the field (work with H. Gao, and work with E. Zaslow). Then I will delve into new and in-progress results on the Legendrian Contact DGA (work with L. Ng). Finally, I will report on how pseudo-holomorphic curves might help us classify Lagrangian fillings in certain cases. During the course of the talk, I will try to highlight some of the interesting open questions and new methods that arise from our current work as well as future directions.

• Speaker: Samantha Allen, Dartmouth
• Title: Using surgery to study unknotting with a single twist
• Date: 03/23/2021
• Time: 2:50 PM - 3:40 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Honghao Gao (gaohongh@msu.edu)

Ohyama showed that any knot can be unknotted by performing two full twists, each on a set of parallel strands. We consider the question of whether or not a given knot can be unknotted with a single full twist, and if so, what are the possible linking numbers associated to such a twist. It is observed that if a knot can be unknotted with a single twist, then some surgery on the knot bounds a rational homology ball. Using tools such as classical invariants and invariants arising from Heegaard Floer theory, we give obstructions for a knot to be unknotted with a single twist of a given linking number. In this talk, I will discuss some of these obstructions, their implications (especially for alternating knots), many examples, and some unanswered questions. This talk is based on joint work with Charles Livingston.

• Speaker: Charlie Frohman, U Iowa
• Title: A Geometric Kauffman Bracket
• Date: 04/06/2021
• Time: 2:50 PM - 3:40 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Honghao Gao (gaohongh@msu.edu)

This is joint work with Joanna Kania-Bartoszynska and Thang Le $\\$ I will discuss the representation theory of the Kauffman bracket skein algebra of a finite type surface at a root of unity whose order is not divisible by 4. $\\$ Specifically, the Kauffman bracket skein algebra is an algebra with trace in the sense of De Concini, Procesi, Reshetikhin and Rosso, so it has a well defined character variety of trace preserving representations, which can be identified with a branched cover of the SL(2,C)-character variety of the fundamental group of the underlying surface. $\\$ In the case of a closed surface the branched cover is trivial so its just the character variety of the fundamental group of the surface. $\\$ The skein algebra is also a Poisson order, so the character variety representations of the Kauffman bracket skein algebra of a closed surface decomposes into representations corresponding to irreducible, abelian and central representations of the fundamental group of the underlying surface. The irreducible representations of the fundamental group of the surface correspond to irreducible representations of the skein algebra. $\\$ We then use this as basic data to define an invariant of framed links in a three-manifold equipped with an irreducible representation of its fundamental group. The invariant satisfies the Kauffman bracket skein relations. $\\$ Such a representation could be the lift of the holonomy of a hyperbolic structure on the three-manifold, hence the title : A Geometric Kauffman Bracket. $\\$