Seminars and Colloquia by Series

On amphichirality of symmetric unions (Virtual)

Series
Geometry Topology Seminar
Time
Monday, October 18, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ceren KoseThe University of Texas at Austin

Symmetric unions are an interesting class of knots. Although they have not been studied much for their own sake, they frequently appear in the literature. One such instance regards the question of whether there is a nontrivial knot with trivial Jones polynomial. In my talk, I will describe a class of symmetric unions, constructed by Tanaka, such that if any are amphichiral, they would have trivial Jones polynomial. Then I will show how such a knot not only answers the above question but also gives rise to a counterexample to the Cosmetic Surgery Conjecture. However, I will prove that such a knot is in fact trivial and hence cannot be used to answer any of these questions. Finally, I will discuss how the arguments that go into this proof can be generalized to study amphichiral symmetric unions.

Invariants of rational homology 3-spheres from the abelianization of the mod-p Torelli group (Virtual)

Series
Geometry Topology Seminar
Time
Monday, October 4, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Ricard Riba GarciaUAB Barcelona

Unlike the integral case, given a prime number p, not all Z/p-homology 3-spheres can be constructed as a Heegaard splitting with a gluing map an element of mod p Torelli group, M[p]. Nevertheless, letting p vary we can get any rational homology 3-sphere. This motivated us to study invariants of rational homology 3-spheres that comes from M[p]. In this talk we present an algebraic tool to construct invariants of rational homology 3-spheres from a family of 2-cocycles on M[p]. Then we apply this tool to give all possible invariants that are induced by a lift to M[p] of a family of 2-cocycles on the abelianization of M[p], getting a family of invariants that we will describe precisely.
 

Invariance of Knot Lattice Homology

Series
Geometry Topology Seminar
Time
Monday, September 27, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Seppo Niemi-ColvinDuke University

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots in rational homology spheres, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.

Mitsumatsu's Liouville domains are stably Weinstein

Series
Geometry Topology Seminar
Time
Monday, September 20, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Austin ChristianGeorgia Tech

In 1995, Mitsumatsu constructed a large family of Liouville domains whose topology obstructs the existence of a Weinstein structure.  Stabilizing these domains yields Liouville domains for which the topological obstruction is no longer in effect, and in 2019 Huang asked whether Mitsumatsu's Liouville domains were stably homotopic to Weinstein domains.  We answer this question in the affirmative.  This is joint work-in-progress with J. Breen.

A curve graph for Artin groups

Series
Geometry Topology Seminar
Time
Monday, September 13, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
online
Speaker
Rose Morris-WrightUCLA

Please Note: Meeting URL https://bluejeans.com/770198652/3456?src=join_info Meeting ID 770 198 652 Participant Passcode 3456 Want to dial in from a phone? Dial one of the following numbers: +1.408.419.1715 (United States (San Jose)) +1.408.915.6290 (United States (San Jose)) (see all numbers - https://www.bluejeans.com/numbers) Enter the meeting ID and passcode followed by # Connecting from a room system? Dial: bjn.vc or 199.48.152.152 and enter your meeting ID & passcode

Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. While specific types of Artin groups have many of the same properties as braid groups, other examples of Artin groups are still very mysterious. Braid groups are can be thought of as the mapping class groups of a punctured disc. The combinatorial and geometric structure of the mapping class group is reflected in a Gromov-hyperbolic space called the curve graph of the mapping class group. Using the curve graph of the mapping class group of a punctured disc, we can define a graph associated to a given braid group. In this talk, I will discuss how to generalize this construction to more general classes of Artin groups. I will also discuss the current known properties of this graph and further open questions about what properties of the curve graph carry over to this new graph. 

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