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February 2019

Applying Quantum Representations of Mapping Class Groups (Wade Bloomquist, UCSB)

February 28 @ 12:00 pm - 1:30 pm
Roberts North 104, CMC, 320 E. 9th St.
Claremont, CA 91711 United States
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One foundational pillar of low dimensional topology is the connection between link invariants and 3-manifold invariants.  One generalization of this has been given by Reshetikhin and Turaev to a surgery theory for colored ribbon graphs.  Then to complete the analogy rather than 3-manifold invariants we now have a 2+1 dimensional topology quantum field theory (TQFT).  For this talk we will only be focusing on one corner of a TQFT, in particular the representations of mapping class groups which are afforded…

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March 2019

Non-existence of epimorphisms between certain genus two handlebody-knot groups (Ryo Nikkuni, Tokyo Woman’s Christian University)

March 7 @ 12:00 pm - 1:30 pm

For two genus $g$ handlebody-knots $H_{1}$ and $H_{2}$, we denote $H_{1} \geq H_{2}$ if there exists an epimorphism from the fundamental group of the handlebody-knot complement of $H_{1}$ onto the one of $H_{2}$. In the case of $g = 1$, this order is a partial order on the set of prime knots and has been determined up to $11$ crossings by Kitano-Suzuki and Horie-Kitano-Matsumoto-Suzuki. In this talk, we consider the case of $g = 2$ and exhibit a lot of…

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April 2019

Geometry of quotient varieties and the algebra of conformal blocks (Han-Bom Moon Fordham University)

April 11 @ 12:15 pm - 1:30 pm
Roberts North 104, CMC, 320 E. 9th St.
Claremont, CA 91711 United States
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An important question in classical representation theory is when the tensor product of two irreducible representations has another representation as a factor. In this talk, I will introduce a quantum generalization of this question and explain how we may relate this question to geometry of quotients of certain complex manifolds. This is joint work with Sang-Bum Yoo.

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Enhancements of the quandle coloring invariant for knots (Karina Cho, Harvey Mudd College)

April 18 @ 12:00 pm - 1:30 pm
Roberts North 104, CMC, 320 E. 9th St.
Claremont, CA 91711 United States
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Quandles are algebraic structures that play nicely with knots. The multiplicative structure of finite quandles gives us a way to "color" knot diagrams, and the number of such colorings for a given knot and quandle is called the quandle coloring invariant. We strengthen this invariant by examining the relationships between the colorings, which are given by endomorphisms. This can be visualized using a directed graph that we call the quandle coloring quiver. We will show that the quandle coloring quiver…

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A (Z⊕Z)-family of knot quandles (Jim Hoste, Pitzer College)

April 25 @ 12:00 pm - 1:30 pm

Suppose K is an oriented knot in a 3-manifold M with regular neighborhood N (K). For each element γ ∈ π 1 (∂N (K)) we define a quandle Q γ (K; M) which generalizes the concept of the fundamental quandle of a knot. In particular, when γ is the meridian of K, we obtain the fundamental quandle. The collection of all such quandles gives a (Z⊕Z)-family of quandles. If K is a knot in M and γ is a primitive…

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September 2019

Topology Seminar: Sam Nelson (CMC)

September 17 @ 3:00 pm - 4:00 pm
Millikan 2099, Pomona College, 610 N. College Ave.
Claremont, CA 91711 United States
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Title: Biquandle Brackets and Knotoids Abstract: Biquandle brackets are a type of quantum enhancement of the  biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this talk we use biquandle brackets to enhance the biquandle counting matrix invariant of knotoids. This is joint work with Neslihan Gugumcu (Izmir Institute of Technology, Izmir, Turkey) and Natsumi Oyamaguchi (Shumei University, Tokyo, Japan).

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October 2019

Topology Seminar: Jesse Levitt (USC)

October 1 @ 3:00 pm - 4:00 pm
Millikan 2099, Pomona College, 610 N. College Ave.
Claremont, CA 91711 United States
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Title: Understanding Structure in the Single Variable Knot Polynomials Abstract: We examine the dimensionality and internal structure of the aggregated data produced by the Alexander, Jones, and Z0 polynomials using topological data analysis and dimensional reduction techniques. By examining several families of knots, including over 10 million distinct examples, we find that the Jones data is well described as a three dimensional manifold, the Z0 data as a single two dimensional manifold and the Alexander data as a collection of two dimensional manifolds. We confirm each of these…

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Topology Seminar: Mauricio Gomez Lopez (U. Oregon)

October 8 @ 3:00 pm - 4:00 pm

Title: Cobordism Categories and Spaces of Manifolds. Abstract: Cobordisms have been one of the central objects in topology since the pioneering work of Rene Thom, which provided the first link between manifolds and homotopy theory. In more recent years, there has been much focus on cobordism categories. These play a fundamental role in the study of topological quantum field theories and automorphism groups of manifolds.  A fundamental result in this field is the theorem of Galatius, Madsen, Tillmann, and Weiss,…

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November 2019

Paper Strip Knots (David Bachman)

November 5 @ 3:00 pm - 4:00 pm
Millikan 2099, Pomona College, 610 N. College Ave.
Claremont, CA 91711 United States
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I will discuss joint work with Jim Hoste, where we prove that a unique folded strip of paper can follow any polygonal knot with odd stick number. In the even stick number case there are either infinitely many, or none.

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Topology Triple-Header!

November 12 @ 2:30 pm - 4:00 pm
Millikan 2099, Pomona College, 610 N. College Ave.
Claremont, CA 91711 United States
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This triple-header of topology talks will include three speakers: First, Hyeran Cho from The Ohio State University will speak about Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams. In this talk, we show that the dual (1, 1)-diagram of a (1, 1)-diagram (a.k.a. a two pointed genus one Heegaard diagram) D(a, 0, 1, r) with 1 ≤ r < 2a + 1 and gcd(2a + 1, r) = 1 is given by D(1/2r, 0, 2a+1-1/r, 1/r) when 1/r…

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