Claremont Topology Seminar

Founded 1989

The Claremont Topology Seminar meets on-line using Zoom on Tuesday's from 3:00-4:00 pm. If you would like to attend, request a Zoom link from Sam Nelson .

For more information about the Seminar, to suggest speakers, or to volunteer to speak, contact Bahar Acu , Dave Bachman , Sam Nelson , Helen Wong or Vin de Silva.

Spring 2022 Schedule

special days, times or locations are in red

Date Speaker Title and Abstract

Tuesday
Jan 18

Tuesday
Jan 25
Organizational Meeting

Tuesday
Feb 1

Tuesday
Feb 8
Sam Nelson
Claremont McKenna College
Title: Experimental Knot Music v2
Abstract: In this talk I will recount the history of my knot theory-based music project and show an example of my method for creating music from knot homsets.

Tuesday
Feb 15
Seonmi Choi
Kyungpook Natl U in Korea
Title : On invariants for surface-links in entropic magmas via marked graph diagrams
Abstract : M. Niebrzydowski and J. H. Przytycki defined a Kauffman bracket magma and constructed the invariant P of framed links in 3-space. The invariant is closely related to the Kauffman bracket polynomial. The normalized bracket polynomial is obtained from the Kauffman bracket polynomial by the multiplication of indeterminate and it is an ambient isotopy invariant for links. In this talk, we reformulate the multiplication by using a map from the set of framed links to a Kauffman bracket magma in order that P is invariant for links in 3-space. We define a generalization of a Kauffman bracket magma, which is called a marked Kauffman bracket magma. We find the conditions to be invariant under Yoshikawa moves except the first one and use a map from the set of admissible marked graph diagrams to a marked Kauffman bracket magma to obtain the invariant for surface-links in 4-space.

Tuesday
Feb 22

Tuesday
Mar 1
Thomas Mattman
California State University, Chico
Title: Title: Two-bridge knots admit no purely cosmetic surgeries
Abstract: (Joint with Ichihara, Jong, and Saito). We show that two-bridge knots admit no purely cosmetic surgeries, ie no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument is based on a recent result by Hanselman and a study of signature and finite type invariants of knots as well as the SL(2,\C) Casson invariant.

Tuesday
Mar 8
Hugh Howards
Wake Forest University
Title; Systematically Detecting Flypes and Hexagonal Mosaics
Abstract: We talk about building knots using mosaics which were as introduced as a way of modeling quantum knots by Lomonaco and Kauffman and a newer variant, hexagonal mosaics, introduced by Jennifer McLoud-Mann. In the process we find a new bound on crossing numbers for hexagonal mosaics and find an infinite family of knots which do not achieve their hexagonal mosaic number while also in a projection which achieves their crossing number, extending a result of Lew Ludwig et al. In the process we introduce a new tool which makes it easier to systematically recognize when two knots differ by a sequence of Flypes (for example, giving a process to recognize that the Perko Pair were in fact the same knot). No background with mosaics or flypes is necessary. This is joint work with Jiong Li* and Xiotian Liu* (* indicates undergraduate students).

Tuesday
Mar 15
No Meeting Spring Break

Tuesday
Mar 22
Aaron Mazel-Gee
California Institute of Technology

Title: Towards knot homology for 3-manifolds
Abstract: The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin--Turaev using quantum groups. Khovanov homology is a categorification of the Jones polynomial of a knot in R^3, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds. In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided (infinity,2)-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.

Tuesday
Mar 29
Rhea Palak Bakshi
Institute for Theoretical Studies, ETH Zurich, Switzerland
Title: Kauffman bracket skein modules and their structure
Abstract: Skein modules were introduced by Jozef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche's generalisation of Witten's conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold. I will also give the exact structure of the KBSM of the connected sum of two solid tori.

Tuesday
Apr 5

Tuesday
Apr 12
Martin Bobb
IHES
Title: Cusps in Convex Projective Geometry
Abstract: Convex real projective structures generalize hyperbolic structures in a rich way. We will discuss a class of manifolds introduced by Cooper Long and Tillmann, which include finite-volume cusped hyperbolic manifolds and other manifolds with well-controlled ends. These manifolds have nice deformation theoretic properties, and we will conclude with an existence theorem for novel structures on some hyperbolic manifolds.

Tuesday
Apr 19

Tuesday
April 26

Tuesday
May 3
Jim Hoste
Pitzer College
Title: On the Non-orientable 4-genus of Double Twist Knots, Part II: Lower Bounds
Abstract: The non-orientable 4-genus of a knot K is the smallest first Betti number of any non-orientable surface in the 4-ball spanning the knot. It is defined to be zero if the knot is slice. In joint work with Patrick Shanahan and Cornelia Van Cott, we attempt to determine the value of this invariant for double twist knots. In an earlier talk at this seminar, I presented methods of determining upper bounds by explicitly describing non-orientable spanning surfaces. In this talk I describe methods for establishing lower bounds using linking forms on 4-manifolds and a major result of Donaldson. These methods suffice to compute the non-oprientable 4-genus of several infinite families of double twist knots.

Archived Schedules

Date	Speaker	Title and Abstract
Tuesday Jan 18
Tuesday Jan 25	Organizational Meeting
Tuesday Feb 1
Tuesday Feb 8	Sam Nelson Claremont McKenna College	Title: Experimental Knot Music v2 Abstract: In this talk I will recount the history of my knot theory-based music project and show an example of my method for creating music from knot homsets.
Tuesday Feb 15	Seonmi Choi Kyungpook Natl U in Korea	Title : On invariants for surface-links in entropic magmas via marked graph diagrams Abstract : M. Niebrzydowski and J. H. Przytycki defined a Kauffman bracket magma and constructed the invariant P of framed links in 3-space. The invariant is closely related to the Kauffman bracket polynomial. The normalized bracket polynomial is obtained from the Kauffman bracket polynomial by the multiplication of indeterminate and it is an ambient isotopy invariant for links. In this talk, we reformulate the multiplication by using a map from the set of framed links to a Kauffman bracket magma in order that P is invariant for links in 3-space. We define a generalization of a Kauffman bracket magma, which is called a marked Kauffman bracket magma. We find the conditions to be invariant under Yoshikawa moves except the first one and use a map from the set of admissible marked graph diagrams to a marked Kauffman bracket magma to obtain the invariant for surface-links in 4-space.
Tuesday Feb 22
Tuesday Mar 1	Thomas Mattman California State University, Chico	Title: Title: Two-bridge knots admit no purely cosmetic surgeries Abstract: (Joint with Ichihara, Jong, and Saito). We show that two-bridge knots admit no purely cosmetic surgeries, ie no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument is based on a recent result by Hanselman and a study of signature and finite type invariants of knots as well as the SL(2,\C) Casson invariant.
Tuesday Mar 8	Hugh Howards Wake Forest University	Title; Systematically Detecting Flypes and Hexagonal Mosaics Abstract: We talk about building knots using mosaics which were as introduced as a way of modeling quantum knots by Lomonaco and Kauffman and a newer variant, hexagonal mosaics, introduced by Jennifer McLoud-Mann. In the process we find a new bound on crossing numbers for hexagonal mosaics and find an infinite family of knots which do not achieve their hexagonal mosaic number while also in a projection which achieves their crossing number, extending a result of Lew Ludwig et al. In the process we introduce a new tool which makes it easier to systematically recognize when two knots differ by a sequence of Flypes (for example, giving a process to recognize that the Perko Pair were in fact the same knot). No background with mosaics or flypes is necessary. This is joint work with Jiong Li* and Xiotian Liu* (* indicates undergraduate students).
Tuesday Mar 15	No Meeting	Spring Break
Tuesday Mar 22	Aaron Mazel-Gee California Institute of Technology	Title: Towards knot homology for 3-manifolds Abstract: The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin--Turaev using quantum groups. Khovanov homology is a categorification of the Jones polynomial of a knot in R^3, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds. In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided (infinity,2)-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.
Tuesday Mar 29	Rhea Palak Bakshi Institute for Theoretical Studies, ETH Zurich, Switzerland	Title: Kauffman bracket skein modules and their structure Abstract: Skein modules were introduced by Jozef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche's generalisation of Witten's conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold. I will also give the exact structure of the KBSM of the connected sum of two solid tori.
Tuesday Apr 5
Tuesday Apr 12	Martin Bobb IHES	Title: Cusps in Convex Projective Geometry Abstract: Convex real projective structures generalize hyperbolic structures in a rich way. We will discuss a class of manifolds introduced by Cooper Long and Tillmann, which include finite-volume cusped hyperbolic manifolds and other manifolds with well-controlled ends. These manifolds have nice deformation theoretic properties, and we will conclude with an existence theorem for novel structures on some hyperbolic manifolds.
Tuesday Apr 19
Tuesday April 26
Tuesday May 3	Jim Hoste Pitzer College	Title: On the Non-orientable 4-genus of Double Twist Knots, Part II: Lower Bounds Abstract: The non-orientable 4-genus of a knot K is the smallest first Betti number of any non-orientable surface in the 4-ball spanning the knot. It is defined to be zero if the knot is slice. In joint work with Patrick Shanahan and Cornelia Van Cott, we attempt to determine the value of this invariant for double twist knots. In an earlier talk at this seminar, I presented methods of determining upper bounds by explicitly describing non-orientable spanning surfaces. In this talk I describe methods for establishing lower bounds using linking forms on 4-manifolds and a major result of Donaldson. These methods suffice to compute the non-oprientable 4-genus of several infinite families of double twist knots.