Claremont Topology Seminar

The Claremont Topology Seminar meets on Tuesday's from 3:00-4:00 pm in Millikan 211, Pomona College. Millikan is located on the NE corner of College Avenue and 6th Street. Following the talk we go for coffee/tea in downtown Claremont. Sometimes we meet a little later and then go to dinner. (Look for special times on the calendar.) Click here for a map of the Pomona Campus.

To reach Pomona College from the 10 freeway, exit at Indian Hill, go north, turn right (east) on 6th street and then turn left (north) on College. To reach Pomona College from the 210 freeway, if traveling East, exit at Towne Avenue, turn right (south) on Towne, turn left (east) on Foothill Blvd, and turn right (south) onto College Avenue. If traveling West on the 210 freeway, exit at Baseline/Padua, turn right (west) onto Baseline, turn left (south) onto Padua at the first light, turn right (west) onto Foothill Blvd at the third light, turn left (south) onto College Avenue.

Parking on College Avenue is free.

For more information about the Seminar, or to suggest speakers, contact Jim Hoste , Dave Bachman , Erica Flapan or Vin de Silva.

Spring 2008 Schedule

special days, times or locations are in red

Date Speaker Title and Abstract

Tuesday
Jan 29

3:00 pm
Dave Bachman
Pitzer College
Title: Counter-examples to the stabilization conjecture for Heegaard splittings
Abstract: 74 years ago Reidemeister and Singer proved that any pair of Heegaard splittings are equivalent after some number of stabilizations. We present the first example of a pair of splittings that require more than one stabilization to be equivalent.

Tuesday
Feb 5

3:30 pm
Ron Stern
UC Irvine
Title: The countdown to the complex projective plane.
Abstract: A key problem in 4-manifold topology is to understand whether "standard manifolds" admit exotic smooth structures, i.e. given a smooth 4-manifold, if there are manifolds homeomorphic but not diffeomorphic to it. In the last several years significant progress has been made in understanding this problem for the manifolds obtained by blowing up the complex projective plane at a small number of points. I will describe the problems in this area, the techniques that have been used to study them, and the results that have been obtained leading to some outstanding results of young mathematicians in this last year.

Tuesday
Feb 12

3:00 pm
Mohamed Ait Nouh
UC Riverside
Title: CP^2 genera of knots.
Abstract: The CP^2-genus of a knot K is the minimal genus over all isotopy classes of smooth, compact, connected and oriented surfaces properly embedded in CP^2 - B^4 with boundary K. We compute, for the first time, the CP2-genus and realizable degrees of a finite collection of torus knots. The proofs using embeddings of surfaces in 4-manifolds, blow-ups and twisting operations on knots.

Tuesday
Feb 19

3:00 pm
Sam Nelson
Pomona College
Title: Racks and Counting Invariants
Abstract: Where quandles define counting invariants for knots and links, racks define counting invariants for framed (classical) knots and links. We will see how to use finite racks to get an easily computed invariant of classical knots and how to incoporate quandle 2-cocyles.

Tuesday
Feb 26

3:00 pm

Tuesday
Mar 4

3:00 pm

Tuesday
Mar 11

3:00 pm
Ryan Blair
UC Santa Barbara
Title: Bridge Number and Conway Products
Abstract: I will define the generalized Conway product of links and give a tight lower bound for the bridge number of this product in terms of the bridge numbers of the two factor links.

Tuesday
Mar 18
no meeting, Spring Break

Tuesday
Mar 25

3:00 pm
Mike Williams

UC Davis
Title: Lens space surgeries on tunnel number one knots

Abstract: In the 1980's and 1990's, John Berge showed that a certain class
of tunnel number one knots in the 3-sphere, the so-called "double
primitive" knots, admit lens space surgeries. Then Cameron Gordon
conjectured that if a knot in the 3-sphere admits a lens space surgery,
then that knot is double primitive. After giving some background on lens
space surgeries, I will discuss an approach to prove this conjecture for
all tunnel number one knots in terms of genus 2 Heegaard splittings.

Tuesday
Apr 1

3:00 pm
Robin Wilson
Cal Poly Pomona
Title: Almost normal surfaces in knot complemets

Abstract: It was shown independently by Stocking and Rubinstein that any strongly irreducible Heegaard splitting for an irreducible 3-manifold is isotopic to an almost normal surface. In the study of bridge surfaces for knots and links the idea of a weakly incompressible bridge surface is immediately analogous to the idea of a strongly irreducible Heegaard surface for a 3-manifold. In this talk I will present recent work that gives an analog of the result of Stocking and Rubinstein by showing that any weakly incompressible bridge surface in a knot complement is isotopic to an almost normal bridge surface.

Tuesday
Apr 8

3:00 pm

Tuesday
Apr 15

3:00 pm

Tuesday
Apr 22

3:00 pm
Fabiola Manjarrez-Gutierrez

UC Davis Title: Knot exteriors and circular handle decompositions

Abstract: A circle-valued Morse function on the knot complement $C_K= S^3 \setminus K$ is a function $f: C_K \rightarrow S^1$ which is Morse and behaves \textit{nicely} in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior $E(K)= S^3 \setminus N(K)$, with the property that every regular level surface contains a Seifert surface for the knot. In this talk we will discuss nice properties that can be obtain from such decomposition.

Tuesday
Apr 29

3:00 pm
Jim Hoste
Pitzer College
Title: On the partial ordering of 2-bridge knots
Abstract: A partial ordering of knots in the 3-sphere, due to Silver and Whitten, is given by declaring K greater than or equal to J if the fundamental group of the complement of K maps onto the fundamental group of the complement of J, preserving peripheral structure. In the case of 2-bridge knots, Ohtsuki, Riley and Sakuma exhibit a construction that, for a given knot J, will produce infinitely many knots K with K greater than or equal to J. It is has been shown by Gonzalez-Acuna and Ramirez that this construction creates all possibilities when J is also a torus knot. If the Ohtsuki construction produces all possibilities for all 2-bridge knots, then it appears that (non-torus) 2-bridge knots with small numbers of distinct boundary slopes must be minimal, that is, only greater than the unknot. We prove this is true for 2-bridge knots with three distinct boundary slopes. This is joint work with Tomasz Przytycki and Pat Shanahan.

Tuesday
May 6

3:00 pm
David Futer

Michigan State

Title: The Jones polynomial and surfaces far from fibers
Abstract: Experimental evidence suggests that the volume of a hyperbolic knot or link is coarsely determined by the coefficients near the head and tail of its Jones polynomial. I will discuss some recent work that proves this is indeed the case for a large family of links. In fact, the volume estimate relies on the claim that coefficients of the Jones polynomial detect that a particular surface is very far from being a fiber. This is joint work with Effie Kalfagianni.

Archived Schedules

Date	Speaker	Title and Abstract
Tuesday Jan 29 3:00 pm	Dave Bachman Pitzer College	Title: Counter-examples to the stabilization conjecture for Heegaard splittings Abstract: 74 years ago Reidemeister and Singer proved that any pair of Heegaard splittings are equivalent after some number of stabilizations. We present the first example of a pair of splittings that require more than one stabilization to be equivalent.
Tuesday Feb 5 3:30 pm	Ron Stern UC Irvine	Title: The countdown to the complex projective plane. Abstract: A key problem in 4-manifold topology is to understand whether "standard manifolds" admit exotic smooth structures, i.e. given a smooth 4-manifold, if there are manifolds homeomorphic but not diffeomorphic to it. In the last several years significant progress has been made in understanding this problem for the manifolds obtained by blowing up the complex projective plane at a small number of points. I will describe the problems in this area, the techniques that have been used to study them, and the results that have been obtained leading to some outstanding results of young mathematicians in this last year.
Tuesday Feb 12 3:00 pm	Mohamed Ait Nouh UC Riverside	Title: CP^2 genera of knots. Abstract: The CP^2-genus of a knot K is the minimal genus over all isotopy classes of smooth, compact, connected and oriented surfaces properly embedded in CP^2 - B^4 with boundary K. We compute, for the first time, the CP2-genus and realizable degrees of a finite collection of torus knots. The proofs using embeddings of surfaces in 4-manifolds, blow-ups and twisting operations on knots.
Tuesday Feb 19 3:00 pm	Sam Nelson Pomona College	Title: Racks and Counting Invariants Abstract: Where quandles define counting invariants for knots and links, racks define counting invariants for framed (classical) knots and links. We will see how to use finite racks to get an easily computed invariant of classical knots and how to incoporate quandle 2-cocyles.
Tuesday Feb 26 3:00 pm
Tuesday Mar 4 3:00 pm
Tuesday Mar 11 3:00 pm	Ryan Blair UC Santa Barbara	Title: Bridge Number and Conway Products Abstract: I will define the generalized Conway product of links and give a tight lower bound for the bridge number of this product in terms of the bridge numbers of the two factor links.
Tuesday Mar 18	no meeting, Spring Break
Tuesday Mar 25 3:00 pm	Mike Williams UC Davis	Title: Lens space surgeries on tunnel number one knots Abstract: In the 1980's and 1990's, John Berge showed that a certain class of tunnel number one knots in the 3-sphere, the so-called "double primitive" knots, admit lens space surgeries. Then Cameron Gordon conjectured that if a knot in the 3-sphere admits a lens space surgery, then that knot is double primitive. After giving some background on lens space surgeries, I will discuss an approach to prove this conjecture for all tunnel number one knots in terms of genus 2 Heegaard splittings.
Tuesday Apr 1 3:00 pm	Robin Wilson Cal Poly Pomona	Title: Almost normal surfaces in knot complemets Abstract: It was shown independently by Stocking and Rubinstein that any strongly irreducible Heegaard splitting for an irreducible 3-manifold is isotopic to an almost normal surface. In the study of bridge surfaces for knots and links the idea of a weakly incompressible bridge surface is immediately analogous to the idea of a strongly irreducible Heegaard surface for a 3-manifold. In this talk I will present recent work that gives an analog of the result of Stocking and Rubinstein by showing that any weakly incompressible bridge surface in a knot complement is isotopic to an almost normal bridge surface.
Tuesday Apr 8 3:00 pm
Tuesday Apr 15 3:00 pm
Tuesday Apr 22 3:00 pm	Fabiola Manjarrez-Gutierrez UC Davis	Title: Knot exteriors and circular handle decompositions Abstract: A circle-valued Morse function on the knot complement $C_K= S^3 \setminus K$ is a function $f: C_K \rightarrow S^1$ which is Morse and behaves \textit{nicely} in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior $E(K)= S^3 \setminus N(K)$, with the property that every regular level surface contains a Seifert surface for the knot. In this talk we will discuss nice properties that can be obtain from such decomposition.
Tuesday Apr 29 3:00 pm	Jim Hoste Pitzer College	Title: On the partial ordering of 2-bridge knots Abstract: A partial ordering of knots in the 3-sphere, due to Silver and Whitten, is given by declaring K greater than or equal to J if the fundamental group of the complement of K maps onto the fundamental group of the complement of J, preserving peripheral structure. In the case of 2-bridge knots, Ohtsuki, Riley and Sakuma exhibit a construction that, for a given knot J, will produce infinitely many knots K with K greater than or equal to J. It is has been shown by Gonzalez-Acuna and Ramirez that this construction creates all possibilities when J is also a torus knot. If the Ohtsuki construction produces all possibilities for all 2-bridge knots, then it appears that (non-torus) 2-bridge knots with small numbers of distinct boundary slopes must be minimal, that is, only greater than the unknot. We prove this is true for 2-bridge knots with three distinct boundary slopes. This is joint work with Tomasz Przytycki and Pat Shanahan.
Tuesday May 6 3:00 pm	David Futer Michigan State	Title: The Jones polynomial and surfaces far from fibers Abstract: Experimental evidence suggests that the volume of a hyperbolic knot or link is coarsely determined by the coefficients near the head and tail of its Jones polynomial. I will discuss some recent work that proves this is indeed the case for a large family of links. In fact, the volume estimate relies on the claim that coefficients of the Jones polynomial detect that a particular surface is very far from being a fiber. This is joint work with Effie Kalfagianni.