Claremont Topology Seminar

Fall 2007 Schedule

Date	Speaker	Title and Abstract
Tuesday Sept 11	Organizational Meeting
Tuesday Sept 18	Jim Hoste Pitzer College	Title: Torus knots are Fourier-(1,1,2) knots. Abstract: Fourier-(1,1,k) knots are knots that can be parameterized by a single cosine function for both the x and y-coordinate and by the sum of k cosine functions in the z-coordiinate. Fourier-(1,1,1) knots are Lissajous knots. Not all knots can be Lissajous. In fact, no torus knot can be Lissajous. All twist knots and all 2-bridge knots to 14 crossings are known to be Fourier-(1,1,2). There is no known example of a knot that is not a Fourier-(1,1,2) knot. I will explain the proof of the theorem given in the title as well as the context for the theorem and how I found the theorem.
Tuesday Sept 25	Dongping Zhuang Cal Tech	Title: Irrational stable commutator length in finitely presented groups. Abstract: We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups.
Tuesday Oct 2	Sam Nelson Pomona College	Postponed to next week due to building evacuation drill!
Tuesday Oct 9	Sam Nelson Pomona College	Title: Quandle Polynomial Invariants Abstract: A finite quandle has a unique two-variable polynomial which expresses the way in which trivial action in the quandle is distributed throughout the set. We can take advantage of this to define quandle polynomial link invariants, which are jazzed-up versions of the quandle homomorphism counting invariant.
Tuesday Oct 16	Rupert Venzke Cal Tech	Title: Braid Forcing, Hyperbolic Geometry, and Pseudo-Anosov Sequences of Low Entropy Abstract: We view braids as automorphisms of punctured disks and define a partial order on pseudo-Anosov braids called the "forcing order." The order measures whether one automorphism induces another given automorphism on the surface. Pseudo- Anosov growth rate decreases relative to the order and appears to give a good measure of braid complexity. Unfortunately it appears difficult computationally to determine explicitly the partial order structure by hand. We use several computer algorithms to study the bottom part of the partial order when the number of braid strands is fixed. From the algorithms, we build sequences of low entropy pseudo- Anosov n-strand braids that are minimal in the sense that they do not force any other pseudo-Anosov braids on the same number of strands. The sequences are an extension of work done by Hironaka and Kin, and we conjecture the sequences to achieve minimal entropy among certain nontrivial classes of braids. In general, the lowest entropy pseudo-Anosov braids appear to have mapping tori that come from Dehn surgery on very low volume hyperbolic 3-manifolds and we begin to analyze the relation between entropy and hyperbolic volume.
Tuesday Oct 23	no meeting, Fall Break
Tuesday Oct 30	Danielle O'Donnol UCLA	Title: Multiplicity of a space over another space. Abstract: The multiplicity of one space over another space is a new concept introduced by Kouki Taniyama at Intelligence of Low Dimensional Topology this summer. I will define this for a general category and show how it gives rise to a pseudo metric. Next we will look at the special case of the multiplicity of a knot over another knot and some basic results.
Tuesday Nov 6	Jennifer Schultens UC Davis	Title: Width complexes for knots and 3-manifolds Abstract: The advances in the study of complexes and their geometry by geometric group theorists shed light on pertinent questions in 3-dimensional topology, most notably in the case of the curve complex. We here explicitly define complexes implicit in many discussions of knots and 3-manifolds. These complexes, though unwieldy, allow a reinterpretation for old insights concerning knots and 3-manifolds and a motivation for new insights.
Tuesday Nov 13	Ben Benoy UC Santa Barbara	Title: A projective version of the Poincare polyhedron theorem Abstract: I will discuss a generalization of Poincare's polyhedron theorem from the constant curvature geometries to the projective setting. Given a collection of convex polyhedra in Real Projective space, and a scheme for gluing faces via projective transformations, I will give conditions for the resulting quotient to have a real projective structure compatible with the gluings. The main condition concerns the holonomy around a codimension two face and is a direct analogue of the angle sum condition in the constant curvature version of the theorem.
Tuesday Nov 20
Tuesday Nov 27	Chan-Ho Suh UC Davis	Title: 2^{15^4 n} Reidemeister moves suffice to unknot. Abstract: Given an unknot diagram D with n crossings, Joel Hass and Jeffrey Lagarias proved there existed a function, depending only on n, bounding the mininum number of Reidemeister moves to take D to the standard unknot. Their proof relied on normal surface theory but with complications related to drilling out a regular neighborhood of the knot. We will describe a new type of normal form for surfaces with boundary in the 1-skeleton of a triangulated 3-manifold. This normal form allows a very quick and dirty proof of the Hass--Lagarias theorem. It also lets us "improve" the Hass--Lagarias upper bound of 2^{10^11 n} to 2^{15^4 n}.
Tuesday Dec 4	Ron Stern UC Irvine	postponed until Feb 5, 2008
Tuesday Dec 11

Spring 2008 Schedule

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.