| 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 |
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| 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 |