Topology Seminar

Volume of links and the colored Jones polynomial

05/03/2016 - 3:00pm
05/03/2016 - 4:00pm
Speaker: 
Anastasiia Tsvietkova, UC Davis
Abstract: 

Since quantum invariants were introduced into knot theory, there has been a strong interest in relating them to the intrinsic geometry of a link complement. This is for example reflected in the Volume Conjecture, which claims that the hyperbolic volume of a link complement in 3-sphere is determined by the colored Jones polynomial. In the work of M. Lackenby, and of I. Agol and D. Thurston, an upper bound for volume of a hyperbolic link complement in terms of the number of twists of a link diagram is obtained. We will discuss how to refine this bound, and how to generalize it from hyperbolic to simplicial volume of links. We will also show how to express the refined bound in terms of the three first and three last coefficients of the colored Jones polynomial for alternating links. The talk is based on joint work with O. Dasbach.

Where: 
Millikan 2099, Pomona College

Seiberg-Witten & Turaev torsion invariants of 3-orbifolds

04/26/2016 - 3:00pm
04/26/2016 - 4:00pm
Speaker: 
Biji Wong, Brandies University
Abstract: 

We construct a torsion invariant of 3-orbifolds with singular set a link. It generalizes the Turaev torsion invariant of 3-manifolds, and gives more information than Baldridge's Seiberg-Witten orbifold invariant. Furthermore, when the singular set is a nullhomologous knot, the invariant not only recovers the Turaev torsion invariant of the underlying 3-manifold, but also detects the isotropy group associated to the knot, and the Turaev torsion invariant of the exterior.

Where: 
Millikan 2099, Pomona College

Rationally null-homologous knots and rational Seifert surfaces

04/19/2016 - 3:00pm
04/19/2016 - 4:00pm
Speaker: 
Katherine Raoux, Brandeis University
Abstract: 

Seifert surfaces have long been a useful tool for defining invariants of knots in the 3-sphere. More generally, Seifert surfaces exist for any null-homologous knot. However, in a general 3-manifold, not every knot bounds a surface since the first homology group may be non-trivial. On the other hand, if a knot represents a torsion homology class, some multiple of the knot will bound a surface. Such a surface is called a rational Seifert surface. I will explain this construction in detail and give examples. In addition, I will show how certain classical knot invariants, such as the genus and self-linking number, can be defined and studied for this more general class of knots. If time permits, I will also discuss how to define tau-invariants from Heegaard Floer homology for this class knots.

Where: 
Millikan 2099, Pomona College

Legendrian graphs with all cycles unknots of maximal Thurston-Bennequin number

10/25/2011 - 3:00pm
10/25/2011 - 4:00pm
Speaker: 
Elena Pavelescu, Occidental College
Abstract: 

A Legendrian graph in a contact structure is a graph embedded in such a way that its edges are everywhere tangent to the contact planes. In this talk we look at Legendrian graphs in $ R^3 $ with the standard contact structure. We extend the invariant Thurston-Bennequin number (tb) from Legendrian knots to Legendrian graphs.

We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with $ tb=-1 $ if and only if it does not contain $ K_4 $ as a minor. 

Where: 
Millikan 211

TBA

11/08/2011 - 3:00pm
11/08/2011 - 4:00pm
Speaker: 
Ismar Volic, Wellesley College
Abstract: 

 tba

Where: 
Millikan 211, Pomona College

Bridge numbers of links and meridian ranks of link groups

11/01/2011 - 3:00pm
11/01/2011 - 4:00pm
Speaker: 
Yeonhee Jang, Hiroshima University, Japan
Abstract: 

A question posed by Cappell and Shaneson asks whether the bridge number of a given link equals the minimal number of meridian generators of its group. We give a positive answer to this question for certain links containing all arborescent links.

Where: 
Millikan 211, Pomona College

TBA

10/11/2011 - 3:00pm
10/11/2011 - 4:00pm
Speaker: 
Mark Kidwell, US Naval Academy
Abstract: 

TBA

Where: 
Millikan 211, Pomona College

An Algebraic Approach to Knot Floer Homology

10/04/2011 - 3:00pm
10/04/2011 - 4:00pm
Speaker: 
Allison Gilmore, Princeton
Abstract: 

Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. Starting with a braid diagram for a knot, one singularizes or smooths each crossing, then associates an algebra to each resulting singular braid. These can be arranged into a chain complex that computes knot Floer homology. After introducing knot Floer homology in general, I will explain this construction, then outline a fully algebraic proof of invariance for knot Floer homology that avoids any mention of holomorphic disks or grid diagrams. I will close by describing some potential applications of this algebraic approach to knot Floer homology, including potential connections with Khovanov-Rozansky's HOMFLY-PT homology.

Where: 
Millikan 211, Pomona College

Twisted Alexander Polynomials of 2-bridge Knots

09/27/2011 - 3:00pm
09/27/2011 - 4:00pm
Speaker: 
Jim Hoste, Pitzer College
Abstract: 

We show how to compute the twisted Alexander polynomial of 2-bridge knots using the Fox coloring. For genus 1 2-bridge knots, we verify that the polynomial has the form conjectured for all 2-bridge knots by Hirisawa and Murasugi. This is joint work with Pat Shanahan.

Where: 
Millikan 211, Pomona College

No Meeting

10/18/2011 - 3:00pm
Speaker: 
No Meeting
Abstract: 

None

Where: 
Millikan 211
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