Date: Tuesday Feb 7th and April 25th 2017
Speaker: Abhijit Champanerkar
Title: An application of non-positively curved cubings of alternating links
Abstract : We will give an overview of a recent paper by Sakuma-Yokota
and discuss plans to read through this paper and related background.
Reference paper:
An application of non-positively curved cubings of alternating links by
Sakuma and Yokota
Date: Tuesday Feb 14th 2017
Hyperbolic ideal tetrahedra and volume
Daniel Berlyne
Abstract: We will discuss the parametrization of hyperbolic ideal tetrahedra
and discuss its volume.
References:
- Book: Lectures on Hyperbolic Geometry by Benedetti & Petronio: Section C.2
- Article: Computation of Hyperbolic Structures in Knot Theory by Jeff Weeks: Sections 1 and 4
- Book: Hyperbolic Knot Theory by Jessica Purcell: Chapter 3
Date: Tuesday Feb 21st & March 7th 2017
Ideal triangulation and gluing equations
Alice Kwon
Abstract: We will give examples of ideal triangulations for knot and link complements and describe Thurstons gluing equations.
References:
- Book: Hyperbolic Knot Theory by Jessica Purcell: Chapter 4
Date: Tuesday Feb 28th 2017
Non-positively curved cube complexes
Jacob Russell-Madonia
Abstract : We will give a quick introduction to non-positively curved
cube complexes, mention Gromov's link condition and show that the Dehn
complex of a link diagram is NPCCC if and only if the diagram is
prime, reduced and alternating diagram.
References:
- From Riches to Raags: 3-manifolds, Right-angled Artin Groups, and Cubical Geometry by Dani Wise
- An introduction to RAAGs by Ruth Charney
- MSRI GGT Summer school video lectures by Jason Manning
- Geometry and combinatorics of cube complexes by Mark Hagen
Date: Tuesday March 14th 2017
Octahedral decomposition of link complements
Rachel Popp
Date: Tuesday March 28th 2017
SnapPy
Daniel White
Abstract : We will show a demo of the program SnapPy which computes hyperboic
structures on 3-manifolds and link complements, and computes many invariants.
References:
SnapPy
Date: Tuesday May 9th 2017
Lehmer's Question and Knot Theory
Dan Silver (University of South Alabama)
Abstract : In 1933 D.H. Lehmer asked whether the product of moduli of
roots of a monic integral polynomial can be arbitrarily close but not
equal to 1. The question remains open despite extensive effort. We
discuss its relationship with knot and 3-manifold theory.
References:
- Lehmer, D.H. (1933). Factorization of certain cyclotomic functions . Ann. Math. (2). 34: 461-479.
- Smyth, Chris (2008). The Mahler measure of algebraic numbers: a survey In McKee, James; Smyth, Chris. Number Theory and Polynomials. London Mathematical Society Lecture Note Series. 352. Cambridge University Press. pp. 322-349.
- Silver, Daniel S. and Williams, Susan G. (2007). Lehmer's question, knots and surface dynamics Mathematical Proceedings of the Cambridge Philosophical Society. 143: 649-661.
Date: Tuesday May 16th 2017
Angle structures on hyperbolic 3-manifolds
Alice Kwon
Abstract : A hyperbolic ideal tetrahedron is determined by its three
dihedral angles. In the 1990s , Andrew Casson and Igor Rivin
discovered a technique for solving Thurston's gluing equations using
angle structures on an ideal triangulation for a hyperbolic
3-manifold. Using angle structures the gluing equations separate into
a linear part and a non-linear part. The solutions to the linear
system of equations form a convex polytope, and the solutions to the
non-linear part is a critical point of a certain volume functional on
this polytope. This talk will focus on the main theorem that a
critical point of the volume functional produces a complete hyperbolic
structure. We will illustrate this method on many examples and give
some applications.
References:
- From angled triangulations to hyperbolic structures
by Dave Futer and Francois Gueritaud,
Contemporary Mathematics 541 (2011), 159-182.
ArXiv