MATH 4997

Vertically Integrated Research: The A-Polynomial and its Relatives

This course originated in the VIGRE program in the Dept. of Mathematics at LSU. The Vertically Integrated Research or VIR courses are a new research experience for graduate and undergraduate students. These are the main educational instrument of the VIGRE activities. These courses run as Math 4997 and are intended to provide opportunities for students to learn about mathematical research in a vertically integrated learning and research community.

Spring 2014

The instructors for this course are: Profs. Dasbach and Stoltzfus, Dr. Kearney.

Prerequisites: Topology I (MATH 7510) or equivalent, or permission of instructors.

The A-polynomial of a knot roughly measures the eigenvalues of those representations of the fundamental group of the torus tube about a knot which extend to the entire knot complement. We will study different known ways to effectively compute using the various presentations of fundamental groups of knot complements. We will also have a visiting expert in the field during the semester.

Section 2 Instructor: Dasbach, Stoltzfus; Kearney Email: stoltz@lsu.edu
Class: 137 Lockett: 3:30 MWF Office: 258/306 Lockett URL: www.math.lsu.edu/~stoltz/Courses/S14/4997/
Lockett Office Hours: 10:30am TTh & TBD Office Fax: 225.578.4276 Office Phone: 225.578.1656

Course Information: The latest course information will be kept on this site.

Syllabus:

Here is the syllabus for the course.

Course Materials

The over-arching theme is to study topological invariants surrounding the A-polynomial of a knot (as the fall semester was devoted to the study of topics related to the Alexander polynomial.)

The A-polynomial measures which two-by-two (det =1) representations of the fundamental group of the torus tube about a knot extend to the fundamental group of the knot complement. As the fundamental group of a torus has two independent generators and is commutative, the representation is determined by its eigenvalues. The classic paper on the A-polynomial written by Cooper-Culler-Gillet-Long-Shalen(CCGLS) combines two approaches (one topological, the other algebro-geometric) into one paper: This paper and more recent computations are available on Marc Culler's web site: http://www.math.uic.edu/~culler/

I. Cord Algebra invariants (developed by Lenny Ng)

The cord algebra studies an algebra constructed from the set of cords of a knot; that is, the set of path between two points on a knot (embedded circle) under a "skein-like" equivalence relation. This attached picture illustrates cords for the trefoil.

This topic can be approached from both a presentation and group theoretical perspective: a presentation of the cord algebra is given in Lenny Ng's first paper referenced below and the group theoretical perspective is found in the appendix of the second.

In addition there is a strong connection between the cord algebra and the A-polynomial elucidated by Lenny Ng in the papers below.

References:

arXiv:math/0302099 Knot and braid invariants from contact homology I. Lenhard Ng. Geom. Topol. 9(2005) 247-297. math.GT (math.SG).

arXiv:math/0303343 Knot and braid invariants from contact homology II, with an appendix written jointly with Siddhartha Gadgil. Lenhard Ng. Geom. Topol. 9(2005) 1603-1637. math.GT (math.SG). The appendix connects the cord ring to the fundamental group and peripheral structure of a knot exterior.

II. Abhijit's Thesis: Relationship between the A-polynomial and Eigenvalues of the meridean and longitude of a faithful hyperbolic representation of the fundamental group of a hyperbolic knot

From the abstract: "Using combinatorics of an ideal triangulation of a hyperbolic 3-manifold N, a complete, orientable, finite-volume, one-cusped Abhijit constructs a plane curve in C×C which contains the squares of eigenvalues of PSL(2,C) representations of the meridian and longitude. We show that the defining polynomial of this curve is related to the PSL(2,C) A-polynomial and has properties similar to the classical A-polynomial."

http://www.math.csi.cuny.edu/abhijit/papers/apoly_bloch.pdf

III. Knot Contact Homology & Augmentations

During the spring semester there will be a visit by an expert in the area of augmentation polynomials: Christopher Cornwell who will also talk in the Virtual Seminar.

Christopher Cornwell: arXiv:1310.7526 KCH representations, augmentations, and \$A\$-polynomials

Abstract: "Knot contact homology is an invariant of knots in R^3, defined via the geometry of the cotangent bundle. Augmentations of knot contact homology are quite useful for relating it to other knot invariants. A correspondence between augmentations and certain representations of the knot group implies the 2-variable augmentation polynomial equals a generalization of the classical A-polynomial."

Office Hours: These times are especially reserved for you, but other times can easily arranged by appointment. If you just drop by my office, Lockett 258, when I am there I am happy to discuss things, unless I have pressing deadlines. Please introduce yourself!

Communication: The best mode of communication is to ask questions in class. Electronic mail and telephone messages are also good for after-class short answer questions and emergencies.

Advice to Students: A good motto is: Focus on Conceptual Understanding! Plan on spending two hours each day reading the section(s) covered, reviewing your class notes and working homework exercises. The material in this course builds cumulatively and the amount covered in one week is substantial.

I expect you to attend each regularly scheduled class and to keep up with the assigned work. Please discuss with me any emergencies requiring you to be absent, to be late for class or to leave early.

Neal W. Stoltzfus   Spring 2014