LSU

Mathematics

Mathematics

Description: 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.

**MATH 4997: Vertically Integrated Research: The Alexander Polynomial and its Relatives.**- Instructor: Profs. Dasbach and Stoltzfus, Dr. Kearney.
- Prerequisites: Elementary Topology (MATH 4039) or equivalent or permission of instructors.

Description: One of the most studied knot invariants is the Alexander polynomial. There are many different ways known to effectively compute it. Some are combinatorial in nature, some are more closely related to the topology of the knot complement. We will discuss the Alexander polynomial as well as some of its relatives, such as knot signatures, the A-polynomial and Heegaard Floer knot homology.

**MATH 4997-7: Vertically Integrated Research: Algorithms and computations in knot theory.**- Instructor: Profs. Dasbach and Stoltzfus, Drs. Kearney and Tsvietkova.
- Prerequisites: Math 2057 (Calculus of Several Variables)

Description: We will explore computational methods in knot theory, particularly of hyperbolic knots, using open source tools: SnapPea, Snap and Bar Natan's Mathematica package KnotTheory.

We particularly welcome undergraduate participants for this VIR course as we will be developing geometric concepts of three-dimensional hyperbolic geometry related to the visualization and design of solid geometric objects which can now be printed on 3D printers.

*Total Enrolled - 18, Graduate - 10, Undergraduate - 8*

**MATH 4997-4: Vertically Integrated Research: Algorithms and computations in knot theory.**- Instructor: Profs. Dasbach and Stoltzfus, Drs. Kearney and Tsvietkova.
- Prerequisites: Math 2057 (Calculus of Several Variables)

*Total Enrolled - 16, Graduate - 13, Undergraduate - 3*

Description: We will explore computational methods in knot theory, particularly of hyperbolic knots, using open source tools: SnapPea, Snap and Bar Natan's Mathematica package KnotTheory.

We particularly welcome undergraduate participants for this VIR course as we will be developing geometric concepts of three-dimensional hyperbolic geometry related to the visualization and design of solid geometric objects which can now be printed on 3D printers.

- MATH 4997-4:
**Vertically Integrated Research: Knot invariants coming from Lie algebras** - Instructors: Profs. Dasbach, Stoltzfus, and Dr. Kearney.

We will study knot invariants arising from Lie algebras including the proof of the MMR conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander-Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the "colored" Jones polynomial). encapsulated in the paper of Bar-Natan and Garoufalidis

In addition, we will read the paper of Saleur & Kauffman relating the Lie algebra GL(1|1) to the Alexander Polynomial and other approaches to the MMR conjecture by XS Lin and A. Vaintrob.

*Total Enrolled - 11, Graduate - 10, Undergraduate - 1*

**MATH 4997-4: Vertically Integrated Research: Links on a Torus and Dimer Invariants**- Instructors: Profs. Dasbach, Stoltzfus and Dr. Kearney.

The knot theory VIR course will study the properties of links with diagrams which project to the torus fiber of the Hopf link. Specifically, we will study the dimer models for the zig-zag links introduced by Stienstra, their Kasteleyn matrices and work to develop related link invariants for these links.

*Total Enrolled - 12, Graduate - 11, Undergraduate - 1*

- MATH 4997-4:
**Vertically Integrated Research: Representations of knot groups.** - Instructors: Profs. Dasbach, Stoltzfus, and Dr. Russell.
- Prerequisite: a fundamental understanding of abstract algebra.

We will study representations of fundamental groups of knot complements and their combinatorics. Topics will be: The A-polynomial of knots, representations of knot groups into SU(n), combinatorial interpretations of certain knot group representations.

*Total Enrolled - 12, Graduate - 10, Undergraduate - 2*

- MATH 4997-4
**Vertically Integrated Research: The colored Jones polynomial.** - Instructors: Profs. Dasbach, Stoltzfus and Dr. Russell
- Prerequisite: Undergraduate Topology or permission of instructor.

The colored Jones polynomial is one of the more mysterious objects in knot theory. We will start with various definitions of it and will try to develop some of its properties. The methods will be elementary.

*Total Enrolled - 13, Graduate - 13, Undergraduate - 0*

**MATH 4997-3: Vertically Integrated Research: Dimer models and knot theory.**- Instructors:
**Profs. Dasbach, Stoltzfus, Dr. Russell** - Prerequisite: Math 2057 and Math 2085 for undergraduates.

A dimer covering of a graph is a subset of edges that covers every vertex exactly once. Dimer models are important objects of study in statistical mechanics, probability theory and more recently

*Total Enrolled - 9, Graduate - 8, Undergraduate - 1*

**MATH 4999-3: Vertically Integrated Research: Visualization of knots on surfaces**- Instructors:
**Prof. Dasbach**with Profs. Heather Russell and Neal Stoltzfus

Over the last 20 years knot theory became one of the central areas in mathematics. One studies properties of knots, like the unknotting number which measures the easiest way to unknot a knot. Our interest will be in the surfaces on which knots project in some nice way, and what those projections tell us about the knot.

We will learn and apply methods in computer graphics, differential geometry, knot theory and other areas of mathematics. The course is intended for both undergraduate and graduate students.

*Total Enrolled - 8, Graduate - 6, Undergraduate - 2*

**MATH 4999-3: Vertically Integrated Research: Differential geometry of curves and surfaces in space.**- Instructors:
**Prof. Dasbach**and**Prof. Stoltzfus**.

We will discuss the differential geometry of curves and surfaces in space. The course will also incorporate the use of Mathematica in the computation and visualization. As applications we will offer projects in Computer Graphics and 3-dimensional positioning systems (location detection in space with the help of two cameras).

Project: Each student would complete a project (possibly in groups) and make a presentation on a geometric topic.

*Total Enrolled - 9, Graduate - 2, Undergraduate - 7*

**Vertically Integrated Research: Three dimensional positioning systems.**- Instructors:
**Prof. Dasbach**and**Prof. Stoltzfus**.