Mathematics Summer Program
Knots at McKendree University
This is a summer program in mathematics for under-represented groups in the mathematical sciences. The program is a Mathematical Association of America activity funded by National Security Agency (grant H98230-06-1-0156) and the National Science Foundation (grant DMS-0552763).
The goal of this program is to expose undergraduate students to mathematical research and assist them in developing a comprehensive career plan. Participants will perform mathematical research and participate in a variety of professional development activities.
Organizers:
Dr. J Alan Alewine
Dr. Heather A. Dye
Participants:
David Etheridge from Augustana College
Irina Garduno from The University of Illinois at Chicago
Amber Ramos from McKendree University
The Mathematical Research:
The students will explore one of two possible problems. The first problem is motivated by two papers by Jozef Przytycki and Qi Chen: The Gram matrix of a Temperly-Lieb algebra is similar to the matrix of chromatic joins (arxiv:0806.0878) and The Gram determinant of the type B Temperly-Lieb Algebra. The second problem is based on a paper by Louis Kauffman and Sam Lomonaco: Quantum knots and Mosaics (arxiv:0805.0339v1). In this paper, the authors define knots based on a grid structure. To study these problems, students should complete the following courses or their equivalent: Introduction to proof and Linear algebra. These courses should introduce the students to the concepts of equivalence classes, bilinear pairings, and determinant. The motivating problems involve number theory, algebra, and knot theory so that the research focus can be directed based on student interest.
Students will study the problems described below:
- Mosaic Knots
- What size grid do you need to construct all 4 or 5 crossing mosaic knots?
- What is the maximum number of crossings that an mosaic knot can contain?
- Temperly-Lieb Algebras in Surfaces
- Consider a disk with m punctures and 2n points around the outermost boundary. We denote this surface as . Find the set of all diagrams in (up to isotopy) with n non-crossing chords.
- Form a surface with genus m by gluing two copies of together. Tabulate the number of curves that bound a disk and the number of curves that do not bound a disk for each pair of diagrams from question 1.
- Compute the determinant of the bilinear pairing found in part 2. How does the determinant change as m or n changes?
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