California State University Chico

Title: Contemporary Research in Knot Theory

Director(s): Thomas Mattman, Department of Mathematics

Dates of Program: June 14 - July 23, 2004

Summary: This project will give three undergraduate students the opportunity to work intensively on contemporary research in knot theory. There are three possible problems, with the student free to choose one project for the six weeks. The first is entitled Trip Matrix Formulation of the Jones Polynomial. When Jones announced his polynomial in 1985, he revolutionized the field of knot theory. The Jones polynomial is much better at distinguishing knots than anything that had gone before. It has been suggested that many properties of the Jones polynomial are in fact algebraic properties arising from the form of the trip matrix. Investigating which features of the polynomial come from the topology of the knot and which are algebraic is an important open question that the students could attack effectively. The second project is entitled Irreducible Trace Polynomials. Following the revolution sparked by the Jones polynomial, many new kinds of knot polynomials have been introduced in the last few years. The trace polynomial defines the trace field of a knot. Dr. Mattman discovered a family of polynomials that, in all likelihood, are the trace polynomials for the (-2, 3, n) pretzel knots. They are defined inductively by the formulae: p_-1(x) = x³ + 2x² + x + 1; -3(x) = -x⁵ - 3x⁴2 - 4x - 2; and n = -((x² + x + 2) p_n+22n+4) (for n odd and negative) What remains is to show that these polynomials are irreducible. It is easy to verify that any particular p_n is irreducible using a computer algebra program such as Mathematica. It will be a good challenge for my students to come up with an argument that applies to all polynomials in the infinite family. The third project is A-polynomial of Pretzel Knots. The A-polynomial is another knot polynomial associated with 2(C)-representations. The calculation of A-polynomials is, in principle, straightforward and can be carried out for any knot. However, for a knot of n crossings, this involves beginning with a system of n polynomial equations in n+1 indeterminates and using elimination theory to reduce to a single equation in two unknowns. To date, only about 30 A-polynomials have appeared in literature, So, simply calculating the polynomials of several knots would be an important contribution to current research. The students are encouraged to pick one of these three projects as a team, and therefore combine their efforts throughout the research program.

Student Researchers: