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Title: Contemporary Research in Knot Theory

Director(s): Thomas Mattman, Department of Mathematics

Email: tmattman@csuchico.edu

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) = x^{3}
+ 2x^{2} + x + 1; -3(x) = -x^{5}
- 3x^{4 }2
- 4x - 2;
and
n = -((x^{2}
+ x + 2) p_{n+2}2n+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:

- Yara Alcala, CSU Chico
- Roberto Raya, CSU Chico
- Matt Rodrigues, CSU Chico
- Dan Tating, CSU Chico

Program Contacts:

Bill Hawkins

MAA SUMMA

bhawkins@maa.org

202-319-8473

Michael Pearson

MAA Programs & Services

pearson@maa.org

202-319-8470