# St. Mary's College of Maryland

Title: Unit Stick Knots and the Shape of a Spinning Soap Film

Director(s): Katherine Socha, Sandy Ganzell, Alex Meadows, Susan Goldstine, Cindy Traub

Email: ksocha@smcm.edu

Dates of Program: June 4 - 13, 2007

Summary:

### 1. Unit Stick Knots (Professor Ganzell)

A stick knot is a closed, knotted loop made from straight, rigid sticks (as opposed to flexible rope). The "stick number" of a knot is the fewest number of sticks necessary to make that particular knot. It is an open question in knot theory whether the stick number always equals the "unit stick number", i.e., the fewest number sticks all of the same length necessary to make the knot. This project will test the unit stick number question for several proposed counterexamples. The methodology will be computational. We will program a computer to model a given stick knot and shrink or lengthen its sticks to obtain a unit stick knot. Various algorithms will be employed that will either eliminate the proposed knots as potential counterexamples, or provide computational evidence (but not mathematical proof) that they indeed settle the unit stick question. (Primary reference: Rawdon, Eric J.; Scharein, Robert G. Upper bounds for equilateral stick numbers. Physical knots: knotting, linking, and folding geometric objects in R3 (Las Vegas, NV, 2001), 55-75, Contemp. Math., 304, Amer. Math. Soc., Providence, RI, 2002.)

### 2. The Shape of a Spinning Soap Film (Professor Meadows)

This project will focus on a class of differential equations inspired by the geometrical problem of finding the shape of a spinning soap film. It is a continuation of one of Meadows' projects from the Cornell REU program in the summer of 2004, where he worked with Zhuan Pei, an undergraduate student from Carleton College. Here, the calculus of variations problem is to find a surface of revolution with parallel coaxial circle boundaries that minimizes an energy equal to the sum of its area and its "centrifugal energy" due to rotation about the axis at a fixed angular velocity. We will study several conjectures and new directions that arose in the previous program. Part of the project will be to develop computational tools and run experiments to analyze the mathematical model. For the preliminary educational component of the project, we will focus on curves in the plane and the basic ideas in the calculus of variations. We will read from "The catenary and the tractrix (classroom notes)" by Robert Yates (Math. Monthly 1959) and from "A new minimization proof for the brachistochrone" by Gary Lawlor (Math. Monthly 1996).

Student Researchers Supported by MAA:

• Lauren Blount
• Michael Firrisa
• Darren McCuchen
• Esrael Seyum

Program Contacts:

Bill Hawkins
MAA SUMMA
bhawkins@maa.org
202-319-8473

Michael Pearson
MAA Programs & Services
pearson@maa.org
202-319-8470