Individual Research Projects Section 16.5

Project 16.3 What In the World?

“I think I’m going crazy, Bill,” said George.  “I can’t figure out a good rating system for the state league.  I’ve got over a hundred teams and I need to come up with a statewide rating system to rank all of those teams.  Any ideas?” ”

As a matter of fact, yes!” said Bill enthusiastically.  “Have you ever heard of the Harbin Football Team rating system?  I think they use it in Ohio.  They use it to determine the high school football teams that are eligible to compete in post-season playoffs.  Each team earns points for games it wins and for games that a defeated opponent wins.  Here’s how it works:  Level 1 points are awarded for each game a team wins.  A level-2 point is awarded for each game a defeated opponent wins.” “I hear ya, but I still don’t get it. How do you put all that together?” asked George.

“Well,” said Bill, “it’s very easy.   We simply use a matrix.” Rank teams A, B, C, D, E, and F if you are given the following information.
Team A beats F, and ties C;
Team B beats A, C, and F;
Team C beats E and F;
Team D ties A and beats F;
Team E beats A and F;
Team F beats C and ties D.  If two teams tie, enter 0.5 in the zero-one matrix instead of 1.

Project 16.4 Knot Theory

Get a piece of string with two free ends, and tie those ends  together with a knot. Some knots that you can tie will hold the  ends of the string together and other knots will not  (no pun intended!) In mathematics, there is a branch of mathematics  known as knot theory. Mathematically, a knot is defined as a closed  piecewise liner curve in R3. Two or more knots together is known as  a link. Knots can be cataloged according to the number of crossings  (ignoring mirror reflections). There is only one knot with crossing  number three (called the cloverleaf knot), one knot with crossing  number four, two with crossing number five, and three with crossing  number of six.

* This research project is adapted from R. E. Kohn, “A Mathematical  Programming Model for Air Pollution Control,” Science and Mathematics,  June 1969, pp. 487-499.<P>

How many knots are possible with crossing number of seven? How many knots are possible with crossing number of eight? Write a paper on knot theory.