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!). There is a branch of mathematics known as knot theory. Mathematically, a knot is defined as a closed piecewise linear curve in \(\mathbb{R}^{3}\). 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), and one knot with crossing number four, and there are two with crossing number five, and three with crossing number six.

a. How many knots are possible with crossing number seven?

b. How many knots are possible with crossing number eight?

Write a paper on knot theory.