As we discovered in our first class, crossings are one of the first ways for us to understand the connections between knots and algebra: somehow, if we can say "enough" about how a strand crosses itself, we can characterize the essential nature of a knot.
So we'll begin by focusing as much as possible only on crossings, by studying objects known as tangles, in which crossings are created between two strands by twisting up their endpoints.
- Discover how different types of twists on a tangle determine its tangle number.
- Argue for why the arithmetic of the rational numbers makes certain relationships among tangle numbers necessary.
- Calculate the fraction of a rational tangle in two different ways, and argue for why the fraction is an invariant of rational tangles.