1: Rational Tangles
- 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.
"What I like doing is taking something that other people thought was complicated and difficult to understand, and finding a simple idea. So that any fool - and, in this case, you - can understand the complicated thing. ---John Conway"
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.
References
- Davis, T. (2017). Conway's Rational Tangles . Accessed at http://www.geometer.org/mathcircles/tangle.pdf .
- Kauffman, L. H., & Lambropoulou, S. (2004). On the classification of rational tangles. Advances in Applied Mathematics , 33(2), 199-237. Available on arXiv at http://arxiv.org/pdf/math/0311499.pdf .
- Tanton, J. (2012). Understanding Rational Tangles . Accessed at http://mathteacherscircle.org/assets/session-materials/JTantonRationalTangles.pdf .
Thumbnail: A knot diagram of the trefoil knot, the simplest non-trivial knot. (Public Domain; Marnanel via Wikipedia )