[1]

Davis, T. (2017).

*Conway's Rational Tangles*. Accessed at`http://www.geometer.org/mathcircles/tangle.pdf`.\(\newcommand{\identity}{\mathrm{id}}
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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.

[1]

Davis, T. (2017). *Conway's Rational Tangles*. Accessed at `http://www.geometer.org/mathcircles/tangle.pdf`.

[2]

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`.

[3]

Tanton, J. (2012). *Understanding Rational Tangles*. Accessed at `http://mathteacherscircle.org/assets/session-materials/JTantonRationalTangles.pdf`.