1: Rational Tangles

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Learning Objectives

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)