# 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

Thumbnail: A knot diagram of the trefoil knot, the simplest non-trivial knot. (Public Domain; Marnanel via Wikipedia)

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