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Euler's Gem by David S. Richeson

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CHAPTER 18

A KNOTTY PROBLEM

O Time, thou must untangle this, not I. It is too hard a knot for me t’untie.—William Shakespeare, Twelfth Night1

One of the earliest topological investigations was the study of knots. We are all familiar with knots. They keep our boats secured to shore, our shoes snug on our feet, and the cables and wires hopelessly tangled behind our computers. These are not, strictly speaking, mathematical knots. A mathematical knot has no free ends; it is a topological circle living in 3-dimensional Euclidean space. (To turn an electrical extension cord into a mathematical knot, simply plug the two ends together.)

In figure 18.1 we see the projections of six mathematical knots: the unknot, trefoil knot, figure eight knot, pentafoil ...

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