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Explorations in Topology by David Gay

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13

Knots

This chapter introduces the concept of a knot, a simple closed curve in three-space, and begins the process of distinguishing one knot from another. Along the way, a knot picture is introduced along with language to identify its features: thread, overpasses, and underpasses. “Moves” for a knot picture are spelled out and the notion of knot equivalence is defined in terms of them. Three-colorability (and later, n-colorability) for a knot picture is introduced and is shown to be invariant under the moves. The notes discuss the history of knot theory and some issues surrounding the classification of knots. This discussion brings into contrast the relatively straightforward classification of surfaces and the much richer and more complicated ...

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