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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.44 Knot Polynomials

W. B. R. Lickorish

1 Knots and Links

A knot is a curve in three-dimensional space that is closed (in other words, it stops where it began) and never meets itself along its way. A link is several such curves, all disjoint from one another, which are called the components of the link. Some simple examples of knots and links are the following:

Image

Two knots are equivalent or “the same” if one can be moved continuously, never breaking the “string,” to become the other. Isotopy is the technical term for such movement. For example, the following knots are the same:

The first problem in knot theory is how to decide whether two ...

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