From Simons Science News (find original story here).
The mathematics of knot theory says that a simple loop and a trefoil are fundamentally different knots. But is that all there is to the question? How can you take a rubber band and turn it into a knot?
Knot theory is the study of knots and their diagrams. It includes techniques for characterizing how knots can be transformed in three-dimensional space and tests for determining when two different-looking knots are actually equivalent. A good introduction is “The Knot Book” by Colin Adams, which explains in detail the “Jones polynomial” mentioned in the video.
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More videos from the Mathematical Impressions series.