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Dan Silver


Daniel S. Silver
Professor of Mathematics
Department of Mathematics and Statistics
ILB 325
University of South Alabama
Mobile, AL 36688-0002

(251) 460-6264/ 460-7969 FAX
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Five Minutes of Knot Theory History

Click When Smoke Gets in Your Eyes to watch a 5-minute video of my dangerous attempt to recreate an experiment that P.G. Tait performed for Lord Kelvin in 1867. More about the poisonous session can be found in an article that I wrote for American Scientist Knot Theory's Odd Origins. (E-reprints are available. Please ask!) Co-starring are Susan Williams and University of South Alabama chemistry professor Andrzej Wierzbicki.

Helmholtz worked out the interaction of two smoke rings traveling in the same direction: "If they have the same direction, the foremost widens and travels more slowly, the pursuer shrinks and travels faster, till, finally, if their velocities are not too different, it overtakes the first and penetrates it. Then the same goes on in the opposite order, so that the rings pass through each other alternatively." Click here to see the smokey pas de deux in a great gif produced by artists at American Scientist.

Kelvin incorporated Helmholtz's ideas into his theory of the vortex atom. According to this theory, chemical elements are small, knotted vortices of ether. Classify knots and links, and you classify the elements.

Maxwell attempted to understand how three rings of ether would pass through each other. On October 6, 1868, he wrote to Kelvin: "[Helmholtz's] 3 rings do as 2 rings in his own paper that is those in front expand and go slower those behind contract and when small go faster and thread through the others. I drew 3 to make the motion more slow and visible not that I have solved the case of 3 rings more than to get a rough notion about this case and to make the sum of the three areas [constant] I have made them fat when small and thin when big." [Maxwell's lack of punctuation was not typical of his writing, which was generally excellent.]

Now for the best part! Maxwell drew the figures to be viewed on a zoetrope ("wheel of life") of his own design. The Cavendish Museum, Cambridge owns the device. Below on the left is a photograph reproduced from The Scientific Letters and Papers of James Clerk Maxwell, edited by P.M. Harman, published by Cambridge University Press, 1995. On the right is a picture of Professor Gordon Squires and me in 1996, during a visit to the museum (photo by Susan Williams). Professor Gordon was then the museum curator. He was very kind to Susan and me during our visit. Sadly, he passed away in 2010.


At my request, Cavendish Museum very kindly put together a gif so that we can all see what Maxwell saw when spinning his zoetrope. My great thanks to Dr. Lisa Jardine-Wright, Educational Outreach Officer and Professor Malcom Longair, current curator, for making this happen. [Copyright is owned by Cavendish Museum, and the image is reproduced here with their permission.]


Now we step into the 20th century. Here is a photograph of topologist Ralph Fox that was given to me by my friend and collaborator Wilbur (Red) Whitten. Fox was a pioneer and tireless promoter of knot theory when the subject was still young. An altered version of the photograph appears in John Milnor Collected Papers, Volume 2 (Publish or Perish Press). If you look closely, you can see Fox's signature.


Click here for a short, elementary exposition on knots and Fox colorings, intended for undergraduates. It is titled ``Why Knot?"


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