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| About site: People/Personal Homepages/K - Krebes, David A |
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| About site: http://www.telusplanet.net/public/dkrebes |
Title: People/Personal Homepages/K - Krebes, David A Includes a thesis work on knot theory and information on schizophrenia.
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David A. Krebes' Personal Home Page
 Dave Krebes
 
// text for first two pictures
I have devoted much of my life to the study ofknots. In my Ph.D. thesis(math), I show that a seamless unknotted loop (circle) of rope cannotbe twisted, contorted or deformed in such a way as to intersect theinterior of a sphere in the pair of arcs shown in the first picture(this is commonly known as a square knot). Equivalently: If a curveintersects the interior of a sphere in a square knot then it isgenuinely knotted (ie. unalterably different from a circle). Forexample in the second picture the loop (follow it around-- it isindeed a single loop) is knotted: A long rubber band cannot bemanipulated into this shape without breaking it and gluing the endsback together. When we say that this a "topological" property of thesquare knot we mean that it is a geometric property that is independent oflengths, angles, or rate of curvature (The size of the circle doesn'tmatter. In fact even an ellipse or a heart shape would do). Themathematical field of "algebraic topology", one of the greatscientific achievements of the twentieth century, expresses many suchproperties in terms of boundaries. Thus to state the result inyet another way: Take a sphere (it could be egg-shaped) and atransparent, rubbery disc with a thick blue opaque circumference(boundary) and try to manoeuvre it (and the sphere too if you like) so that the picture inside thesphere (you can't see what the transparent interior of the disc isdoing) is exactly as shown inthe first picture, ie. a square knot. You will not succeed. However,even after you have tried many times and gained considerableexperience with any of these variations of the problem, you must stillorganize your experience into a mathematical proof before you canconclude with certainty that you didn't stop just one twist and tugtoo soon! As a matter of fact, mathematicians were not satisfieduntil they had proven that the simplest of knots, the overhand knot (you can see twoof these in the square knot above) is not in fact a slip-knot! This issomething we all know from practical experience. Would you ever havethought of trying to prove such a statement?As anotherexercise, try showing that if you replace each of the two strands ofthe square knot with two strands that run side-by-side like a pair ofrailroad tracks through the sphere, for four strands in total, thenthe situation is opposite to that described above: There is indeedsuch a disc, or a way to "tie" ("wrap" might be a better word; thereare no free ends with which to tie) the circle.
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On the bottom we see a knot which, in contrast, canindeed be tied in a rubber band. Press the buttons to see how to tie anduntie it. (For more interesting knot graphics, see the KnotPlot site by Rob Scharein of the University of British Columbia, who authoredthese. Copyright 1998.) If you find this topic interesting please have a look at my problem list. Knots make a great hobby, too. Have a look at Peter Suber's page. Three of the knot theory research mathematicians who have inspired meare Prof. Louis Kauffman (The University of Illinois at Chicago), Prof. Martin Scharlemann (TheUniversity of California at Santa Barbara)(Publications 25 and 26), and mythesis advisor, Prof. Peter B. Shalen (TheUniversity of Illinois at Chicago). Here is a list of some of my favourite poems. For famous literary events (mainly birthdays), arranged by day of the year, please visit the Waterboro Public Library site, the University of Toronto site, or a brief listing of my own. This is excellent information if you're looking for something special to celebrate on a given day. Does anyone know of a similar listing for mathematics? Miscellany: http://www.interlog.com/~r937/doomsday.htmlpresents some ideas on the problem of figuring out the day of the week agiven date falls on, due to the Princeton mathematician JohnH. Conway. Personally, however, I prefer to remember which day of theweek the multiples of seven (7, 14, 21, 28) fall on in the month. Here are my views on the future of text-formatting. Say hello! Send e-mail here, quoting "knots" in the subject field. |
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Includes | a | thesis | work | on | knot | theory | and | information | on | | schizophrenia. |
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http://www.telusplanet.net/public/dkrebes
Krebes, David A 2008 October
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Includes a thesis work on knot theory and information on schizophrenia.
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