Author Topic: The fig.8., the 3 fig.9, and 45 more stoppers.  (Read 1599 times)

xarax

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The fig.8., the 3 fig.9, and 45 more stoppers.
« on: March 11, 2014, 11:24:08 PM »
  When I was telling Derek Smith that I had spent my day tying 49 stoppers, I was not joking !  :)
  As proved by J.W.Alexander (*), every knot can be transformed to and represented as a braid.
  Now, there is an interesting most simple class of braids that are called "two bridge" braids - and the knots that can be transformed to / represented as such braids, are called "rational knots". To the practical knot tyer, they are the familiar fig.8, fig.9, etc., knots, that can also serve as stoppers.
   However, although most of the simpler knots can be turned into two bridge braids, and thus serve as stoppers, there are some that can not - the 8_5 and 8_10, for example (3), are knots with 8 crossings that can not be transformed into such fi.8 / fig.9 like stoppers. 
   Herman Grubber (1) offers diagrams of those knots, from 4 to 16 crossings, in his site :
   http://www2.tcs.ifi.lmu.de/~gruberh/
   I reproduce his 49 diagrams of the knots from 4 to 9 crossings here, in two pages, for convenience. ( See the attached pictures ). The continuous red line in the lower part of each diagram should be "cut", so the braid of the "open" knot will be "open", with two ends and three entangled legs along its length.
   P-V. Koselef (2) shows the same knots, and he also shows their so-called Chebyshef (tying) diagrams, which are very interesting to the practical knot tyer, because they represent an easy way of how to actually tie those stoppers.
  http://www.math.jussieu.fr/~koseleff/knots/kindex1.html
 
 1. http://www.hermann-gruber.com/
 2. http://www.math.jussieu.fr/~koseleff/
 3. http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table
   
  (*) Alexander was also a noted mountaineer, having succeeded in many major ascents, e.g. in the Swiss Alps and Colorado Rockies. The Alexander's Chimney, in the Rocky Mountain National Park, is named after him. When in Princeton, he liked to climb the university buildings, and always left his office window on the top floor of Fine Hall open so that he could enter by climbing the building
This is not a knot.

xarax

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Re: The fig.8., the 3 fig.9, and 45 more stoppers.
« Reply #1 on: April 14, 2014, 07:18:10 PM »
   A related picture, with a few simple "rational" "closed" knots, which can be turned into "closed" two-bridge braids, which can be turned into "open" stoppers.

http://homepages.warwick.ac.uk/~maaac/images/3_1to7_7.jpg
This is not a knot.