Day 16, MAT115

• Reminder: You have an assignment (concerning knots) which is due Thursday after fall break (next time).

• Links:
  • Borromean rings: "We must all hang together, or assuredly we shall all hang separately."
  • Hopf link: obtained by cutting a mobius band "in half", then in half again.
  • Solomon's "Knot" (which is related to the torus knots, starting from four points on a square).

• Knots:
  • The unknot (circle, or simple closed loop) -- zero crossings
  • The trefoil knot (as a torus knot: can be reduced to three crossings)
  • The five-crossing torus knot (can be reduced to five crossings)
  • The five-crossing twist knot
  • The four-crossing twist knot (the "figure-eight" knot)

We talked about how essential it is to make sure that you can distinguish overs-and-unders: we need to know which strand is above, and which is below, every time there is a crossing.

• We now know five knots. How do we show that two squiggle knots are the same ("knot equivalence")? That's the job of the Reidemeister moves.
  1. What are the Reidemeister moves?
     • Tools for showing that two projections of knots are really the same knot.
     • Names of the moves reflect the number of strands involved. A Strand is a piece of a knot picture that goes from undercrossing to undercrossing, with only overcrossings inbetween.
     • Reidemeister moves preserve the knot -- they don't change the knot.

     • Type I	Type II	Type III

  2. Using the Reidemeister moves
     1. to turn a projection of an unknot into another more complicated appearing unknot:

     2. That was a pretty easy one. Let's use all three moves, starting with a circle, to turn an unknot into something that looks complicated -- but is actually the same.

        I'll go over it with Microsoft's Paint program. I've saved off a version of an earlier attempt at this here.

        Here's our set of moves.

     3. Turning a more complicated appearing unknot into a simple circle unknot. (Here it is as a powerpoint presentation.)
     4. Here's my knot, but with commentary. I'm very flattered that someone would steal my knot!:)
     5. A figure-eight knot is equivalent to its mirror reflection, which one can show by Reidemeister moves. However the trefoil knot is not equivalent to its mirror reflection.

  3. Reidemeister moves and Tricolorability
     1. Tricolorability: "A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:
        1. At least two colors must be used,
        2. at most three colors are used, and
        3. at each crossing, the three incident strands are either all the same color or all different colors.

     2. Tricolorability is preserved by Reidemeister moves:

        Reidemeister Move I is tricolorable.	Reidemeister Move II is tricolorable.	Reidemeister Move III is tricolorable.

     3. Therefore, if the projection of one knot is tricolorable and the other isn't, they're different knots (because we can't get from one to the other via Reidemeister moves).

     4. Theorem: there are at least two different knots.
        1. The unknot is not tricolorable ("At least two colors must be used");
        2. The trefoil knot is tricolorable:
        3. The figure-eight knot is not tricolorable (it requires four colors):
        4. Danger: many knots are tricolorable -- being tricolorable doesn't mean that your knot is the trefoil knot -- but it does mean that your knot is not the unknot!

           Links can be tricolorable, too -- for example, the unlink is tricolorable! (That's just two circles, one lying on top of the other, as in Borromean rings.)

• Trefoil knots are in your heart (the following images are from a paper by Art Winfree at the University of Arizona, on "excitable media"):
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• My dad carved mathematical objects, including this surface for Art Winfree (related to heart arrhythmias):