Day 22, MAT115

We're now in the final stretch, and have only a few new topics left. This week we're going to finish up knots.

As you can see, there were lots of grades in the C and above, but too many below that.

The mean and median were 73.

Today I'll show you that the knots we're discussing are found in your heart!:)

• 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.

  • Therefore, if the projection of one knot is tricolorable and the other isn't, they're different knots.

  • 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.)

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.

      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.