Day 32, MAT115

• Tomorrow is election day -- I hope you'll get out there and vote!

• I'm sorry -- haven't graded your assignment yet. I'll have it for you Wednesday.
• You have a homework assignment due Friday.
• You have two new reading assignment:
  • a summary of some knot concepts.
  • Please read Idea #23, Topology (p. 92)

• Question of the day: What makes a knot tricolorable?

• Last time we learned how to show that two knots are the same ("knot equivalence")? That's the story of 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. Turning a more complicated appearing unknot into a simple circle unknot.

  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.)

• Moving on to Moebius bands: the Band that keeps on playing.
  • Moebius bands are a part of geometry, but relate to a special area called "topology".

    Riddle: What's the difference between a donut and a coffee cup?

    Answer: There isn't any! (to a topologist....)

  • What is a Moebius band?
    • Let's make some!
    • How do we know that the moebius band is not "topologically equivalent" to a regular band?

  • What happens when you cut a moebius band "in half"? Let's check out a bug on a band....

  • What if you cut it into "thirds"?

  • By the way, did you know that there are two different Moebius bands? They're mirror images of each other! (Symmetry, again....)
    • Check yours against someone else's, and convince yourself that they're really different.
    • Make one of each!

  • My dad loved moebius bands so much....
  • Applications of Moebius bands:
    • Conveyor belts
    • Kazakhstan's National Library in Astana
    • Recycling Topology
      • a history of the recycling symbol
      • Recycling -- Moebius, or only Seemingly Moebius?
    • A joke: "Why did the chicken cross the Mobius band?"

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