Day 24, MAT115

Last Time

Next Time

Announcements:
- You have assignments due both Tuesday and Thursday of next week, including a last reading assignment from your text (about fractals and infinity -- our last two topics).
- Your homework on bands/links is returned.
- Your homework on Platonic solids is returned.
- Cinnamon Schummer is presenting a poster on Tuesday next week: "If you have time you should come see! Although I have to warn you, there is no fun math involved.... Lots of awesome stats and talks of p-value (nerd cool stuff) It is in the student union ballroom. It addresses the Heroin Epidemic and heroin related attitudes." 12:30-2:45.
- April 26th, Aaron will be skipping out for an event: "On Tuesday, April 26th, I will be missing class due to a school performance of Into the Woods at the Fine Arts Center here on campus." I tried to get him to tell me what role he'll be playing, but he's holding out on me. Details, Aaron!
- Don't forget that your logo day is coming!
  Evidently NKU is up in arms over its new Logo! There's apparently not enough mathematical content in the logo, to the displeasure of the students.
Today's Question of the day:

"How can we tell two knots apart, or decide that they're the same?"
Before we do that, however, I want to introduce you to the RCO -- Really Cool Object -- which brings together the Borromean rings, golden rectangles, and Platonic solids (an icosahedron). It turns out that there are golden rectangles in the heart of an icosahedron, interlocked as Borromean rings!
We now know five knots. How do we show that two knot projections ("squiggles") are the same knot ("knot equivalence")? That's the job of the Reidemeister moves.
1. What are the Reidemeister moves?
  - Tools for showing that two projections (pictures) of knots really show 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. We're not cutting -- we're just going to shove parts of the knot around.
  - 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:
    We'll just try a few with a string, to see what we can learn.
  2. A figure-eight knot is equivalent to its mirror reflection (a trefoil knot is not equivalent to its mirror!)
  3. Turning a more complicated appearing unknot into a simple circle unknot. (Here it is as a powerpoint presentation.)
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.)

Website maintained by Andy Long. Comments appreciated.


Type I	Type II	Type III