Day 21, MAT115

• I hope that your second exam went well. I haven't looked at them yet, but should finish by Thursday, for sure. Hopefully sooner. So I'll post the results and any curve and a key by Thursday.

• This week we have a couple of days off for "Late Spring Break" (to coincide with Easter, I guess). So there will be no new material on Thursday.

• I will give you a new homework assignment, however, so that you can enjoy working on your knots. This is something that a lot of folks incorporate into their logos. Remember your logos?

• Your logos will be due at the end of the semester. Here's the text from our syllabus:

  LOGO: Companies have logos; you should have one, too! Your math logo will be something (like a family crest) that represents you. It will be created using elements from this course (or other mathematical elements of your own choosing). You will type up a one-page sheet, illustrating and explaining your choice. These will be exhibited on-line.

• We're in the home stretch folks: stick with me. This new material (starting from Mobius bands) will be covered on the final, along with a review of the material covered on the first two exams. The final will be half new material, half material from the first two exams.

• There are only a few more topics left. We'll spend another three class sessions on knots and links; then fractals; then infinity (and maybe a fourth).

• Your squiggles and knots and links homework is graded.
  1. Here's a "key".
  2. Of course two of these questions were entirely open (although I expected you to pay attention to what Vi was doing); squiggle graphs and snake knots.
  3. The first questions was just a check to make sure that you knew Euler's formula for planar graphs (what I call "ran-2": regions minus arcs plus nodes = 2). Most of you aced that one. It should have also solidified in your minds some details about the Platonic solids.
  4. The fourth one was about the Borromean rings, and that was an "over-under" problem for a lot of you.
• Materials for today:
  1. Knots and Tricolorability (video).
  2. The "Knot work" and tricolorability

• Let's recall:
  • Links are like the links you find in chains: constructed by "linking" loops (or rings) together. There are more than one ring (more than one snake, if you're Vi Hart).
  • Knots: one closed ring, or loop (or ssssssnake).

• 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 "figure-8" knot (at least four crossings), which is what we call a "twist 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.

• The "five knots":
  • The five-crossing torus knot (can be reduced to five crossings)
  • The twist 5-knot (at least five crossings)

• These are just the first few knots. More generally, we have the Rolfsen Knot Table

• Knot Theory: so now we have five knots:
  • We begin with the "unknot", of course: a circle. This is the simplest knot, but not very interesting.
  • Trefoil Knot -- the simplest interesting knot. It has three crossings, and there's no way to make it any simpler! We will also call it the 3-knot.
    • Can we make a simpler knot -- one with either 1 or 2 crossings? Why not?
    • There are two different ways to think about making the trefoil knot: as a
      1. twist knot, and as a
      2. torus knot.

  • figure-eight knot (a twist knot)
  • There are two distinctly different kinds of 5-knots
    • The cinquefoil knot
    • The 3-twist knot

• How can we tell knots apart? Are they really different? For that we need to study knot equivalence. our primary tool, the Reidemeister moves, but we're going to save that for next time. Today we'll talk about tri-colorability.

• 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.)