Day 25, MAT115

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Announcements:
- You have a homework due on Tuesday, related to knots and links.
Last time we practiced our five basic knots, and talked about tricolorability: today we use that to distinguish between knots. We also use Reidemeister moves to change one "knot projection" into one that is familiar, and so identify a knot.
Knot Theory
- 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.
  Let's draw it in these two different ways.
  One other thing: there are two different trefoil knots -- mirror images of each other! You can't turn one into the other without breaking it.
  Another thing: the trefoil is found in the "guts" of Borromean rings.
- figure-eight knot
- Something fun to do with a few friends: can we make a human knot in the shape of a trefoil knot? If so, then we won't be able to undo it -- you and your friends will be knotted together forever! I hope that's a good thing....
- 5-Knot
  - Actually, there are two kinds of 5-knots
  - To help you, we have the Rolfsen Knot Table
  - The cinquefoil knot
  - The 3-twist knot
  - How can we tell them apart? Are they really different? For that we need to study knot equivalence, and our primary tool, the Reidemeister moves.
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. Turning a more complicated appearing unknot into a simple circle unknot. (Here it is as a powerpoint presentation.)
  3. Here's my knot, but with commentary. I'm very flattered that someone would steal my knot!:)
  4. A figure-eight knot is equivalent to its mirror reflections: I'm having a hard time with this one -- can you help me?
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?"

Links:

Symbolic Sculpture and the Borromean Rings
Some resources
A handout for tricolorability and Reidemeister moves
A nice "human knot" project, describing various facets of knot theory.

Website maintained by Andy Long. Comments appreciated.
Updated on 01/05/2015 14:56:19


Type I	Type II	Type III