Day 24, MAT115

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Announcements:
- Exam stuff:
  - First, there is a curve: add 10 points to your score at the top.
  - Exam 2 is returned.
  - A key is available.
- You have a homework due in one week, related to this knot stuff.
We'll review the notions of links and knots (the first five), and then move on to answer this
Question of the day: What makes a knot tricolorable?

Links:

Links are like the links you find in chains: constructed by "linking" loops (or rings) together.
Borromean Rings: "Is it possible to create a link of three loops (or rings) in such a manner that they are indeed linked, yet if we remove any one of the rings the other two become unlinked?"

Yes, indeed! Let's try it with rubber bands first.

Standard diagram of the Borromean rings

A realization of the Borromean rings as ellipses

Coat of arms (of the municipality of Hallsberg, Sweden) showing padlocks in Borromean rings configuration

(Thanks Wikipedia!)

icosahedra, golden rectangles, and interlocking rings. It turns out that there are golden rectangles in the heart of an icosahedron

Let's make one using 4x6 index cards.
- Use symmetry!
- How far off golden are we?
- Do you see the icosahedron?

More on the Borromean rings:

I always think of them in the context of this quote:
"We must all hang together, or assuredly we shall all hang separately." Benjamin Franklin, at the signing of the Declaration of Independence.
Led Zeppelin, in the album sleeve of their album Led Zeppelin IV used it:

"Drummer John Bonham's symbol, the three interlocking rings, was picked by the drummer from [Rudolf Koch's Book of Signs]. It represents the triad of mother, father and child, but also happens to be the logo for Ballantine beer." (from the Wikipedia article on Led Zeppelin IV).

(thanks Matt!).


If you want to draw the Borromean rings, you'd draw three circles, as in the first figure above:	but you'd want to indicate, somehow, that one ring is below another ring (aka the "Irish Trinity"):

One more:

Scene from Stora Hammar stone

with a closeup of the rings:

Let's start with a video from Vi Hart, about snakes and graphs.
- Notice her graphs in the first part of the video: those are the two non-planar graphs, and the graph of an octahedron!
- By making a closed-curve squiggle graph, you're creating a knot.
- "Borromean snakes": "No two snakes are actually linked to each other" (1:22).
- Just before the Borromean snakes there is a trefoil snake (1:14).
- "What kind of knots are you drawing and can you classify them?"
We began our discussion of knots with "Knot Mosaics":

(an example of Solomon's Knot -- which you might notice is actually a link!)
- Celtic Knot Mosaic Examples: (not quite the same idea, but pretty! And again, you'll see that there are links contained therein -- including Borromean rings)
  - Triangular paper (in case you want to try making some of those).
- Knot Mosaics
  - help us to draw Borromean Rings, for example: use your Mosaic Graph Paper to create your own version of the Borromean Rings.
  - Rectangular Mosaic Tiles and symmetry
  - We can use them to make any knots: try your hand at
    - the trefoil knot
    - the figure-eight 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. 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.)