Day 15, MAT115

• You had a reading for today: Knots: a handout for math circles. We'll take a look at that before we're done today.
• You have an assignment which will be due Thursday after fall break.

• 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 five-crossing torus knot (can be reduced to five crossings)

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.

Question of the Day:

• 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.
  • Knots: one closed ring, or loop

• Today we begin our discussion of knots with two more knots (the last two of those that you'll be expected to know, and that we've not yet encountered). Then we'll consider "knot mosaics", and discuss some ideas about how to distinguish knots.
  • The family of twist knots
    • The trefoil as a twist knot (at least three crossings)
    • The "figure-8" knot (at least four crossings)
    • The twist 5-knot (at least five crossings)

  • "Knot Mosaics" make great examples in which we find knots, and can even be used to help us to visualize knots.
    • 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: 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 them apart? Are they really different? For that we need to study knot equivalence, and our primary tool, the Reidemeister moves.

    • To help you, we have the Rolfsen Knot Table

• 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.)