Day 23, MAT115

We're going to look at a gallery of all the knots we know, their characteristics, how we can tell them apart, etc.

In particular, we're going to be talking more about the Reidemeister moves, and tricolorability.

1. A Gallery of Knots
2. Which knot is this?
3. Talking you through some other knots
4. What knot is this? A big squiggle knot.

1. Google Drive:
   

2. PNC bank:
   

• 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.

• 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

Also we want to look at some characteristics. So let's visit the knot gallery.

We can also throw some bands in here, too, to remind ourselves of how the bands related to the knots and links.

• Last time I took you from an unknot to a more complicated version via Reidemeister moves, and then back again.

  That was a pretty easy one. Let's use all three moves, starting with a circle, to turn an unknot into something that looks complicated -- but is actually the same.

  Here was our set of moves.

• Now we'll have a go with another one that's not too complicated.

• Then I'll talk you through a couple more:
  • Turning a more complicated appearing unknot into a simple circle unknot. (Here it is as a powerpoint presentation.)

  • Here's my knot, but with commentary. I'm very flattered that someone would steal my knot!:)

  • A figure-eight knot is equivalent to its mirror reflection, which one can show by Reidemeister moves. However the trefoil knot is not equivalent to its mirror reflection.

• Finally, let's have a go at a somewhat complicated knot. Then I'll give you a few of your own for homework....