- Our quiz today will involve knot identification,
distinguishing them via the Reidemeister moves and
tricolorability.
You need to know your knots and links for the final!
- For next Monday: I'd like you to provide me with a one
paragraph description of your project and a one
paragraph description of your logo; I want to make
sure that everyone has something to share on the
following Monday and Tuesday - Logo and Project days!
- As mentioned, Danielle is running for city council in Norwood. She
is having a fund-raiser on
May 4th:
Comedy for Progressive Candidates: May the Fourth/Farce Be With You! - I noticed a hand-written note on the board about a production with
"JB" on it: I assume that it's our JB:
You can do the twofer: see JB's show in the afternoon on the fourth, and then some comedy with Danielle in the evening!:)
- We reviewed the Reidemeister moves, and explored using them a
little bit more. The objective is to use them to
show that two pictures of knots are the same knot.
- We learned via a video that there is no good algorithm for
figuring out how to use Reidemeister moves to transform one picture of
a knot into another picture.
It's an art form, at the moment; one which quantum computers will perform better than we will!
- Then we discussed the concept of tricolorability:
- Tricolorability:
"A knot is tricolorable if each strand of the
knot diagram can be colored one of three
colors, subject to the following rules:
- At least two colors must be used,
- at most three colors are used, and
- at each crossing, the three incident strands are either all the same color or all different colors.
- tricolorability is preserved by the Reidemeister moves:
Type I Type II Type III So that "once tricolorable, always tricolorable" is a rule.
- Thus we learned that the unknot and the trefoil must be different knots, because one is tricolorable and the other isn't.
- Tricolorability:
"A knot is tricolorable if each strand of the
knot diagram can be colored one of three
colors, subject to the following rules:
- Among our collection of knots and links, only the trefoil and the unlink are tricolorable.
- More review, prior to our quiz
- Another different way to draw the toroidal knots.
- A tricolorability exercise
- Let's start with the lover's knot on the back
- Test for links/knots, and known versions of each.
- Exploit symmetry
- "Smart" trial and error
- Knot tangles: three dimensional knots
- Your quiz!
- My collection of knots and links
- Tricolor These Handout: just two of the knots/links are not tricolorable; all the rest are!
- A much more thorough introduction to both Reidemeister moves and tricolorability is in this article: Tricolorability of Knots, by Kayla Jacobs.