Day 23, MAT115

• You have a new reading assignment, about the material we are discussing today.

• You had a homework that was to be a small part of the exam: to make some pretty knots, and take a picture, and send to me.

  Some of you evidently didn't realize it, and so have not submitted that. If you would like to, submit it by Thursday.

  Then we'll have another art contest.

• In light of our Math/Art Pop-up Conference, I want to share a quote that I found a few year's back, in an interesting opinion piece in the Washington Post:

  You would also know -- if you were better at this than I am -- that sentences are music. And that both sentences and music are math. Equations. Beats separated by pauses. Microbursts of energy clustered and cut and culled to find balance. You would know that sometimes "ain't" just fits in a way that "isn't" or "is not" does not. Same with "them" instead of "those." You would know that even the choice of "does not" at the end of the above sentence instead of "doesn't" was intentional, because of the repetitious rhythm of "does not" existing immediately after "is not." You would know that short phrases lead to shorter sentences, which punch in a way that longer ones sometimes can't. Like this just did.

• We had an exam! Let's talk about that.
  • I enjoyed digging up the local logos, and finding out which ones have some interesting mathematics in them. I found too many to include them all: for example, the Duke Energy logo is a Hopf link (if the bands weren't cut...:)
    

  • Key

  • I found out that wallpapers were tricky, and I decided to handle the scoring of your exams to minimize the damage. So you have two scores at the top of your exam: the score for the first four pages of your exam, and the score on the last problem (the wallpaper problem). (There is also the five-point knot picture points, depending on whether you submitted your knot picture or knot -- I haven't included that part).

    Now, if

    • your wallpaper score was a better percentage than your score on the first portion, it got included into your score (which will then be out of 100);
    • it was worse than the first part, it got dropped, and your score will be the percentage on the first part (out of 80, where your knot picture score will be included).
    So the wallpapers can help, but not hurt.

    They were tricky!

    On this problem I ended up just looking for good reasoning: Both of these wallpapers actually have \(R_2\) symmetry, and the one on the left has two lines of reflection (whereas the one on the right has none).

    Then we look into the types of centers of rotations....

How do we distinguish knots?

• Before we get to that, however, I want to introduce you to the "RCO" -- Really Cool Object -- which brings together the Borromean rings, golden rectangles, and Platonic solids (an icosahedron).

  This is the most beautiful piece of mathematics I know:

  It turns out that there are golden rectangles in the heart of an icosahedron, interlocked as Borromean rings!

  

  We'll make some using 3x5 (or 4x6) cards -- which are not quite golden. Their corners are the 12 vertices of the icosahedron. The tricky part is locking them together!

• A little review, and a preview of what's ahead: trying to distinguish this collection of knots (and links, too!).
  • The figure-eight knot has four crossings, as does Solomon's "knot" -- but Solomon's "knot" a link, not a knot. This is the first (interesting) instance of a knot and link with the same (minimal) number of crossings.

    But we can easily figure out which is which, because the knot is a single continuous piece of material, whereas the link is two separate pieces of material.

  • We know that there are two distinct 5-crossing knots (we can see them in Rolfsen Knot Table)
    • The cinquefoil knot
    • The five-crossing twist knot is actually known (somewhat confusingly) as 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.

• How do we show that two knot projections ("squiggles") are the same knot ("knot equivalence")? That's the job of the Reidemeister moves.
  1. What are the Reidemeister moves?
     • Tools for showing that two projections (pictures) of knots really show 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.

       It's all about overs and unders!

     • Reidemeister moves preserve the knot -- they don't change the knot. We're not cutting -- we're just going to shove parts of the knot around.

     • Type I	Type II	Type III

  2. Here's a summary of the moves.

     The third images on that page stems from this picture from a recent Science issue:

     In order to consider the picture on the left a knot, we have to know what its ends are doing. In the figure at right, I assumed that they are just connecting to each other in the simplest way.

     The succession of steps then go on to show that the knot is actually an unknot! That's good news for the hypebusy people!

  3. Using the Reidemeister moves
     1. to turn a projection of an unknot into another more complicated appearing unknot:

        Here's another picture of an unknot, which could trick you -- but knowing the R1 move saves you:

        

        We'll just try a few with a string, to see what we can learn.

     2. Now let's use these three moves to create equivalent versions of the trefoil knot, but using each of the three Reidemeister moves.

     3. A figure-eight knot is equivalent to its mirror reflection (a trefoil knot is not equivalent to its mirror!)

     4. Turning a more complicated appearing unknot into a simple circle unknot. (Here it is as a powerpoint presentation.)

  4. 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.)

• Some Solid (Three-dimensional) Geometrical Facts about the Golden Section