- Your graph homework is (over)due -- my bad, I forgot to collect it last time. Please turn it in now.
- Your mobius band homework is due next time.
- Your first links and knots homework (assigned today) is due next Monday, before Thanksgiving.
The Topologist's favorite riddle: What's the difference between a donut and a coffee cup?
Answer: There isn't any! (to a topologist....)
- Last time we made some. A Mobius band is made from a regular band
using only a single twist (of 180 degrees).
- There are two different Mobius bands? They're
mirror images of each other! (Symmetry, again....)
- Properties of a Mobius band:
- A Mobius band has only one side.
- A Mobius band has only one edge.
- Applications of Mobius bands:
- Conveyor belts
- Kazakhstan's National Library in Astana
- Recycling Topology
- The official recycling symbol is Mobius, but over
time an alternative has crept in:
- Mobius, or only Seemingly Mobius?
- My dad loved Mobius bands so much....
- a history of the recycling symbol (and an AARP article on the history of the creation of the recycling symbol).
- The official recycling symbol is Mobius, but over
time an alternative has crept in:
- There are two different Mobius bands? They're
mirror images of each other! (Symmetry, again....)
- Cutting a band:
- When you cut a Mobius band "in half", it becomes a longer band, with two twists. Cut that in half, and you have two bands, but they're linked together.
- What happened when you cut it into "thirds"? It results in two interlocked rings, one a one-third width Mobius band, the other a twice-twisted, one-third widthed, twice-lengthed band.
- The Mobius handout
- Mobius Bacon....
- Mobius music
- For the next three days we will be studying links and knots. How
does the Mobius band relate to these?
- Draw the edges of "the wedding band". This is the unlink.
- Draw the edge of a Mobius band. (Maybe we can get a good artist to do it!) When you cut it "in half", it remains whole, and results in another band (called the "unknot").
- Draw the edge of a twice-twisted band (the alternative recycling symbol). Now cut a twice-twisted band down the middle. What do you get? (a Hopf-link)
- Draw the edge of a thrice-twisted band. Now cut a thrice-twisted band down the middle. What do you get? (a trefoil knot)
- The four-twist band leads to something called Solomon's "knot" (but it's actually a link). We'll draw these without drawing the band, using a general strategy for drawing what are called "Toroidal" knots (and links).
- The five-twist band leads to the "Cinquefoil" knot.
- Links are like the links you find in necklaces:
constructed by "linking" bands (loops, 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.
Some examples of representations of Borromean rings:
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!) - 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:
But we can't actually tell from these images exactly what we have: we need to know more. We need the pattern of over-n-unders. That's essential. Pay close attention to those.
"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).
Ironically I just discovered John Paul Jones's symbol in another brewing company (while enjoying one of their products); Arcadia Brewing company of Kalamazoo, Michigan, has this as their logo:
Maybe every Led Zepellin symbol is on a beer somewhere?
-
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
- I always think of them in the context of this quote:
- The Duke Energy logo is a Hopf link:
Our next question of the Day:
- We began our discussion of knots with "Knot Mosaics" (key point: it's all about "overs and unders"!):
(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
- 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)
- 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
- twist knot, and as a
- torus knot.
- 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 begin with the "unknot", of course: a circle. This is the simplest knot, but not very interesting.
