Day 14, MAT115

• Reading for next time: Knots: a handout for math circles

• What is a Mobius band?
• 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:
  • The official recycling symbol is mobius, but over time an alternative has crept in:
    

Question of the Day:

• Links:
  • Links are like the links you find in chains: constructed by "linking" loops (or 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.

    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:

      

      "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).

      

      (thanks Matt!).

      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

      with a closeup of the rings:

  • Our next question of the Day:
    How do we draw the first five knots?

  • We began our discussion of knots with "Knot Mosaics":
    

    (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
    • 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.
      Let's draw it in these two different ways.

    • 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.

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