Day 31, MAT115

• There is an assignment due today.
• You have a new homework assignment, which will be due next Friday -- concerning knots, of course.

• Question of the day: How can we tell two knots apart?

• Last time we encountered the two distinct five-knots -- there are two different ones:
  • The cinquefoil knot
  • The 3-twist knot

• Let's start by reviewing how to draw all the knots you'll be expected to know on a first name basis: in addition to those two, there's the four-knot (or Figure-eight knot), and the three knot (Trefoil knot), plus the unknot of course.

  They come in two families: the twist knots (e.g. trefoil, figure-eight, and 3-twist), and torus knots (trefoil and cinquefoil). Notice that the trefoil knot is in both groups!

• How can we show that two knots are different? Let's see how we can establish that the two five knots are different.

• Now a related question: how can we show that two knots are the same ("knot equivalence")? That's the story today... the story of Reidemeister moves
  1. What are the 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.
  2. Definitions
     • Strand: a piece of a projection that goes from undercrossing to undercrossing, with only overcrossings inbetween.
     • Tricolorability:
       • Each strand is colored just one color
       • At crossings either one color or three different colors meet.
  3. More on Tricolorability
     1. Definition
     2. Preserved by Reidemeister moves
     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;
        2. The trefoil knot is!
  4. Using the Reidemeister moves
     1. Turning a projection of an unknot into another more complicated appearing unknot
     2. Turning a more complicated appearing unknot into a simple circle unknot.

aaaaaaaeuaaaaaaaaaaaaaaadpbfdpaaaaaaaaaaaevfcmmkbjuaaaaaaaaadpacmmkadpaaaaaaaaaaevfaabjuaaaaaaaaaaaaaadpaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa