PSET 1

  1. Show that the following knot is the trefoil:

My work

  1. Show that the following is the unknot

    • I worked this out by hand, but I’m worried I might have messed up a step or broken a rule when undoing the knot.

  1. Show that the three projections of the figure-eight knot from Adams Figure 1.5 are all the same knot

    • I got the first two projections to match, but I got stuck on the third one.

  2. At first I thought I had gotten the trefoil knot, because I physically couldnt put this together with string so I drew it out and it seemed like the trefoil. However, I repeated this process and it seems more like the figure-8? At some point I keep getting stuck.

  3. The possibilities for connecting strands: (I looked this up, but now that I did, I understand)

    • Leaving the strands unlinked (no interaction).

    • Swapping pairs to create a twist.

    • Looping one strand back on itself.

    • Looping both strands back to form a clasp.

  4. With only two crossings, you can always untie them back into an unknot, making them trivial. I noticed this while working with strings.

  5. It’s still the same knot, just more complex. You can undo extra crossings, and I tested this myself.