Introduction to Knot Theory

I consider myself to be a knot theorist, so I will work knot theory into any topology class I teach. I personally use this to have a fun day - such as a day right before/after an exam or holiday when I expect the students to be less focused than usual.

Timing

• Prep time: A couple hours
• Class time: One half class day
• Curriculum location: Wherever

Supplies

Quantities depend on the number of students in your class. Given my classroom setup, I ran this with students working in pairs, but I imagine groups of 3-4 would work similarly well.

• Wikki Stix (or any brand of wax-covered string), 6 different colors, 2 sticks of each color per group
• Sandwich size Ziplock bags, one per group
• This worksheet
• LaTeX file
• knot-1.jpg
• knot-2.jpg
• knot-6.jpg
• knot-7.jpg
• knot-8.jpg

Instructor Instructions

Preparation

1. Tie the knots shown in the worksheet, being consistent with which color you use for which knot.
• The upper left diagram is an unknot. First, connect two sticks together by twisting and rolling the ends together as tightly as you can - it helps if your hands are warm - until the wax melds into itself to make a smooth connection. Then connect the ends the same way to form a circle. Then, twist it 3 times in the correct direction to form the diagram shown.
• The middle left diagram is a trefoil. Connect two sticks together, then tie a loose overhand knot like you're tying your shoes, and connect the ends. Move it around until it looks right. If your crossings all go the wrong way, switch it with the next diagram.
• The lower left diagram is the other trefoil - tie another overhand knot, but this time start with crossing the strings the unnatural way (if you usually put left over right, start with right over left).
• The upper right diagram is second trefoil again. Tie it the same way, then grab the lower right part of knot (oriented as in the diagram) and flip it up away from you. Straighten out the knot to make it look symmetrical.
• The middle right diagram is the unknot. Just connect two sticks, then connect the ends to form a circle. You could also very feasibly just do this with one stick.
• The lower right diagram is the culprit diagram of the unknot. I find it easiest to start with two connected sticks, then follow the diagram around pushing the end over and under to match the diagram. It can help to use the fact that the wax is slightly sticky, so you can anchor it to your paper so you don't accidentally undo your work as you go around. Double check your work - just one wrong crossing changes this knot!
• Gently smash all your crossings together so the knots stay in place.
• Put one of each knot into the Ziploc bags.
• Make note of which color corresponds to which diagram.
• Print and make copies of the worksheet

In Class

1. Explain to students what knots are, and how to read knot diagrams.
2. Give them the worksheet in groups, and have them spend some time trying to guess which diagrams correspond to the same knots. You may want to make them vote as a class which ones they think are the same.
3. After they feel settled, tell the class you're going to hand out physical knots. Their first task is to match the diagram to the knot. If they skip this step they won't know when they were right or wrong!
4. After they decide which knot goes with which diagram, they're welcome to manipulate the knots to see which ones they can make match!
5. Walk around and see how they're doing. Question whether or not they're sure they got them right.
6. This can lead into a discussion about the Reidemeister moves and knot invariants if you want.