Knot Arithmetic

Description

If students have already done the activity A Classroom Menagerie of Mathematical Knots, they will have plenty of knots to work with in this activity, and a familiarity with knots that will help them understand the concept of knot addition better.

Materials

Instructions

Ideas for discussion

Materials

• Knots that were made in the activity A Classroom Menagerie of Mathematical Knots or for each group of students, the knots in the top row of the collection of knot diagrams. (NOTE: Since adding two knots together involves cutting the original knots, and "transforming" them into a third, students may want to make new knots for this activity so that they can still have the ones they have already made as samples.)
• Ropes.
• Paper and pencil.

Instructions

1. Draw the figure below on the chalk board.

   

   Explain that knots are added together by doing a little bit of surgery. First, make a cut in each one of the knots that will be added. Do this carefully, without untying. Then the cut ends from the first knot are attached (taped) to the cut ends from the second knot.

2. Help the students to generate questions about adding knots. Some possible questions are:
   • Does it matter how you join the two ends of rope when you add two knots?
   • Can we make some of the knots on the collection of knot diagrams by adding knots?
   • Can we make the knots that we are most familiar with?
   • Is there any knot that will be different when we add the zero knot to it?
   • If we switch the position of the two knots (i.e., put the left one on the right and the right one on the left) how does that affect the result?
   • If you add three knots together, does the order in which they are added affect the result?
   • When we add two knots together, do we get a different result if we cut a different part of the knot when we do the surgery?
   • When we add two knots together, will we get a different result if we use the mirror image of one of the knots instead?
   • What if we use the mirror image of both knots?
   • What happens if we add two knots together, then take the same two knots, flip one them over, and add them. Will the result of both additions be the same?
   • What if we flip both knots over?
   • Are there two knots that you can add together whose result will be the zero knot ?
   • Can you add a knot to the left- hand trefoil knot to turn it into the right-hand trefoil knot ? What about other mirror image knots?
   • If this is knot addition , what about knot subtraction , knot multiplication , knot division or other operations? Could there be knot fractions? negative knots? odd knots and even knots? prime knots?
3. After the students have generated a list of interesting questions about knot addition, discuss how you could go about answering one of the questions. A suitable plan might look like the following:
   • Clearly state the question and the things that you will be doing to try and find an answer.
   • Decide which knots will be used for the investigation.
   • Plan how to keep track of information, such as making lists and tables.
   • Draw the knots that will be added as part of the investigation.
   • Show in the drawing where the knots will be cut and taped when the addition is performed.
   • Do the addition with knotted rope.
   • Twist and pull the rope into equivalent knots that are not like the first drawing and record those with knot drawings.
   • Identify related questions. (Maybe someone else is working on them.)
   • Locate other people doing similar work. Compare and discuss results.
4. Have the students work individually or in groups to answer one or more of the questions that they generated.

Ideas for Discussion

1. Have the students present to the rest of the class some information about the question(s) that they investigated, how they went about their investigation, and what they found out.
2. Encourage students to be active listeners to the other students presentations. Point out how such simple objects as knots become complicated once we start comparing them, adding them, and doing other things with them to understand more about them. It is easy to make mistakes, or to get half way through what you were explaining and find out you have confused yourself. This happens to mathematicians all of the time.
3. Explain that mathematicians rely on one another to listen carefully to their ideas and help them figure out if they have done as good a job as they possibly can when they investigate a question. When mathematicians present their discoveries to each other, they aren't thinking, "Oh, I hope they will be impressed by how smart I am to have thought about all this complicated stuff." Their attitude is very different. They are thinking, "I hope they will listen carefully to what I am explaining. When they don't understand, they will ask me questions so that I can help explain it better. If they think I have made a mistake, they will tell me. I want to make sure that this is all done right."
4. Encourage students to think about the mathematical truth of their conclusions. A conjecture is a statement which, after experimentation with many examples seems plausible. Once a conjecture is made, the next step is to try to prove it rigorously. If students have difficulty finding rigorous proofs for their conjectures, they shouldn't be discouraged. Today,in all fields of mathematics, there are unproved conjectures which have eluded proof or disproof for years. Occasionally, when new mathematical techniques are developed, mathematicians can apply them to conjectures that have been made and either prove or disprove them.