Braids and Links

Description

Materials

Instructions

Ideas for discussion

Materials

• For each student:
  • Ropes.
  • Braid Information Page
• Optional: Large braids for demonstration and display.

Instructions

1. Introduce the idea of a braid, and connect it to what students already know about braids.
2. Referring to the Braid Information Page below , show students how braids are built up from their components.

   

   Notice that there are four possible ways to cross three strands. These are called the components for building braids. An infinite number of braids can be made from these components--many more than the familiar braid that we know for braiding three strands of hair!

3. Point out on the Braid Information Page how a finished braid is turned into a knot by joining the ends. Notice, however, that you can join the ends in several different ways, and not all of them produce the kinds of knots that were studied in other activities. When two or more closed loops of rope are tangled and knotted, the resulting object is called a link. Counting the number of closed loops in the link is a good way to begin classifying links. The number of closed loops is called the link number.
4. Experiment with some three-strand braids and keep track of the results. The possibilities for interesting questions multiply fast. Some ideas include:
   • What are all the possible braids that can be made from two components ? three? four?
   • Using the four components for braiding three strands, what familiar knots can you make?
   • Can you arrange braid components in such a way that when you join the ends the resulting knot is the zero knot ?
5. Ask students to figure out what the components are that they can use to build braids made of four strands. Using these components, experiment with braids that have four strands. What about more strands?
6. Every knot is a closed circular braid. This is a famous and surprising theorem of Knot Theory. It means that no matter how much a knot appears to be tangled, looped and twisted, it can be rearranged as a closed circular braid made of some number of strands. Have students experiment with some knots to see if they can always rearrange them into closed circular braids. How might this be useful for writing down descriptions of knots?

Ideas for Discussion

1. Comment on what the idea of braids adds to the study of knots.
2. Have students present the results of their investigations with braids to one another.
3. Does the idea of every knot being a closed circular braid help in the classification of knots?