The Challenge: Can you find the 4, 5 and 6 crossing knots?

Materials needed: Patience, creativity, paper, pencil, maybe some strips of t shirts and a safety pin? Math conceptsyou could explore with this challenge: Combinatorics & permutations, counting, graph theory, knot theory, topology, vertices/intersections

If you follow that thread, you’ll get loads of instructions, but the gist is this: A mathematical knot is a closed loop, and it’s called a #-crossing knot based on the minimum number of crossings. Untangle it as much as you can. Two knots are the same if you can mess them around and get them to have the same projection. These two, for example, are the same:

Can you find the 4, 5 and 6 crossing knots?

Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

1.) Make sense of problems and persevere in solving them. What does it mean for a knot to have an “intersection” or a “crossing”? How can you use use that definition to help you discover the different knots?

8.) Look for and express regularity in repeated reasoning. What sorts of crossings are common? Can you sort the knots by similar types of crossings?