#MathArtChallenge 96: Not so much rational tangles as… a 2 link challenge.

2 link knots
A grid arrangement of a variety of green and teal crocheted links.

The Challenge: How many 2 link knots can you find? See the examples above to help you get started.

Materials Needed: knot materials (these could be crocheted like I have, or shoe laces, or electrical cords which you can plug into themselves)
Math concepts you could explore with this challenge: combinations & permutations, graph theory, knot theory, topology, vertices/intersections.

Soooooooo, I planned on having this be rational tangles, but that didn’t work out. If you want to skip to the challenge, scroll down to THE CHALLENGE (it’s obvious, I promise).

Still, I wanted to do rational tangles because I desperately wanted to carve out some time for myself to play more with knot theory and because I think that rational tangles are spectacular (even if I don’t fully understand them yet) But, as they say, I bit off more than I could chew tonight. (Although seriously, if you want to play with me and rational tangles, let me know, I’d love to give poor Professor Noodle Arms a break from my incessant questioning.)

I also wanted to have this be on rational tangles because I’d get the chance to introduce you to Candice Price and Mariel Vázquez, who are both working on biological topology with DNA and knot theory (my heart!)

I was introduced to Dr. Price because she features in an episode of My Favorite Theorem wherein she explains rational tangles and the Conway’s rational tangle theorem. She also happens to be a co-founder of MathematicallyGiftedAndBlack.com which is a wonderful resource for helping to diversify your knowledge of mathematicians. Dr. Price also mentions Dr. Vázquez in the My Favorite Theorem episode, which led me to this gem, and honestly, be still my heart, if I’m ever able to meet either of these women, I’ll act like any normal person meeting Beyoncé.

Unfortunately, I don’t think I can make rational tangles work as a math art challenge tonight (maybe a surprise bonus challenge once I get my feet under me) BUT I can get you closer. At least knot theory is involved.


So here’s the challenge: Using two links – these are CLOSED loops (Use an extension cord for each – you can plug them into themselves, and they’ll work beautifully) – see how many totally unique arrangements you can make. I’ll start you off. Below are a bunch of 2 link knots…(someone’s going to yell at me for that language, just do so nicely in the chat and I’ll adjust)… each unique from the others. Can you come up with others? 

Things to ponder:

  • Can you prove there are an infinite number of 2-link knots?
  • Aren’t Dr Price and Dr Vázquez amazing?
  • If you removed one of the links, (cut and pulled away) how many of the arrangements would leave an unknotted link behind? (This, admittedly, is directly related to my Bridges submission this year. Totally worth your time to peruse the whole gallery.)

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. How can we know that we’ve gotten “all” of them?

7.) Look for and make use of structure. What types of links can we find and classify? Can any of those types be classified in more than one way?

Author: Ms. P

Math Teacher in Minneapolis, MN.

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