#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.

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.

THE CHALLENGE

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 standards. 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?

#MathArtChallenge Day 23: Brunnian Links

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Build a set of 3, 4 or maybe even 5 Brunnian links.

MATERIALS NEEDED: Your call. Joey already did the 3 link one with straws. I used crocheted yarn. Shoelaces?

And then because I realize it’s not nice to not pull apart the 4 for you:

DAY23 MAC STILLS

#MathArtChallenge Day 22: Extending Day 14…

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Dave Richeson came up with a brilliant extension for Day 14

MATERIALS NEEDED: paper (grid?) and pencil!

2020-04-05 20.39.04

#MathArtChallenge Day 19: Alternating knots

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Draw an alternating knot. Then see if there’s any way to draw a looping line that cannot be turned into an alternating knot.

MATERIALS NEEDED: Pencil & eraser, or whiteboard, or markers?

#MathArtChallenge Day 18: Celtic Knots

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Create a Celtic knot, and do some wondering about why you got the number of links that you did. Can you predict how many links you’ll get? (I wrote another blog post on this a while ago, but only go there if you need more examples, because I reveal a lot of the good stuff in it: Knots, Links, & Learning)

MATERIALS NEEDED: Paper, pencil. If you have grid paper, that might help, and here is some special grid paper you can use courtesy of Justin Aion.

Here is a visual set of instructions, and below that is a video tutorial.

Day18 MAC instructions

#MathArtChallenge: Day 2 Looping Colors

Day2 MAC
A looping line, with created spaces alternating blue and red in color.

Materials Needed: writing utensil, writing surface
Math Concepts: intersections, knot theory, structure, functions

Draw a long, looping, self-intersecting line that meets back with itself at the start. Avoid having any 3 lines cross at the same intersection (although after a bit it may be fun to play with this). Then, select a color, and start coloring in the spaces created by your line’s intersections.

Can you color it such that you end with every section alternating colors?

Depending on how you use this activity, you may engage with different standards. 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. Why can any graph be 2 colored? As you add intersections, how many new spaces are created? Can you create a function to describe the relationship between the number of intersections and the number of enclosed spaces?
2.) Reason abstractly and quantitatively. How many new spaces does each crossing create?
6.) Attend to precision. Ensuring that a 2-coloring is correct by methodically working through your design.