#MathArtChallenge Day 68: Unknotting Number

The Challenge:

Look. I get it. Every time I post a knot theory #mathartchallenge, it’s crickets. But I take great pride in my inability to take a hint.

Any (?) knot can be untangled if you change the right (?) crossings from over to under or under to over. So get a knot, and untangle! pic.twitter.com/WBqTKameNR

— Annie Perkins (@anniek_p) May 24, 2020

Materials needed: whiteboard?, pencil/paper?
Math concepts you could explore with this challenge: circles counting, knot theory, topology, vertices/intersections

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 to change an intersection? Does one move from over to under always result in a simpler knot? How do you know?

3.) Construct viable arguments and critique the reasoning of others. Once you have your solution for a given knot, create an argument to convince a skeptic that you’ve found the fewest possible uncrossings necessary.