#MathArtChallenge 79: Knot Surfaces & “Why” Diversity

A Seifert knot surface.

The Challenge: Inspired by this tweet (below), the goal here is to make a knot surface*.

Materials Needed: Can be done with tape and paper, also through crochet.
Math concepts you could explore with this challenge: Knot theory, topology

To my great shock, although it seems sensical in hindsight, these knot surfaces (at least this one) function like a möbius one sided strip**. There’s only one edge! Seems like a rather egregious failing on my part that, knowing these strips exist, I hadn’t even considered other surfaces that may have a single edge. Here is my crocheted version.

I followed these instructions to crochet this. Will definitely be trying for some general knot surfaces soon. Replications of some of Hanne Kekkonen‘s work is likely soon to follow.

This particular challenge came about because I’m spending some of my summer working with a friend on knot theory. We spent a fair bit of our last conversation talking about how our different “natural” ways of approaching knots complement and can build on one another. To make sense of knots myself, I immediately needed to hold and manipulate them – I crochet them so I can stretch and move them to see their various projections and start “knowing” them in 3-space. Donnie, on the other hand, imagined an electron traveling the length of the knot, and how it might be acted upon by forces from the various crossings. It was a wonderful discussion and has expanded our ability to consider knot crossings (which is what we’re investigating).

That, right there, is why we need an actual diverse field of mathematicians. Two different angles for the same problem that cracked it more open for both of us. We decided that there has to be a way to discuss the relationship between the over/under crossings in a knot, which led us to the space between them…hence knot surfaces.

And that’s just two people. Who already share a lot in common. Imagine where we could get with 3, 4, 10…50 people, especially if they bring vastly different lived experiences and perspectives to the table.

I am routinely in class with 30+ other humans, but I don’t often have these kinds of conversations in class. Do kids come up with “new” ways to solve problems? Absolutely. But the “different” methods in class are often just different algebraic paths – rarely as disparate as crochet and electrons. I want to dig into why that is. Why there aren’t broader solutions/methods coming out of class. I can think of 3 main contributors to that:

  1. Expectation of where knowledge already exists: One thing that encourages open conversation between Donnie and myself is that neither of us know knot theory that well, so no one has an upper hand – I’m not waiting for him, nor is he waiting for me to “reveal” the path. That’s a hurdle for us to overcome in math class. Students are often (I know because they tell me this explicitly) waiting for me, the teacher, to “reveal the path”.
  2. Limitations rooted in what is “known” & what the goals are: Donnie and I have no set goal beyond “learn more about knot theory”. That means we can move in whatever direction we want to go in. In math class, I may get kids exploring symmetry, and it’s often because I want to get them to describe the difference between rotational and reflective symmetry. End of story. There’s a fixed goal that limits our ability to stretch beyond it.
  3. Time: With as much material as we’re supposed to cover, we rarely let students actually play with math unless we see that play headed in a productive direction. That stinks.

So the reflective questions I’m currently considering with this knot surface are:

  • Why do we name mathematical ideas after people? Rather stupid to imagine they’re the first or only person who independently explored that idea.
    • How might our ideas about mathematicians and who “does” math change if we were to abandon these names?
  • How do I better set myself and my students up to challenge the ideas of what we “know”?
  • How can I introduce more play into class and actually leverage the diversity of my student’s experience and knowledge?

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 many edges will each of these knot surfaces have? Can you prove it?

5.) Use appropriate tools strategically. What methods can you use to build these such that you can “see” the knot as well?

*They’re often referred to as “Seifert surfaces”, and, fine, credit where it’s due, as best I can tell Seifert was the first to describe them, but I’m going to call them “knot surfaces” because that’s why I’m interested in them here, and I’m frankly it’s exhausting to always credit the European “discoverer” of something that naturally exists in the universe. Especially when so many of those “discoveries” eventually are found to have also been discovered by someone else before or independently of the man who gets credit. 

**Blech. And of course möbius strips are named after a guy who “discovered” them in 1858, except that of course he didn’t discover them

Author: Ms. P

Math Teacher in Minneapolis, MN.

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