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Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and I'm joined as always by your other and let's be honest, better, host.



Evelyn Lamb: I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And tomorrow is my 40th birthday. So everything I do today is the last time I do it in my 30s. So, like, having my last mug of tea in my 30s, taking out the compost for the last time in my 30s, going for a bike ride for the last time in my 30s. So I'm, I'm kind of enjoying that.



KK: Well, congratulations. Let's not talk about how long ago I passed that landmark. I will say there's a switch that goes off when you turn 40. So riding your bike will be more difficult tomorrow, I assure you.



EL: Well I’d better get one in then.



KK: Any big plans?



EL: I’m actually going to the Janelle Monae concert. She's in town on my birthday. I'm sure that's a causal relationship there.



KK: It must be.



EL: So yeah, I'm excited about that.



KK: Okay, so fun fact, my Janelle Monae number is, is two. So I have a half brother. Very long story. I have a half brother, who also has a brother by — his mother has two children with — my dad was one of them. And then another man was the other one. So this other one, his name is Rico. He was a backup dancer for Janelle Monae.



EL: Wow. So yeah, brush with celebrity there.



KK: I mean, of course I've never met Janelle Monae, but you can — actually if you look him up, so there's a style of dance, sort of Memphis Jook, it’s called. Dr. Rico. He's something else. Amazing dancer.



EL: Wow. Interesting life.



KK: That's right. That's right. So anyway, enough about us. We have guests on this show. So today we're pleased to welcome Allison Henrich. Allison, introduce yourself, please.



Allison Henrich: Hi. Yes. I'm Allison Henrich. Happy birthday. I'm so excited for you.



EL: Yes, you get to be on my last My Favorite Theorem of my thirties!



AH: Yes, awesome! I feel so special. So I'm Allison Henrich. I'm a professor at Seattle University, and I'm also currently the editor of MAA Focus, which is the news magazine of the Mathematical Association of America.



KK: I have one on my desk.



AH: Woo-hoo! Is it one of mine?



EL: Yeah, and when we were chatting before we started recording, you made the mistake of mentioning that you've done some improv comedy. Is that something you do regularly?



AH: So I wasn't an improv artist. This is such a cool event. This science grad student at the University of Washington started this type of improv comedy where they have two scientists give short five-minute talks. And then this improv comedy troupe does a performance that's loosely based on things that they heard in the science talk. And so I gave a talk about some basic knot theory ideas, and it was so funny. I wish everyone could have the experience of an improv comedy troupe doing a whole set about your like research or your job. Yeah, it was so amazing.



EL: Cool. But also, it sounds a little stressful. A little bit. Yeah.



AH: Yeah. You want to not be too boring. And you gotta, like — it's really interesting. The other speaker tried to work in things for them to make jokes about, and they totally didn't take the bait. And they found like more interesting things to make jokes about, but I definitely tried to work in some things that would help them riff off of my talk, and it worked pretty well. Like just referring to knots with quirky names and making jokes about knot theorists and whatnot.



KK: Sure. What-knot. Hahaha.



AH: There are a lot of good knotty puns.



KK: Sure. Okay, so this podcast does have a theme. And the question is, what's your favorite theorem?



AH: Yes! This is a hard question.



KK: Of course.



AH: I’ve decided to tell you about my second favorite theorem. Should I admit that?



KK: Sure.



EL: I’m sorry, that’s a different podcast, My Second Favorite Theorem. It has two slightly worse hosts.



AH: It’s the cheap knockoff.



KK: No, it's the sequel, once we get rid of this one, we're gonna move on.



AH: Just, we're all out of mathematicians, we’ve got to go through them again. So my, let's call it my favorite theorem.



KK: Sure.



AH: My favorite theorem is the region crossing change theorem. So I have to tell you a bunch of stuff before I can explain what this theorem is.



KK: Sure. But it must be about knots.



AH: It is about knots. So, you know, knots we represent, typically, with two-dimensional pictures called knot diagrams, where you have ways of representing when a strand is going over and when a strand is going under at a crossing. And so every type of knot that there is has infinitely many diagrams you can draw of it. But no matter how you draw a diagram of whatever your favorite knot is, it can always be unknotted if you're allowed to do a special kind of move called a crossing change. So if you have your favorite knot diagram, and you're allowed to switch the over and under strands on whichever crossings you want, you can always turn that knot diagram into the diagram of an unknot, which is like a trivial knot that'll fall apart if you unravel it a little bit.



EL: Basically just a circle, right?



AH: Yeah, a circle. I mean, all knots are circles, so I have trouble. Like, a geometric circle.



EL: A boring circle.



AH: Yeah, a boring circle.



EL: And so this theorem, does it come with like, a number of how many of these crossing changes?



AH: Ah, so this is not my favorite theorem. This is a theorem that's going to help us understand my favorite theorem.



EL: Okay.



AH: So this theorem has a really interesting proof that Colin Adams calls “proof by roller coaster.” So the the theorem that says you can unknot any not diagram by changing crossings, you can accomplish unknotting using a certain algorithm where you choose a starting point to travel around a knot, and you decide that every time you encounter a crossing for the first time, you're going to go over it. So the fact is that you're kind of like always traveling downwards. And then when you get to the very end, you take a little elevator back up to where you started. So this will always create an unknot. So it's not that surprising that this is true, that if you're allowed to change whatever crossings you want, you can unknot things. What is surprising is my favorite theorem, which is that region crossing changes can unknot any knot diagram. So let me tell you what a region crossing change is. So you have your knot diagram in the plane. A lot of us kind of imagine that this plane is on a big sphere. So can we picture not diagram on a ball? Is that okay?



KK: Sure. Make it a big enough ball, and it looks like a knot diagram.



AH: Exactly. Yup. So we've got a knot diagram on a ball, and the knot diagram basically separates the surface of the ball into different regions, right? So this amazing theorem uses this operation called a region crossing change. And what a region crossing change is, is you choose a region in the diagram, and you change every crossing along the boundary of that region. So in my head, I'm picturing kind of like a triangular region in the diagram. And if I do a region crossing change on that region, I'm going to change all three crossings that are kind of around that region. So this is the amazing result: every not diagram can be unknotted by region crossing changes. So you no longer, seemingly, have control over individual crossings, you can only change groups of crossings at a time.



KK: Okay.



EL: But you can still do it.



AH: Yes, you can still do it.



KK: Right. That seems less likely. The other one, you told us and I thought, Well, yeah, I can kind of see, before we even saw the proof, I could sort of imagine, well, yeah, you just lift them up basically.



AH: Exactly. You lift it up, and then if it gets stuck, you know, change that crossing. But you can only change groups of crossings with the region crossing change. But amazingly, it's still an unknotting operation. So that just blew my mind when I heard that.



KK: Okay, so now I have questions. So, more than one, right? You can't expect to be able to just do one of these, right?



AH: Right. I mean, so if you have a region that just has one crossing on it, it's like a super boring region, because it's just a little loop.



KK: Yep.



AH: And that's actually called a reducible crossing.



KK: Sure.



AH: If you just have a little loop, it doesn't matter which way, which is going over and which is under.



KK: No, but I guess I meant, so you know, you've got one region, right?



AH: Yeah.



KK: So there might be multiple regions, you might have to change many of these, right?



AH: Yes, yes.



KK: What if two are adjacent, then you do one flip on one and one flip on the other, then you're undoing some of the flips from the other.



AH: Exactly.



KK: Is that why it works, maybe?



AH: That is why it works. So it’s a really cool proof. It's actually a proof by induction, which is so cool, that you can have like a proof on knot diagrams that's a proof by induction. But it's by induction on the number of reducib