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 reducible crossings. So the number of these crossings that you could sort of flip out of the diagram. They're not really necessary for the knotedness of the knot. But the base case is the most interesting part of the proof, where you have a knot diagram that has no reducible crossings. So no extraneous little loops or flips on it. But it's very constructive, and it uses things like checkerboard colorings, and it uses splices, or smoothings, which is where you take a crossing and you turn it into — like, you basically get rid of the crossing by cutting it and reattaching ends so that it's just — I’ve got this picture in my head, how do I say it? What's the best way to say that? So you have a crossing, and you want to get rid of it by cutting it and reattaching ends so that there's no crossing anymore. Does that make sense?

KK: Well, it’s sort of like a braid, right? I mean, so you imagine sort of a braid cross, you just clip the string above and below and then you just reattach, then you don't have it, right? Is that what you’re doing?

AH: Okay, yeah, what you just said totally makes sense because I could see your fingers.

KK: This would be a better video podcast, I suppose.

AH: I know. Yeah, at least for topology, or geometry. But the proof basically creates a checkerboard coloring that tells you how to find a collection of regions where you can basically control which crossing you're going to change. So I can change just one crossing, by carefully selecting a group of regions where exactly one, or exactly three of the regions involved in that crossing are going to get changed, but every other crossing in the diagram is next to either zero, two, or four regions that are being changed. So if it gets changed, it'll get changed back and look like it like it started.

KK: Right. Okay. All right.

EL: So I have not thought about knot theory, probably since we talked with, like, Laura Taalman on this podcast years ago. It's not something I think about a whole lot. And so I was not expecting this induction to be on the number of reducible crossings because they're so silly, you can just undo it, and then your diagram doesn't even have it anymore. So, yeah, why not the number of crossings or the number of regions or something?

KK: Yeah.

AH: So the reason reducible crossings are annoying for region crossing changes is because at a reducible crossing — you know, at any crossing, if you zero in on it, it looks like there are four different regions involved in the crossing, but with a reducible crossing, two of those four regions are actually the same region.

KK: Right.

AH: So it can look locally like you're changing two regions, so that you know, the crossing shouldn't flip. But you're really changing one, so the crossing does flip. So that's why reducible crossings are the annoying thing that you need to carefully control.

KK: Okay. All right.

AH: Yeah. And so once you get into the inductive step, you basically want to take a reducible crossing, change it so that you have two pieces, one has one fewer reducible crossings, and you know how to deal with that. And then one is a totally reduced diagram of a knot.

EL: Yeah. But the base case is the hard part, it sounds like.

AH: Yes, yes, yes. Totally.

EL: Interesting.

AH: Yeah. So yeah, so one of the reasons I love this is because I love unknotting. In general, I find unknotting questions really interesting. And I highly recommend everyone go listen to Laura Taalman's My Favorite Theorem podcast because she talks a lot about unknotting problems. But also the woman who proved this result is named Ayaka Shimizu. She’s a Japanese mathematician, probably my age, maybe a little bit younger, maybe she's about to have her 40th birthday or something, I don't know. But she is one of the coolest mathematicians I've ever met. She's definitely the cutest mathematician, and her talks are so cute that you're like, oh my gosh, I'm watching such a cute talk! And then you realize, oh my God, this result that she just proved is really amazing! So she's just super, super cool. I love her so much, and I think it's amazing that she proved this result that, you know, the Japanese math community wondered about for a long time, but no one came up with a proof before her. And she must have, maybe she was even a grad student at the time, or she was definitely a very young mathematician when she proved this result. So I love it.

KK: So here's a question: why would you want to allow such operations? I mean, because physically, changing the crossing, I mean, that would be great when your shoes are knotted, right? Like, you could just go Oh, snap, that's unknotted. Right. Is there a practical reason? And by practical, it could be including things like, it doesn't change the knot invariants or something? Or I don't know, it must if you get to the uknot, but I mean, is it… or is it just fun?

AH: Well, so the other thing — yeah, it's just fun. The other thing you need to know about me is that I study games that you can play on knot diagrams.

KK: Okay.

AH: And this result enabled this Lights Out-type game, they actually have a website. You can search for this game called Region Select. It's a really fun solitaire game that's a lot like Lights Out if you've heard of that game. And basically, the fact that the region crossing change is an unknotting operation basically means that any lights out game that you can think of or any region select game you can think of is playable, so you can have a knot diagram. Basically, at each crossing, instead of a crossing, you have a light. So it looks a lot like a graph, actually. You have a light and the lights are, some of them are on and some of them are off, and you need to select regions to try and turn all of them on or turn all of them off. And it's a really fun solitaire game that comes from this.

KK: Okay.

AH: But I’ve actually use the region crossing change to invent one of the many games that I've studied. It's called the region unknotting game. And basically, I'm super interested in these types of two-player games, where you start with a knot diagram, or maybe the shadow of a knot diagram. And you have two players doing something to the diagram, and one player wants to create the unknot and the other player wants to create something knotted. And so we have many games of this variety we've invented. One is the knotting-unknotting game. There's the region unknotting game. I’m about to publish a paper with some students called the arc unknotting game. And there are more. I could go on and on listing games.

EL: Kind of like you know that you can always unknot these things, but it's like, can you unknot it faster than someone can knot it? Is that sort of what's hard about playing this game?

AH: It doesn't have to be faster, necessarily. So the game, these games always are of the form, each player is going to move and they're going to go back and forth until everything is completely determined. And then at the very end, you see whether you have a knot or an unknot. And so you could be playing the long game, like, Oh, I'm just gonna wait it out playing on these little crossings over here to force the other player to play in this region of the knot diagram first, so that I can, you know, have the last move and turn it into a knot at the very end. So yeah, they're combinatorial games, topological combinatorial games, which is cool, because then you, then there is a player who has a winning strategy. And so your goal is to figure out which player is it? And what is a strategy that will always allow them to win?

EL: You said that the proof is constructive. So does that mean that given a knot diagram, you — someone who knew the proof — could actually say, okay, I can, you know, look at this knot diagram and do some sort of wizardry on it and say, Okay, the second player is definitely going to have a way to win this game. Or first.

AH: Yes, it can help. But of course, when you're playing two-player games, the other player can always thwart it. Like, let's say, I have to change these three regions in order to make this unknotted. Well, the other player knows that too. And so they're going to make it so that I can't change one of those regions. But actually, the the constructive way that the proof goes for region unknotting, the region unknotting operation, like basically, there's a complementary set of regions that will have the same effect. You can either do all the moves on this set of regions, or you can do all the moves on this other set of regions, and it will have the same effect on the diagram. So that does help inform game strategy, although we haven't looked at the types of diagrams that are terribly difficult to see how to unknot, because those are already hard enough to figure out strategies for.

KK: Right, right, right. Maybe it's like NIM, right? Like, if when you're playing, and if you have a huge numbers or piles of toothpicks, or whatever, you just kind of play randomly for a while, right?

AH: Yeah.

KK: And then when it gets small enough to where you can kind of analyze it, then you start to do it.

AH: Yes. Actually, I was giving a talk on not games at the Canada-USA math camp. And John Conway was in the audience. And he and all the students who were obsessively playing with him got really excited about calculating numbers for these topological combinatorial games. It's kind of a funny story. He said, he doesn't usually come to talks that speakers give at the math camp, or he didn't. And he said, normally he would leave before the speaker started speaking, because he was afraid to make speakers nervous. Like he didn't want them to be too nervous with him in the audience. But he was so intent on thinking about some problem that he was thinking about with a student there, that he just accidentally ended up in the room until it was like too late to leave. And so he told me afterwards about this dilemma he had, like, would it be worse for him to stay? Or worse for him to get up in the middle of my talk?

EL: Oh yeah, I’m glad he stayed. It would feel like a snub.

AH: Yes, he did stay. And we had a nice conversation about it afterwards, which was amazing. Because if you don't know about John Conway, he was, like, the king of knots and games and all of these things that I care about. So it was very cool.

KK: Yeah, that is cool. All right. So part two.

EH: Yes.

KK: What does this theorem pair with?

AH: This is it was such an obvious answer to me. With this paired with, because are you familiar with Nancy Scherich and her math and dance work?

EL: No, I don't think so.

AH: Nancy Scherich is a knot theorist. Actually, she works with braids. And she's also an amazing dancer, aerial acrobatics person. Acrobaticist? Acrobat. Aerial acrobat. And when she was a grad student, she won the Dance Your Ph. D competition, representing cool things about braids with dance. And since then she has recorded a number of other videos demonstrating mathematical ideas. And my husband is a musician, and he makes the music for her videos.

KK: Okay.

EL: Ao she just had a video that came out within the last month that is showing the proof of Alexander's theorem, which is a theorem about braiding. And the music that my husband composed for her dance piece, my husband's name is James Whetzel, was just beautiful, and just beautifully went with this performance that shows how the theorem works. And so I would recommend James Whetzel’s music.

KK: Unbiased, of course.

AH: I’m totally biased, and he has a new, actually, so I always forget if it's under Whetzel or James Whetzel because he has two different music personas. Right, so he has a new EP under Whetzel, W H E T Z E L, and the title track is “I want to go about my day,” and I think “I want to go about my day” would pair very well. Oddly enough, he also has songs called “Reidemeister Moves” and “This Is what Topology Sounds Like” and “Mama Proves a Theorem.” So he has some various songs with mathematical titles.

EL: So interesting that he came up with those and you also have done with things with these. What a weird coincidence.

AH: I know. How strange, isn't it? Yeah. So, but anyway, I think that everyone should go check out Nancy Scherich. I mean, you could probably just go to YouTube. Scherich is S C H, E R I C H. And check out Alexander's theorem. It's so beautiful. She does pole dancing to it.

KK: Okay, cool.

AH: Because it's about how you can turn any projection of a knot into a projection that always revolves in the same direction around a pole. So it works really well with that medium.

EL: That is so neat. So we have some things to watch and listen to after we're finished with this episode.

AH: Yeah.

KK: So we would like to give our guests a chance to plug anything that they're working on, or where we can find you on the intertubes.

AH: Yes, this is very timely because I'm trying to get out the word about an interesting event that I'm cohosting at the Joint Math Meetings.

KK: Okay.

AH: So, last Joint Math Meetings, a bunch of folks associated with Center Minorities in the Mathematical Sciences put on a storytelling event at the Joint Meetings. And it was so amazing and lovely. And they're doing it again, this Joint Meetings. But my friend Aaron Wootton and I were so inspired by this that we decided to also host a storytelling event at the upcoming Joint Meetings. And the theme is, people will be telling stories about some professional rejection that they experienced that was pretty crushing that ended up turning into something even better. So Aaron and I realized we both have stories like this where we didn't get something, we were totally feeling awful about it. And then it ended up being like a way bigger, more awesome thing. So we have a number of speakers lined up, but we need more. And so we have a web form that I created a bit.ly URL for. If you're interested in in telling a story no more than five minutes in length of the Joint Meetings, go to bit.ly/JMM2024STORY, all uppercase. Well, the bit that l y is lowercase, uppercase, JMM2024STORY, and we'd love to have people submit requests to speak, and I really hope that we have a good turnout for the event itself. It's going to be on Friday afternoon at the Joint Meetings. So mark your calendars. And let's see, the session is called Inspiring Stories: How an Academic Rejection Led to Something Amazing.

KK: Okay. In San Francisco, here we go.

AH: Yes. Can I add one more thing?

EL: Of course, you also, you also host a podcast, right?

AH: Yeah.

EL: I don't know if that's the thing you wanted to plug, but you should plug it too, and whatever you were about to say.

KK: You can plug as many things as you want.

AH: Okay, I have a lot. Okay. So we're just finishing up a book that's a handbook for math majors, called Navigating the Math Major: Charting Your Course. And it's going to be published through MAA Press by the AMS, and that should be coming out by MathFest of next year. So be on the lookout for that, especially if you're at a university that has one of these one- or two-credit freshmen seminars for math majors, or, like, an intro to the math major course. But also, it'll be good for just advisors and mentors to recommend to students and for students who might just be starting out in their college career, and they need some advice about, you know, what communities they should try and be a part of, how to apply for an REU, what kind of weird jobs are available for people. And this is what Evelyn was talking about earlier, because she did a wonderful interview for us about science writing, and that career path for math majors. So I definitely want to plug that. And regarding the podcast, it's a collaboration that I do with my friend who's an artist, Esther Loopstra. The podcast is called Flow into Authenticity. But what it's about is, if you're stuck in your life, it could be professionally or it could be personally, how can you use creativity and intuition to get unstuck? And we're actually writing a book on this called Think Like an Artist, Create Like a Mathematician, that's going to be published by 619 Wreath, which is Candice Price and Miloš Savić’s new publishing company. So that might be coming out in 2024, as well.

EL: Yeah. Well, and that sounds like something all of us can probably use it at some point.

KK: Sure.

EL: We always feel a little stuck.

AH: Yes. It's designed — we’re kind of aiming it professional stuckness in the book, but it's really broadly applicable. So very excited about that. And Esther Loopstra is amazing. She's a fine artist. She used to be an illustrator. You know, she used to work for, like, American Greetings and Target and all these places doing illustration. But now she's a fine artist and creative coach, and just super insightful about how we can use creativity to get unstuck.

EL: Cool.

AH: So, yeah, so the podcast is flow into authenticity. And the book is Think Like an Artist, Create Like a Mathematician.

KK: Cool. All right. Well, we'll try to link to everything that we can find links to.

AH: Okay, thank you.

KK: All right. Well, Allison, this has been terrific. Thanks so much for joining us.

AH: Thanks. Thanks for having me. This is such a fabulous opportunity, and I really appreciate getting to talk to you about today.

KK: Sure.

EL: Yeah. It was a lot of fun.

[outro]

On this episode, we talked with our delightful guest Allison Henrich, a mathematician at Seattle University, about the region crossing change theorem in knot theory. Here are some links to things we mentioned that might be interesting for you.
Henrich's website
MAA Focus magazine
Ayaka Shimizu's paper about the region crossing change theorem
Region Select, a game you can play where you try to unknot a knot using region crossings
An article Henrich coauthored about the region unknotting game
Nancy Scherich's YouTube channel, where she shares videos of her dances about math
James Whetzel's song I Want to Go About My Day on Bandcamp
The signup form for the mathematics storytelling event Henrich is cohosting at the Joint Mathematics Meetings in January 2024
Flow into Authenticity, the podcast she cohosts with artist Esther Loopstra