Kevin Knudson: Welcome to My Favorite Theorem. I am Kevin Knudson, professor of mathematics at the University of Florida, and I am joined by my cohost.

Evelyn Lamb: Hi. I’m Evelyn Lamb. I’m a math and science writer in Salt Lake City, Utah.

KK: How’s it going?

EL: Yeah, it’s going okay. It’s a bit smoky here from the fires in the rest of the west. A few in Utah, but I think we’re getting a lot from Montana and Oregon and Washington, too. You can’t see the mountains, which is a little sad. One of the nice things about living here.

KK: Yeah. Well, Hurricane Irma is bearing down on Florida. I haven’t been to the grocery store yet, but apparently we’re out of water in town. So I might have waited a couple days too late.

EL: Fill up those bathtubs, I guess.

KK: I guess. I don’t know. I’m dubious. You know, I lived in Mississippi when Katrina happened, and the eye came right over where we lived, and we never even lost Direct TV. I’m trying not to be cavalier, but we’ll see. Fingers crossed. It’s going to be bad news in south Florida, for sure. I really hope everybody’s OK.

EL: Yeah, definitely.

KK: Anyway.

EL: Fire, brimstone, and water recently.

KK: Anyway, we’re not here to talk about that. We’re here to talk about math. Today we’re thrilled to have Candice Price with us. Candice, want to say hi?

Candice Price: Hi everyone!

KK: Tell us a little bit about yourself.

CP: Sure. I’m currently an assistant professor of mathematics at the University of San Diego. I got my Ph.D. at the University of Iowa, and I study DNA topology, so knot theory applied to DNA, applied to biology.

EL: So that’s knot with a ‘k.’

CP: Yeah, knot.

KK: San Diego is a big switch from Iowa.

CP: Yeah, it is. In fact, I had a stopover in New York and a stopover in Texas before getting here. All over.

EL: You’ve really experienced a lot of different climates and types of people and places.

CP: Yeah. American culture, really.

KK: All right. You’ve told us. Evelyn and I know what your favorite theorem is, and I actually had to look this up, and I’m intrigued. So, Candice, what’s your favorite theorem?

CP: Sure. My favorite theorem is actually John H. Conway’s basic theorem on rational tangles. It’s a really cool theorem. What Conway states, or shows, is that there’s a one-to-one correspondence between the extended rational numbers, so rational numbers and infinity, and what are known as rational tangles. What a rational tangle basically is, is you can take a 3-ball, or a ball, an open ball, and if you put strings inside the ball and attach the strings to the boundary of the ball, so they’re loose in there but fixed, and you add these twists to the strings inside, if you take a count to how many twists you’ve added in these different directions, maybe the direction of west and the direction of south, and if you just write down how many twists you’ve done, first going west and then going south, and then going west, going south, all of those, all the different combinations you can do, you can actually calculate a rational number, and that rational number is attributed to that tangle, to that picture, that three-dimensional object.

It’s pretty cool because as you can guess, these tangles can get very complicated, but if I gave you a rational number, you could draw that tangle. And you can say that any tangle that has that same rational number, I should be able to just maneuver the strings inside the ball to look like the other tangles. So it’s actually pretty cool to say that something so complicated can just be denoted by fractions.

EL: Yeah. So how did you encounter this theorem? I encountered it from John Conway at this IAS program for women in math one year, and I don’t think that’s where we met. I don’t remember if you were there.

CP: I don’t think so.

EL: Yeah, I remember he did this demonstration. And of course he’s a very engaging, funny speaker. So yeah, how did you encounter it?

CP: It’s pretty cool, so he has this great video, the rational tangle dance. So it’s fun to show that. I started my graduate work as a master’s student at San Francisco State University, and I had learned a little bit about knot theory (with a ‘k’) as an undergrad. And so when I started my master’s I was introduced to Mariel Vazquez, who studies DNA topology. So she actually uses rational tangles in her research. That was the first time I had even heard that you could do math and biology together, which is a fascinating idea. She had introduced to me the idea of a rational tangle and showed me the theorem, and I read up on the proof, and it’s fascinating and amazing that those two are connected in that way, so that was the first time I saw it.

KK: Since I hadn’t heard of this theorem before, I looked it up, and I found this really cool classroom activity to do with elementary school kids. You take four kids, and you hand them two ropes. You allow them to do twists, the students on one end of the ropes interchange, and there’s a rotation function.

CP: Yeah.

KK: And then when you’re done you get a rational number, and it leads students through these explorations of, well, what does a twist do to an integer? It adds one. The rotate is a -1/x kind of thing.

CP: Right.

KK: So I was immediately intrigued. This really would be fun. Not just for middle school kids, maybe my calculus students would like it. Maybe I could find a way to make it relevant to my undergrads. I thought, what great fun.

CP: Yeah. I think it’s even a cool way to show students that with just a basic mathematical entity, fractions or rational numbers, you can perform higher mathematics. It’s pretty cool.

KK: This sort of begs the question: are there non-rational tangles? There must be.

CP: Yes there are! It categorizes these rational tangles, but there is not yet a categorization for non-rational tangles. There are two types. One is called prime, and one is called locally knotted. So the idea of locally knotted is that one of the strands just has a knot on it. A knot is exactly what you think about where you have a knot in your shoestring. Then prime, which is great, is all of the tangles that are not rational and not locally knotted. So it’s this space where we’ve dumped the rest of the tangles.

KK: That’s sort of unfortunate.

CP: Yeah, especially the choice of words.

KK: You would think that the primes would be a subset, somehow, of the rational tangles.

CP: You would hope.

EL: So how do these rational tangles show up in DNA topology?

CP: That’s a great question. So your DNA, you can think of as long, thin strings. That’s how I think about it. And it can wrap around itself, and in fact your DNA is naturally coiled around itself. That’s where that twisting action comes, so you have these two strings, and each string, we know, is a double helix. But I don’t care about the helical twist. I just care about how the DNA wraps around itself. These two strings can wind around, just based on packing issues, or a protein can come about and add these twists to it, and naturally how it just twists around. Visually, it looks like what is happening with rational tangles. Visually, the example that Kevin was mentioning, that we have the students with the two ropes, and they’re sort of twisting the ropes around, that’s what your DNA is doing. It turns into a great model, visually and topologically, of your DNA.

KK: Very cool.

CP: I like it.

KK: Wait, where does infinity come from, which one is that? It’s the inverse of 0 somehow, so you rotate the 0 strand?

CP: Yes, perfect. Very good.

KK: So you change your point of view, like when I’m proving the mean value theorem in calculus, I just say, well, it’s Rolle’s theorem as Forrest Gump would look at it, how he tilts his head.

CP: Right. I’m teaching calculus. I might have to use that. That’s good. I mean, hopefully they’ll know who Forrest Gump is.

KK: Well, right. You’re sort of dating yourself.

CP: That’s also a fun conversation to have with them.

KK: Sure. So another fun conversation on this podcast is the pairing. We ask our guests to pair their theorem with something. What have you chosen to pair Conway’s theorem with?

CP: So I thought a lot about this. So being in California, right, what I paired this with is a Neapolitan shake from In n Out burger. And the reason for that is, you’ve sort of taken these three different flavors, equally delicious on their own, right, rational numbers, topology, and DNA, and you put them together in this really beautiful, delicious shake. So the Neapolitan shake from In n Out burger is probably my favorite dessert, so for me, it’s a good pairing with Conway’s rational tangle theorem.

KK: I’ve only eaten at In n Out once in my life, sadly, and I didn’t have that shake, but I’m trying to picture this. So they must not mix it up too hard.

CP: They don’t, not too hard. So there’s a possibility of just getting strawberry, just getting vanilla, just getting chocolate, but then you can at some point get all three flavors together, and it’s pretty amazing.

KK: So I can imagine if you mix it too much, it would just be, like, tan. It would just be this weird color.

CP: Maybe not as delicious looking as it is tasting.

KK: That’s an interesting idea.

CP: It’s pretty cool.

KK: So we also like to give our guests a chance to plug anything they’re working on. Talk about your blog, or anything going on.

CP: Sure. I am always doing a lot of things. I am hoping I can take this time to plug, in February we have a website—we is myself, Shelby Wilson, Raegan Higgins, and Erica Graham—a website called Mathematically Gifted and Black where we showcase or spotlight every day a contemporary black mathematician and their contributions to mathematics, and we’re working on that now. We’ll have an article in the AMS Notices in February coming up. It’s up now so you can see it. We launched in February 2017. It’s a great website. We’re really proud of it.

EL: Yeah. Last year it was a lot of fun to see who was going to be coming on the little calendar each time and read a little bit about their work. You guys did a really nice job with that.

CP: Thanks. We’re very proud, and I think the AMS will put a couple of posters around the website as well.

KK: Great. Well, Candice, thanks for joining us.

CP: Thank you.

KK: This has been good fun. I like learning new theorems. Thanks again.

CP: Yeah, of course. Thank you. I enjoyed it.

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