Unraveling DNA with Rational Tangles | Infinite Series

When you think about math, what do you think of? Numbers? Equations? Patterns maybe? How about… knots? As in, actual tangles and knots?

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Previous Episode:
How Big are All Infinities Combined? (Cantor’s Paradox)

There is a special kind of mathematical tangle called a rational tangle, first defined by mathematician John Conway around 1970 which relates to biology and the study of DNA.

Written and Hosted by Tai-Danae Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow and Linda Huang

Resources:

Modeling protein-DNA complexes with tangles by Isabel Darcy
(the tangle examples in today’s episode can be found here)

Understanding Rational Tangles (Recreational Guide)

The Knot Book by Colin Adams

Knot Theory and Its Applications by Kunio Murasugi

On the Classification of Rational Tangles by Louis Kauffman and Sofia Lambropoulou

DNA Topology by Andrew Bates and Anthony Maxwell

Proof of Conway’s Rational Tangle Theorem

The Shape of DNA (video with Mariel Vasquez)

How DNA Unties its Own Knots (video on topoisomerase with Mariel Vasquez)

Knots and Quantum Theory by Edward Witten

Tangles, Physics, and Category Theory

My Favorite Theorem Podcast

Topics in Knots and Algebra (Online Course at Bridgewater State University)

Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon.

Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!

And thanks to Mauricio Pacheco and Andrew Poelstra who is supporting us at the Lemma level!

[PhilipSportel]

This is your best one yet, Tai-Danae. Keep bringing us awesome math!

[rishabhjain]

I can't easily imagine combining the horizontal and vertical twist easily. Should have put a simulation of it step by step instead of just showing the final tangle

[Drna]

Loved this episode. I am a bioinformatician, but I have not heard of this area of study. Definitely will check links in the description. Thank you!

[AKAIMAX]

This is a really cool topic, I kinda hoped it would go into more detail about stuff in knots. The rational tangles and rational numbers link is really cool but I don't understand where the continued fraction comes from. Why would you be applying division into the tangle if every operation you mentioned only adds 1 or subtracts 1 from the rational number representing the tangle?

[MrTej]

Your presenting skills have been improving. This was arguably the best presented video you have done thus far. Keep at it.

[Tehom]

Just noticed that Tai-Danae has slipped the hoped-for division by zero into this presentation. Specifically, the vertical no-tangle is the (negative) reciprocal of the horizontal no-tangle, which is associated with zero.

[michaelnovak]

Best channel in the history of youtube

[fyermind]

Looking forward to follow-up episodes on the topic. I'm excited, but just as I'm getting ready to dig in, the video ended!

[abhisekswain]

can we extend this system to not just 2 but any number of strings???

[rkpetry]

...confuses horizontal and vertical—rotate a connected pair and it flips sign (±), , and if points are perpendicular, which sign does their lines-twist have...are connections supposed to have directed graph arrows...

[wobh]

This dovetails nicely with the recent team-up from PBS SpaceTime, PBS Eons, and It's Okay to be Smart.

[anastasiia.litkevych]

AWESOME VIDEO, THANK YOU SO MUCH!
I’m doing a project on this topic and your video made a great job for my understanding of topic!!❤

[kenahoo]

One thing that might be helpful to point out - flipping the picture along a diagonal axis, so that the right side becomes the bottom side, corresponds to taking the reciprocal. That's how we get continued fractions: start by twisting the right side a positive or negative number of times (get an integer), then flip (take the reciprocal), then do more twisting on the right side (add integer), then flip again (reciprocal), etc.

That's also how the horizontal starting point and vertical starting point she shows are reciprocals of each other, 0 and ∞.

[Metaknightmare]

These configurations of rational tangles remind me of group theory. The way certain configurations of tangles can be reached by several different continued fractions equal to the same result is like how the same configuration on a Rubik's Cube can be reached by different combinations of twists. Could it be that knot theory is just a type of lower-order form of group theory with more restrictive principles that bound groups to the inside of a sphere? Do higher-dimensional knots exist?

[mannyglover]

Very cool. My favorite mathematics professor at Georgia Tech specialized in knot theory. I only got a minor in mathematics, so I just thought that his field of research was a “pie in the sky“ area of pure mathematics, but this video makes me appreciate how it has really important real world applications. Thanks for the video! Very cool

[Heirborn]

The pair that came up with knot theory were at my school a few weeks ago. So cool!

[zetadroid]

Another great episode. You two are doing better and better every time

[DanieleMarchei]

I'm about to graduate with a thesis on how to factorize knots using tangles and reidemeister moves, I can confirm that the rabbit hole is very deep

[theflaggeddragon]

I second another comment here, this is your best upload so far! Keep bringing us more topology! :D

[BC]

Tai-D you're knocking them out of the park 👌