Unexpectedly Intriguing!
July 10, 2020

Mathematician John Conway passed away of COVID-19 earlier this year. Among his many legacies was a particularly difficult-to-classify knot with 11 crossings that, in the five decades after he had identified it, had defied full mathematical classification for whether or not it was a slice knot.

Before we go any further, let's take a crash course in knot theory to understand why that posed a problem for mathematicians, starting with the gentlest entry into the subject we can find in the form of the following one-and-a-half minute video introduction:

Now that you've gotten your feet wet, let's expand on that introduction with the following 10-minute video description of the tools mathematicians have developed to study knots:

In maths, knots aren't just models of tangled bits of rope and string, but can often represent complex geometries, such as the edges of much more complex surfaces. In real life, those representations have application in molecular analysis, where knot theory has provided insights into understanding the properties of complex molecules, such as DNA.

That's where the concept of slice in knot theory comes into play. If you wrap a knot around a three or four-dimensional sphere and then cut through it to create a disc-shaped "slice" of the sphere while also creating a section cut of the knot in the process, if you can do that and produce simple loops without crossings like the "unknot" featured in the introductory videos, you have a slice knot, which puts it into a class of knots that share certain properties, like the simpler stevedore knot, which only involves six crossings. On the other hand, if you cannot avoid having a tangled crossing no matter how you might take a slice of the knot, then you do not have a slice knot, which puts it into a different family of knots with different properties.

In 2019, University of Texas grad student Lisa Piccarello cracked the classification problem, determining that Conway's 11-crossing knot was "not slice", which is a big deal because of how difficult it has been for mathematicians to arrive at that conclusion, and which has earned no fewer than two articles in Quanta magazine to cover the story. We can't do better in generally describing how she was able to successfully resolve the question than those two articles, which describe how she approached the problem in part by exploring it in four dimensional space.

That's not easy, so to better understand that part of how she determined Conway's knot was not slice, let's start with her own description of why a mathematician would go into the fourth dimension from the University of Texas' press release announcing her achievement:

If an ant living on the earth would like to leave an island without touching the water, it is going to have a hard time. Because on the 2-D surface of the Earth, the water completely surrounds the island. But, if the ant builds a bridge (which rises up, into a third dimension, above the water), then suddenly it has an option to leave the island. We naturally think about 3-D space all of the time. Studying 4-D space is fun because, just as the ant can leave the island once it's allowed to go up, in four dimensions more things are possible.

How does that actually work? Amanda Hager describes how to unknot knots using the fourth dimension in the following 4:22 minute video, in which she starts off by putting you in flatland as one of Piccarello's ants:

But as for what going four-dimensional can do for you in untangling knots, you can get a better sense from Zsuzsanna Dancso's discussion of her work in studying the related concept of braids in higher dimensions in the following 14:26 minute Numberphile video, where the part of her work that is directly analogous to slicing knots is about 8 minutes in:

That concludes this very short crash course introduction to both knot theory and four-dimensional geometry. If you're intrigued enough to learn more, or know someone who would be, we'll conclude by pointing you to Matt Parker's Things To Make And Do In the Fourth Dimension, which is a fun way to start digging deeper into the 4D world.

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