Graham’s number is famous for being the largest number to have ever shown up in a proof. The true story is more complicated, as I discovered by asking Graham. But here’s a much smaller but still respectable number that showed up in knot theory:
It’s 2 to the 2 to the 2 to the 2… where we go on for 101,000,000 times. It appears in a 2011 paper by Coward and Lackenby. It shows up in their upper bound on how many steps it can take to wiggle around one picture of a link until you get another picture of the same link.
This upper bound is ridiculously large. But because this upper bound is computable, it follows that we can decide, in a finite amount of time, whether two pictures show the same link or not. We know when to give up. This had previously been unknown!
Here’s the paper:
• Alexander Coward and Marc Lackenby, An upper bound on Reidemeister moves, American Journal of Mathematics 136 (2014), 1023–1066.
Let me spell out the details a tiny bit more.
A link is a collection of circles embedded in 3-dimensional Euclidean space. We count two links as ‘the same’, or ‘ambient isotopic’, if we can carry one to another by a smooth motion where no circle ever crosses another. (This can be made more precise.) We can draw links in the plane:
and we can get between any two diagrams of the same link by distorting the plane and also doing a sequence of ‘Reidemeister moves’. There are 3 kinds of Reidemeister moves, shown above and also here:
Coward and Lackenby found an upper bound on how many Reidemeister moves it takes to get between two diagrams of the same link. Let n be the total number of crossings in both diagrams. Then we need at most 2 to the 2 to the 2 to the 2 to the 2… Reidemeister moves, where the number of 2’s in this tower is cn, where c = 101,000,000.
It’s fun to look at the paper and see how they get such a terrible upper bound. I’m sure they could have done much better with a bit of work, but that wasn’t the point. All they wanted was a computable upper bound.
Subsequently, Lackenby proved a polynomial upper bound on how many Reidemeister moves it takes to reduce a diagram of the unknot to a circle, like this:
If the original diagram has n crossings, he proved it takes at most (236n)11 Reidemeister moves. Because this is a polynomial, it follows that recognizing whether a knot diagram is a diagram of the unknot is in NP. As far as I know, it remains an open question whether this problem is in P.
• Marc Lackenby, A polynomial upper bound on Reidemeister moves, Annals of Mathematics 182 (2015), 491–564.
As a challenge, can you tell if this diagram depicts the unknot?
If you get stuck, read Lackenby’s paper!
To learn more about any of the pictures here, click on them. For example, this unknotting process:
showed up in this paper:
• Louis Kauffman and Sofia Lambropoulou, Hard unknots and collapsing tangles, in Introductory Lectures On Knot Theory: Selected Lectures Presented at the Advanced School and Conference on Knot Theory and Its Applications to Physics and Biology, 2012, pp. 187–247.
I bumped into Coward and Lackenby’s theorem here:
• Evelyn Lamb, Laura Taalman’s Favorite Theorem, Scientific American, 8 March 2018.
Taalman’s favorite theorem gives a way to know for sure whether a knot is equivalent to the unknot, a simple circle. It shows that if the knot is secretly the unknot, there is an upper bound, based on the number of crossings in a diagram of the knot, to the number of Reidemeister moves you will have to do to reduce the knot to a circle. If you try every possible sequence of moves that is at least that long and your diagram never becomes a circle, you know for sure that the knot is really a knot and not an unknot. (Say that ten times fast.)
Taalman loves this theorem not only because it was the first explicit upper bound for the question but also because of how extravagant the upper bound is. In the original paper proving this theorem, Joel Haas and Jeffrey Lagarias got a bound of
where n is the number of crossings in the diagram. That’s 2 to the n hundred billionth power. Yikes! When you try to put that number into the online calculator Wolfram Alpha, even for a very small number of crossings, the calculator plays dead.
Dr. Taalman also told us about another paper, this one by Alexander Coward and Marc Lackenby, that bounds the number of Reidemeister moves needed to show whether any two given knot diagrams are equivalent. That bound involves towers of powers that also get comically large incredibly quickly. They’re too big for me to describe how big they are.
So, I wanted to find out how big they are!
If you want a more leisurely introdution to the Haas–Lagarias result, try the podcast available at Eveyln Lamb’s article, or this website:
• Kevin Knudson, My favorite theorem: Laura Talman, Episode 14.