How Complex Is a Knot? New Proof Reveals Ranking System That Works

A new way to relate two knots was introduced back in 1981 by Cameron Gordon. He thought that this new relationship could be used to arrange groups of knots according to their complexity.

Ian Agol, a mathematician at the University of California, Berkeley, posted a six-page paper that proved Gordon's conjecture, giving mathematicians a new way to order knots by complexity.

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Ian Zemke, a mathematician at Princeton University, showed how to apply new methods to a problem that wasn't around in the 1980s. Agol started working on the conjecture late last fall after he learned about it a little over a year ago.

Gordon, who is a professor at the University of Texas, Austin, said that other people were thinking about the problem.

Gordon's theory attempts to organize the universe of knots. There is an observation that you can change a knot's appearance by twisting it or sliding it around. The ends of the string are merged to form a closed loop to prevent mathematicians from unraveling the string and retying it. knot theorists try to figure out which drawings of knots are different from each other.

Gordon said that if you can move one around so that it looks exactly like the other, then you should.

It's difficult to figure out which knots are equivalent. A simple loop can be changed into many different things. Two complex knots might appear to be the same, even if they aren't.

concordance is a less stringent concept used by mathematicians. Sitting in four-dimensional space is what is referred to as concordance. New ways of contorting knots suddenly open up when mathematicians add an extradimensionality to the equation.

Think about how you are confined to the ground. You can only walk from one side to the other. You will be stuck if you are surrounded by a brick wall. An eagle does not have this problem. A wall can contain it because it can fly up into the third dimensions.

In the same way, a fourth dimensions gives a knot a lot more freedom. Any knot can be untied in a four-dimensional space. If two knots can be connected by an imaginary cylinder, they are concordant.

You have two rubber bands. If you slip a rubber band around each end of a sheet of paper, you can make a cylinder. The rubber bands are connected by a paper cylinder.

If you switched out one of the rubber bands for a bad one, you would struggle. The rubber band would inevitably cross through the cylinder at some point. It wouldn't be smooth if you moved your cylinder into four dimensions. Two knots can be connected by a smooth cylinder in four-dimensional space.

Gordon suggested a way to compare two knots based on their complexity. He had to add another restriction to the cylinders. In ordinary concordance, mathematicians allow themselves some questionable moves to make things easier, such as pinching off loops or attaching extra ones. Gordon's adaptation doesn't allow you to add extra loops.

Ribbon concordance doesn't necessarily go both ways because of this restriction. If you have two knots, you can use a ribbon concordance to transform the first knot into the second, but not the other way around.

The property could be used to rank the knots. If you need to add loops to the second knot in order to transform it into the first, it is a less complicated knot. That creates an order for any set of knots with ribbon concordances.

The mathematicians had to prove three things to show that ribbon concordance has important properties. Gordon pointed out the first two in his original paper. If two knots are both ribbon concordant to each other, they have to be equal.

After Agol began to study the problem, he had a flash of insight. He posted an answer on the math discussion website MathOverflow that had a connection to the conjecture.

Agol was more pessimistic the following day. That may seem like a particular danger, as Agol is an outsider to the area. He studies objects in three-dimensional space. The fresh eyes of Agol may have worked in his favor.

Agol didn't use any of the modern techniques that caused the problem to surge in popularity. Instead, he studied the ideas that have been around for a long time.

Gordon was surprised by the solution.

The techniques he used could have been proved at the time I wrote the paper. He said that it makes it even more interesting.