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Monday, June 27, 2022

Complexity of a Knot

Saw this measure come up in addressing complexity.

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

“Ribbon concordance” will let mathematicians compare knots by linking them across four-dimensional space.

Leila Sloman

Back in 1981, Cameron Gordon introduced a new way to relate two knots — mathematical constructs modeled after the knots that appear in a single thread or string. In his paper, he conjectured that this new relationship could be used to arrange groups of knots according to how complicated they are.

This winter, 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. “What was really surprising about this paper is, one, that it’s super short,” said Arunima Ray, a researcher at the Max Planck Institute for Mathematics. “And secondly, that it’s using some tools that are, let’s say, unusual to this particular question.”

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Progress on the question was slow until 2019, when Ian Zemke, a mathematician at Princeton University, showed how to apply powerful new methods to the problem that weren’t around in the 1980s. Agol learned about the conjecture a little over a year ago, and he started working on it in earnest late last fall.

“That was nice, other people were kind of thinking about this problem,” said Gordon, who is a professor at the University of Texas, Austin. “And then, somewhat out of the blue, along comes Ian Agol with his beautifully short, beautifully elegant proof.”

Gordon’s conjecture is one of many in knot theory that attempt to organize the infinitely tangled universe of knots. At the heart of this project is the observation that you can drastically alter a knot’s appearance by twisting the strands or sliding them around. (To prevent mathematicians from simply unraveling the string and retying it however they like, the ends of the string are merged to form a closed loop, like a rubber band.) Given drawings of knots, knot theorists try to figure out which ones are truly distinct, and which are different depictions of the same object.  .... ' 

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