Knot Theory Where Mathematics and Physics Meet: Part 2

This is a continuation from last time.

To summarize the main points,
knot theory, which was a research tool for atomic models, developed independently and led to a new encounter with natural science,
is the story.

First, I will mention two key figures in knot theory as a pure mathematics.

One is James Alexander (1888-1971), who devised a mathematical formula to classify knots.
As an aside, he was a super-elite who was chosen as one of the first professors at the Institute for Advanced Study in Princeton, along with Einstein and von Neumann.

For those who do not know von Neumann, I will quote a past post.

This formula is called the "Alexander polynomial," and if this value is different, it means it is a different knot.)

You might think, if the formula is done, isn't it complete?

However, in later years, cases were found where even if they were equivalent, they could be classified separately, and the mathematician who further improved the level of perfection based on that issue is Vaughan Jones (1952-2020).

Jones noticed that the relational expression devised by von Neumann, who appeared briefly above (called von Neumann algebras, and as mentioned in the post above, devised as a mathematical foundation for quantum mechanics), is similar to knot theory.

And as a result of deepening research based on this idea, he invented the "Jones polynomial", which is more precise than the Alexander polynomial. This is a story from the 1980s, so it is already a modern story.

With the advent of this polynomial, it suddenly attracted the interest of other mathematicians, and more precise polynomials were devised one after another. (In other words, should I say it is that profound...)

Not limited to pure mathematics, from this time on, applications to natural science also became intertwined.

In particular, what I personally felt was unique is, surprisingly, "DNA analysis," which falls under the field of life science.

???... I think that is what you are thinking.

It is well known that DNA has a double helix structure.

And DNA is packed with sequence information consisting of four types of bases.

Heredity is the process of converting the base information within this double-stranded DNA into proteins (a bit too rough an expression, though).

To do that, first, it is necessary to "unwind" this double helix in order to replicate and synthesize the base sequence.

Having written this much, I think you can somewhat see what I want to say.

Actually, it is "enzymes" that play the role of cutting and synthesizing DNA, but by deciphering this knot structure, we can infer the reaction rate of the enzymes, or more specifically, their mechanism of action.

This is quite an exciting application. (Or is it just me?)

And one more thing.

Last time, we saw that knot theory first encountered natural science when it was used to model 'atoms,' which were thought to be the smallest elements of matter at the time.

Surprisingly, knot theory will reappear in the search for matter even more fundamental than atoms. In a sense, you could call it a return to its origins.

That will be the topic next time.

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