the possibility of encryption with knot theory

I took this knot theory class this quarter. The professor wasn't able to mention too much applicational aspects of the theory but greatly deepened my understanding of topology and knot theory.

Formally "Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3." ---sciencedaily.com

But it is very easy to understand in layman words. A knot is just a knoted closed string.

You can see more here.

Back to the possibility of encryption. As far as I know of, one of the current methods of encryption is using prime factorization of natural numbers. More generally, there are kinds of math operations whose inverse is much harder to carry out or, rather, there are math structures that are easy to build but very hard to decompose. This property enables encryption because the other party is unlikely to gain the information unless they have, a priori, the answer.

Natural numbers are hard to factorize. You need to make square root of n tries to ensure finding a factor of a natural number, not to mention if the number factors into more than one primes (but in this case, you actually do not need that many tries in the first place). But anyhow, the difficulty to factor a number is what makes encryption by fatorization possible.

With knots, the situation is more amazing. Theorems ensure that knots can be factorized into "prime knots" the way integers can be factorized into prime integers. Furthermore, there's not yet obvious algorithm to do that. With integers, even though complicated, there are still several algorithms that ensures the eventual factorization. But not knots.

Practical difficulties remain, of course, because this is only my primitive thought. To actually use knot for encryption, knots have to be tabulated so that a universal representation is reached in the encrypting and encrypted device. (Possibly by numbers like standard knot table goes now. ) The reason for a universal representation is simply that the same knot can have thousands of different diagrams (pictures) and we want to know which is which. There should be many more, but I just had this thought and want to share.

PS.

I don't have an expedient way to explain how the subject is non-trivial and we can actually understand knots mathematically.

There is this book: C. Adams, The Knot Book (1994, W. H. Freeman) that is supposed to be understandable for high school students. And a lecture note.

But they all require some effort from the reader. So if any of you know of some easy approaches to introduce the interersting results and upshots. Leave a comment. Other discussions are welcomed too.

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