編み物・結び目
[Header Image Credit] via Pinterest
Unexpectedly, inspired by the following article🙏
"Knit" transformed itself to "Knot" in my erratic brain, as usual…😵
I'm sure everyone knows about knots 結び目.
Then, I happened to watch the following YouTube video:
The Insane Math Of Knot Theory
https://www.youtube.com/watch?v=8DBhTXM_Br4
The video refers to Peter Guthrie Tait (1831 - 1901), a Scottish mathematical physicist and early pioneer in thermodynamics and one of the original "encyclopedists" of knots.
Here're images of his hand-drawings of knots:
At first glance, these drawings seemed to resemble Tamil script and Mayan script🤔
Knot Theory is a branch of the geometry of 3 dimensions, therefore, is close to Algebraic Topology, however, I did not have any interest in this theory in the days at an institute of technology (Apr. 1976 ~ Mar. 1982), most likely owing to lack of algebraic methods in my narrow-minded view.
I was completely ignorant❌
While I have been vaguely aware of analogies between knot theory and algebraic number theory such as the following:
Prime ideals in a number ring are analogous to knots in a 3-manifold M,
The ideal class group of a number field corresponds to the first homology group H₁(M, Z),
I have not spent time to look into recent developments in so-called Arithmetic Topology, NOT Arithmetic Geometry.
Just skimming two the following papers, I was tremendously surprised😮
It was only recently that I learned knots are described by Quantum Groups, which are specific types of Hopf algebras. Last year I studied Group Scheme which uses Hopf algebra, however I did not notice Quantum groups. It feels as though I have been avoiding theories related to knot theory…🤔
The following short video was so enlightening while Ed Witten's style of talk is very much restrained.
Knots and Quantum Theory - Edward Witten
https://www.youtube.com/watch?v=cuJY14BYac4
Edward Witten is the first and only physicist to win the Fields Medal (1990).
Thanks to this video, I learned something completely new😮. That said, it's only a superficial understanding.
Knot theory may appear to be a purely mathematical hobby, but its applications are quite remarkable, as shown in the contents of the book
"Knot Theory and Its Applications"
by Kunio Murasugi:
At the end of this article, let me introduce two articles which describe that the following conjecture has been disproved:
Conjecture: If J and K are knots, then u(J#K) = u(J) + u(K).
Here u(K) denotes the unknotting number of a knot K, and J#K is called the connected sum of knots J and K, a new larger know formed from J and K by cutting each knot at one point and joining the cut ends.
To be continued… though on a different topic😎
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