Knot Theory 4: Braids

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Knot Theory: Lecture 4
Andrews University: Math 487 (Spring, 2019)

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I really enjoyed the way he builds the story, starting from a very intuitive example, asking questions, investigating along with the students, then introducing the relevant theorems. You can see that Both the lecturer and the students are excited about the topic!!

navyanthkusampudi

One thing I want to point out that I don't feel was really explained in this video that I figured out by messing around with braids: If you add sigma 1 or its inverse to the beginning of a braid word, you necessarily have to change the values of the rest of the braid word, increasing each by 1. Otherwise you actually do get entirely different links/knots. This makes total sense once you realize it, but I don't think you ever go over an example involving that specific move in this video.

cerberus

It's nice to see that the students are getting a little more involved in the lectures!

cerberus

Thanks a lot, i study about knot theory and quantum invariants and your lessons are very helpful in gaining basic knowledge😊

sudedursun

Some of the positive electrons are connected to grounds and can complete multiple connections. Speaking in terms of our physical environment.

cdgt

I was very surprised by the last theorem. It's incredible.

sudedursun

Thanks for the interesting lectures, I absolutely love them!

Eikosin

I tend to have more visuo-spatial intuition so it bugged me for much of the video that no mention was made of the braids s^1 s^2 and s^2 s^1 producing equivalent knots because you could effectively "slide" the crossings around the loop. After you finished listing all of the knot-preserving moves on braids and said that was all of them, I was confused for a minute before I realized that conjugation was the trick I had in mind. s^1 s^2 == s^-1 (s^1 s^2) s^1 == (s^-1 s^1) s^2 s^1 == s^2 s^1

elendiastarman

What is the topological characterization of a braid? For Alexander's Theorem to be a theorem (rather than a definition) we need to know what it means for braids to be equivalent independent of the word characterization. (Well, okay, if you consider the diagrams as a formal objects, then it would be a (rather trivial) theorem between words and diagrams. We'd like it to be a theorem between words and something akin to string embeddings into S^3).

wdacademia