Summer of Math Exposition

It’s just a minor footnote towards the end, but two distinct knots with the same Jones polynomial would be an interesting exploration. When does it happen, how likely is it?

Excellent use of real-world footage to demonstrate the various knot moves. Consider replacing “ambient isotopy” and “planar isotopy” with simpler terms.

Excellent video!

However I think some of this already well covered in the youtube sphere? Particularly the veritasium video?

Other than that, a solid video overall.

Nice picture and nice topic! Thank you!

The humor from the cameraman’s comments worked. I waas worried they’d get overused and tiresome, but very well done :)

The first half of the video is a self-contained exposition, and the bracket half could be pretty cleanly lopped off into another video with just a quick preface about what Reidemeister moves are. I’m sure I’m not only reviewer who will make this observation, but this is a case where the video really could have been much shorter, reaching a larger audience while still being a quality introduction to knot theory.

I’m probably more knowledgeable about knot theory than your intended audience, which seems to me to be post-calculus* undergraduates. I’ve seen this particular material presented many times, and I think this is a serviceable introduction, elevated considerably by integration of both animated and physical knot manipulations. In particular, it would probably have been a lot less work if you had only stuck to one of those, but I think the exposition would have suffered. Props for going that extra mile.

Some specific comments:

The “ambient isotopy” vocabulary was casually dropped without being defined/described— I was prepared to just gloss over this like I think most viewers would, but then you keep using this term. I wonder how comfortable the intended audience would be; at some point does the question arise “wait, what does that actually mean, again?”

( * This qualifier is not because you use the word “differentiable” but it’s not unrelated. I think that this word for a student currently in a calc class but without strong multivariable intuition, is going to be a bit of a pain point— they’ve seen this word in a technical context, so will want to understand how you’re using it here, but you are not so concerned with how you’re using it here and (rightfully) gloss over it. For folks who haven’t seen that calc definition, I think they’ll be fine at that moment, but for that audience I worry that some of your discussions are a bit too careful, e.g. “obviously any diagram can be made generic, why are we spending so much time on this?”)

Early on in defining the bracket, aaround 17:00 you stop to basically repeat something just said— I think that was a good idea. It’s definitely the most conceptually difficult part of the video, and I appreciated you acknowledging that.

While I can recognize the discussion of Reidemeister I moves for the X-polynomial as “complete and correct”, as a viewing experience I found it not very persuasive. Not sure what could be done about that since it is a genuinely thorny issue (except to acknowledge that fact).

Personally, I thought the idea that Reidemeister moves are the natural consequence of the decisions made in “generifying” a diagram was very cool. I’ve never seen them described that way before, and that was the most memorable part of the video for me.

The shoutout to knotfol.io at the end was great, very cool tool :D

Excellent content! Nice work. One minor nit. The subject matter is fairly advanced, so why the fish cartoons? I found the initial scenes using real rope to be a little hard to follow visually. The later animations were much clearer.

I would say the volume of the music is kind of inconsistent and this can sometimes be distracting. The audio quality, including the occasional bumps to the microphone could also be improved. Perhaps explain why you divide by 2 to calculate the linking number? I understand what you’re going for with describing the knot as a one dimensional object at the start, but perhaps it would be better to say “made out of 1D material”, or “locally 1D”? Otherwise the knot would just be a line with no crossing point information.

Other than that, very awesome video!!

I think knot theory is an acquired taste. I could follow a fair percentage of this video. At times I asked myself what practical use any of this was, since the invariant (the number that distinguishes knots) is just a polynomial variable. It could measure the number of crossings, but without a number were the presenters measuring anything? This was a long video and I managed to watch all of it which says a lot. Again, I suspect that like Boogey Down Bugs millipede videos, this is something you either get or you don’t.

Really love the IRL visualisations. They add so much to the explanations. Some things could be made clearer here, though. For example, when defining links, using ropes of different colours. And while the explanation itself is very standard, the presentation really makes it stand out.

This was simply great.

• motivation: I don’t really see objectively why I should care about this in terms of the “real world”, but this video was pulled off so well I found myself invested despite that. I don’t think you really need to introduce more motivation in that regard (I kind of disagree with this being a category for scoring anyways—many theoretical topics don’t have any real-world application but are really cool regardless)
• clarity: Very clear—I have very little background in knot theory (it consists entirely of a Veritasium video I saw once) and I didn’t have any trouble following along with the explanation
• novelty: I’m sure there are other videos about the Jones polynomial, but yours was exceptionally well-done
• memorability: I loved the off-script moments and the two-person setup. It really lends it a degree of authenticity and conversationality (respectively) that other videos lack

misc:

• I was off-put at first by the video being over half an hour long, but after watching it I can say the length was necessary, and it was utilized well.
• One thing I would take into account (somewhat pedantic to be honest) is that the manim knots suddenly coming in halfway through the video were a bit stylistically jarring. I would have committed to using either the live-action knots or manim knots the whole way through. Again, quite small, but I couldn’t think of anything else in the way of actionable improvements. (This is why I’m giving you .1 points off)

Great video overall, good luck with the competition!

A well done video. The explanation maybe wasn’t the most intuitive but it works.

Very well built, with relatively easy notions at the beginning, then getting really advanced.

I also appreciate the economy of means, it’s really nice to do something so rich in ideas with limited means (do you know that in French there is an expression “it’s made with three pieces of string”?)

I’m giving it a goot mark, not because I KNOW for a fact that it’s good, but because I feel like I need to watch it several times, because I’m lost. Although, it’s good, else I wouldn’t bother watching it later and writing this message.

This was a fairly difficult video for me to enjoy or get into.

I thought we were going into the subject of knot theory with the idea of canceling knots with the idea of the link number. Instead, we covered a lot of jargon and “operations” where I felt there wasn’t enough foundation to go into it. This felt like a student lecture rather than a tutorial since there almost seemed like there was more context that had that perhaps others had as well.

Unfortunately, I couldn’t grasp my head around the use of the operations, and even more unsatisfied with how the algebra was introduced since I didn’t understand why we were using A or even how these $\langle \text{knot} \rangle$ notation was used, or what value it served. Is this conventional notation? What significance is the value of a complex A like in $A = 1, e^{\frac{i \pi}{3}} , e^{\frac{-i \pi}{3}}$ ? Why did we gloss that over when that was very interesting to cover since there weren’t any complex values we had before?

Knot theory can be either very simple to explain or in this case overly complicated without explanation since I was wondering why we’re using these operations and wondering if there was an easier way to do it. I was even wondering why some of these operations weren’t obvious to try.

How are the Reidmemeister moves used in a literal sense? I didn’t understand what it does to the knot in simple terms, since we were talking about how it confines to the rules of the knot.

Also, the rules of the knot were also obvious to the layman but over explained here that I felt even more confused why we didn’t use more diagrams to get the basics down.

I just feel so out of the loop (no pun intended) about this video that it make me question whether I knew enough of the basics beforehand to watch this.

The explanation was very clear and well done, though the audio quality could be improved.

Very nice.

Nice exposition, nice demonstrations nice choice of topic. Proving that the trefoil is not unknotted using Jones’ polynomial’s invariance to Reidemeister moves is an all time classic.

A few cool facts that connect knot theory with complexity theory:

1. The knotting problem is known to be NP hard, and even much harder, something we call #P-hard (pronounced “sharpie hard”).
2. It’s actually much weirder. Imagine a device that given a knot computes its Jones polynomial. That’s easier than unknotting, right? OK, now imaging that it doesn’t compute its Jones polynomial but just its evaluation at some point (this point happens to be a fifth root of unity, but that doesn’t matter). Sounds even easier, right? OK, now say that it doesn’t even give you the actual value of the Jones polynomial, but just a rough approximation. This sounds easy, right? How powerful could a machine with this capability be? Apparently, this modest capability is enough for universal quantum computation! Aharonov–Jones–Landau showed this problem is BQP-complete.
3. The unknot problem (just telling whether a knot is the unknot or not), on the other hand, seems much easier. We don’t have an algorithm for it yet, but we do know that it is in both NP and co-NP. So if it is in P, the entire polynomial hierarchy collapses, which we do not believe is true.
4. There is a very cool categorification of the Jones polynomial using a tool called Khovanov cohomology. There’s a very nice expository paper by Dor Bar-Natan.

Fantastic video! Great explanations and visualizations. Keep up the good work!

Really cool video on knot theory that kind of goes through the basics on the basic moves to untangle a knot and some of the basic invariants. I liked the good presentation through a simpler invariant which then builds to a more complex one, as well as the explanation of a classification of isotopies. I think one thing that could be addressed was why we wanted a much more complicated invariant rather than sticking to the linking number, it wasn’t quite obvious why the new invariant is so much more powerful.

Motivation: 8 / 9 – Generally it’s a very interesting concept you present at the beginning, but it’s a self contained one. I would love to see the topic being introduced for example by some real life problem related and solvable by knot theory, or maybe even some problem from mathematics but other, more known field of maths.

Clarity: 8.5 / 9 – I think your explanations were top notch, loved it! The only thing I want to pick on is the beginning – you defined quite a lot of stuff there (knot, link, ambient isotopes, Reidemeister moves) and simply went with them for the rest of the video – I think some more attention should be given to those definitions (like, showing them in text on screen, or re-explaining what are those things the next few times they appear on the video). It’s very subtle, I know, but I needed to come back a few times to recall those definitons.

Novelty: 7 / 9 – While knot theory is not a very common topic to discuss, there are several videos on it on YouTube already. The introduction is magnificent, but the topic is not as fresh as it could be (it would be better if you would start with an interesting problem that wasn’t shown before, like I proposed in “motivation” section)

Memorability: 9 / 9 – Visuals, humour, audio – all of very high quality. I’m impressed!

Overall: 8 / 9 – I think it’s one of the best introductory videos on knot theory and definitely one of the best from this edition of SoME. Congrats and thank you for creating this! Good luck in the competition! <3

This is an interesting video.

It explains the topic very clearly but is fairly slow. Due to the lack of visuals other than a piece of string the slow speed of the video became a little tedious and some level of animation and speed up could have been possible.

Discussion of Simple Crossings was predictable, given the motivation just before, leaving the astute viewer feeling clever. Discussion of Planar Isotopy & Reidemeister Moves was very simple and laid out well for high school students or collegiate students without prior experience. Some textual components overlapped with caption “safe-zone”. Arguments in background / with cameraman were funny moments. Seem candid. Fish characters added something to look at. Skein Relations were necessary for the Bracket Polynomial discussions, but seemed a bit more difficult. Could explain ~1 minute where Knot Theory is important / has applications: molecular biology is a very prominent example. Open question given at the end for further study encourages curiosity.

Excellent video! I thoroughly enjoyed it as someone who’s always been curious about knot theory. I especially loved that you used a physical rope for the 1st half of the video - it’s extremely cool to see abstract mathematically sophisticated ideas in a concrete physical object.

There were a few moments where I felt like some different design choices would have helped - for example, a higher contrast color than light-blue against yellow-white for the orientation arrows. Or at 4:40, color the smaller loop before the bigger one. I liked the zoomed in and spotlighted demo of the intersection points at 4:58, but I feel the description of rotating the over-strand counterclockwise would have been nice as a text overlay. I was confused at 5:13 trying to calculate all the numbers because I missed that key idea of rotating the over-strand. At this point it also would have helped to rotate the appropriate arrows.

There was a notably funny moment midway through saying “bro that’s sus”, and there were a few more similar asides in the second half of the video. I feel the intro could have used a moment of personality like that - similarly, showing faces at the very end of the video was a nice surprise, as I thought you were deliberately faceless YouTubers. It might have helped to have a brief moment talking directly to the camera in the intro so viewers can connect voices to human faces (increasingly important in age of AI slop, unfortunately).

Pretty good video. The topic was introduced well and, as someone who hasn’t looked much into the field of knot theory, I felt like I walked away with a better general understanding of knots and knot invariants. I especially like how, instead of deriving the Jones polynomial invariant, you two derived a somewhat lesser-known polynomial invariant then linked it to the Jones polynomial invariant so the watcher has an easier time understanding what you’re doing.

Here voting for low-dim top gang!

An excellent introduction to knot theory. I learned a lot. The hands-on demonstrations were fun.

This was a fantastic introduction to knot theory. I was able to follow a pretty good chunk of this. Your explanations were great, and the animations worked very well to underscore the narration. I will have to watch it at least once more to get the details, but this held my interest throughout which is rare. Love to see what else you create!

The humor and double narration works out well, and the physical demonstrations (as are quite common and easy-ish for knot theory) are good. The video is approachable, has a nice pace, and is informative. Good stuff.

Thanks for the practice problems and references. Video seemed a little dry, not a huge problem, just not my subject. Great video, thanks for posting.