Untangling the beautiful math of KNOTS

Check out my MATH MERCH line in collaboration with Beautiful Equations

Suppose you have two tangled knots. How can you tell whether they are the same knot or different knots? Welcome to Knot Topology. A knot is a nice embedding of a circle into three dimensional space, up to "ambient isotropy" where you can wiggle it around but not cut it. We can project this onto a plane to create a knot diagram. Then, two knot diagrams represent the same knot if and only if there is a sequence of Reidermeister moves between them. We next explore various knot invariants like the simpler tricolorability and then expand to the Alexander Polynomial and ultimately compute this polynomial for the trefoil knot.

0:00 Mathematical Knots
0:33 Knot Diagrams
1:48 Reidermeister Moves
4:24 The equivalence problem
6:05 The idea of Knot Invariants
6:27 Tricolorability
9:12 Alexander Polynomial
14:37 Future direction in Knot Topology

COURSE PLAYLISTS:

OTHER PLAYLISTS:
► Learning Math Series
►Cool Math Series:

BECOME A MEMBER:

SOCIALS:

[kanishkachakraborty]

Love the topology t-shirt, and incredibly interesting video as always. Thank you!

[kanishkachakraborty]

My understanding of how the 1st and 3rd Reidermeister moves preserve tricolourability:

1st: A section of the knot is crossing itself, so only 1 colour is used - tricolourability allows a crossing having only a single colour.

3rd: If the two initial crossings satisfied tricolourability, the move preserves it, because sliding a section of the knot can only shift the position of crossings without modifying the nature of the crossings.

[greghearn]

I absolutely love knot math. Great to see such a nice breakdown of it.

[NonTwinBrothers]

Best video explaining the concept of know theory I've ever seen.
Well not that I'd be able to understand it way back then but you know

[HelPfeffer]

0:02 I genuingly thought you were going to say

"Are they the same not, or knot?"

[jcreazy]

I came here because I wanted to know how knots work. Now I'm more confused. Fascinating video. Thanks for making it.

[aashsyed]

The title is so good so how good can the video get ? 10⁹ times better.

[shortsismakingmybrainrot]

Omg this is cool, I absolutely love your channel, thanks so much for helping me get through my uni math courses.

[brazni]

I was tired of being practical all the time so I got into knot theory

[BleachWizz]

Nice video, definetly going into my references for my article!
Also I have something to add:
3:18 - but just Reidemeister moves in sequence is not enough to directly simplify any diagram.
By that I mean it's not always identifiable that you can perform a Reidemeister move to remove a crossing.

I'm not sure though if moves that INCREASE the number of crossings are necessary, I THINK that only movements that keeps the number of crossings the same would have a chance to allow some undoing, but I could be wrong and I'd love to know.

[lgl_noname]

4:20 to 4:25
amazingly, Google subtitle AI managed to not make a spelling mistake in the script.

6:09 I spoke too soon .
7:50 Knot invariant is definitely throwing it a curve ball.

[mahmoudalbahar]

Many thanks for your great videos.

And I suggest making video about non-integer base of numeration.

[seslocrit]

Could you do a video of DNA and Knots?


Also, (on an unrelated topic) could you do a video on non-linear dynamics?

[oriole]

Thanks for your videos! I'd like to bring your attention to the lower volume of some of your videos. If you right click on this video and select "stats for nerds" you can see that the content loudness is -14.6dB, where it should really be closer to 0dB. Since this is a log scale, the gap is substantial. Some of your videos are definitely mixed at significantly quieter volumes than others. In practice it just means that people would have to turn up the volume quite a bit, but that might make their next video play loud in a jarring way. Depending on the software that you use for editing, it may be possible to include a "compressor" filter in the audio chain, which can be used to normalize the audio levels and reduce the audio's dynamic range which will make it easier to hear on devices like phones, laptops etc, in addition to getting the output to be closer to 0dB. There are lots of tutorials on how compressors work (it's standard fair for radio and TV broadcast). Thanks!! -Nick

[crytpcrux]

Just discovered the secret to Picasso art. They sort of look like knots. Doesn't the second one in 1:16 look like a Picasso drawing of Mr. Potato Head?

That gives us a "Mr. Reidermeister Potato Head" by Picasso.

[maxp]

Wow, this video is just knots… apologies, I couldn’t resist it. :)

[Jack_Callcott_AU]

The Reidemeister moves have inverses 2) and 3) are their own inverses and we can create an inverse to 1) then we can say two knots K_1 and K_2 are related by a relation R such that K_1 R K_2 iff there is a sequence of Reidemeister moves from K_1 to K_2. R is reflexive, symmetric and transitive and is therefore an equivalence relation on the set of knots which partitions the set into different knot-types. Am I not correct? BTW, thanks for the video. I have never seen this before.

[man]

I once tried to make preztels with different knots, but it was too hard so I ended up with a plate of tri-knots lol

[hdheuejhzbsnnaj]

Fantastic, but what about a full course! 😉

[OeshenNix]

I had my volume on low and thought this was Zach star