To whom it may concern: If the audio bothers you as it does me, you can go to windows options -> ease of access -> audio -> turn on mono audio. Just remember to turn it back off afterwards.
This is so cool...I got turned on to knot theory by diving into another youtube lecture series on quantum physics (kind of a counter intuitive order, I know lol). But the implications of the parallels between topology and theoretical physics are astounding. I'm definitely going to binge watch the rest of this series/class tomorrow when we're all snowed in. Thanks for making quality education accessible for everyone!!
Glad to hear you're enjoying it! I hope you get a lot out of the videos -- note there are also some problem linked in each video description that you can try. Stay warm!
When finding the determinant, why is one column and one row automatically crossed out to reduce dimension? Is it because the determinant will be the same either way? Is there a proof for further reading?
So, how does the concept of luminiferous aether differ from a concept of quantum vacuum or quantum foam? How does the knot concept differ from that of a closed string? Are we going to get to the Kontsetitch Invariant some day?
Hi sir, very nice video! Could you please tell me where I can find proofs for the theorems you illustrated here and, more in general, a good knot theory book? Greetings from Italy
what if you can cut at a under crossing and make it an upper crossing ? can a multiple crossing knot by this way be transformed successively to an unknot ?
Yes, just by changing crossings any knot can be changed into an unknot! You should think about this until you convince yourself that it is true. It may help to recognize that changing crossings is equivalent to letting the knot "pass through itself".
Amazing video. Thanks a lot. I have a question though: for the last knot you mention (the 7-4), why should we consider a 6x6 matrix when there're 7 crossings? For the Trefoil, there are 3 crossings thus a 3x3 matrix, why should it be different when p goes up?
It's not. After writing the overdetermined 3x3 matrix for the trefoil, you delete a row and column before taking the determinant. We only took the determinant of the 2x2 you'll matrix. The same is done for 7_4.
Thumbs up and subscribed. A question. At 1:01 and other times, you are using some sort of plastic tubing. What is that called and where is it sold? I've used whiteboards and extension cords for practice. Didn't really like the extension cords because they seem to want to return to a previous state. Any help would be greatly appreciated.
is posible to make a matrix using 0, 1, and -2? because x + y == 2z (mod p) is not only equivalent to 2z - x - y == 0 (mod p), but also x + y - 2z == 0 (mod p)
Good question! In a 4-dimensional setting, every knotted circle would be trivial. This is because you have an extra dimension that allows the knot to "pass through" itself with touching itself. (Convince yourself of this.) However, we can think of knotted spheres in 4-dimensional space! That is, in a 4D universe, it would be possible to have a knotted up a basketball!
not my field at all so I got confused when you went from colors (kindergarden) to number (at least high school). So dumb question, could a not be 5 colorable and not 3 colorable? I sense it's not, but I think everything would be clearer if I know why...
No worries! It is easy to get knotted up... Yes, a knot can be 5 colorable but not 3 colorable. For instance, if a know has determinant 5, then it would be 5 colorable but not colorable. Here's an example of such a knot: katlas.org/wiki/5_1
For the first couple videos, only high school algebra. As they advance, some of the ideas will become more advanced and abstract, but I try to explain what is needed along the way.
@@MathatAndrews thank you so much for your quick reply !!! I will watch all of the videos and try to read "the knot book" and ,hopefully, publish a paper.
