Problem Set: drive.google.com/file/d/1Odm4... Knot Theory: Lecture 2 Andrews University: Math 487 (Spring, 2019) Andrews Math Department: www.andrews.edu/cas/math/ Anthony Bosman: www.anthonybosman.com/
The eye contact, interaction, pace and short review in the beginning makes this class great. After practicing teaching for 4+ years I can tell Anthony really puts in effort to make this class engaging and intelligible
I've stumbled across this because I'm doing an undergrad project about the Alexander polynomial and I can't express how helpful your videos have been. Thank you so much!
Every second of this course, is a moment I gasp so loud with all the unexpected turns this course takes. A very thankful word to the professor who clearly looks so devoted in introducing his lesson.
The best time for me to take this class was when I was in college years ago. The second best time is now. Thank you so much for putting up these videos!
I’m learning about knot theory for my senior project right now, these videos are indispensable! Thorough, clear, and neatly displayed. Thanks for making this free to the public! This video is older but if you see this, do you have any recommendations for resources for someone interested in learning more about knot theory?
Thrilled to hear that! Colin Adams "The Knot Book" is a very accessible and comprehensive introduction (and you may be able to find a free PDF). From there, there are a number of more advanced texts that you can begin to branch out into.
started studying knot theory as a side project on my maths class for my computer science degree. I got invested so much even though i didn't have to. Its such an intriguing concept. Such a lively way to teach things like that...
Just here learning about Knot Theory .. in 2023. No reason other than curiosity. Certainly not for school like many of these other commentors, lol. But - what a fabulous presentation and series. Excited for the other videos! Thanks
Great videos, great content. As a high-school student I am very interested in knots and topology and I learnt a lot from the first two lectures. Looking forward to the next ones. Thanks a lot for sharing this high-quality work and making me excited for learning new things about maths. 😊
So glad that you are finding them valuable! Note each video also contains a handout with some problems you can try linked in the video description. Feel free to send me an email as you get through the series some.
Which textbook do you recommend which follows these lectures closely? Your explanations are crystal clear and build upon one another. I just need a textbook with problem sets so I can practice.
31:30 hold on a second... t=-1 gives you the colorizability determinant! because when you plug it in 1-t (where a section is on top) becomes 2 and t (where a section is underneath, on the left) becomes -1. that's really cool
i have a question, how can we know that the determinant of the matrix to a knot (to a prime knot) is positive? when you defined det(k) = |det(M)|, i had this doubt, because a module is always positive, so the det(k) must be positive. but i cant think of a way to proof that det(k) is always positive. (i'm not a native english speaker, hope you understand my question)
To calculate the Alexander polynomial of a knot we need first give an orientation on the knot. I am not clear why changing this orientation doesn't affect the Alexander polynomial?
I think I found a mistake in the first example at 30:00 minutes, if you delete other lines and determine the other determinants some of the results are not alexander polynomials it seems.
I just use glow sticks! They are cheap, easily bend and snap together, come in various colors, and are a lot of fun. Some also find "tangle" toys to serve the purpose well--though they are a bit more pricey.
These work really great too. One packet it enough for about 3 knots with a crossing number 9. www.amazon.com/gp/product/B01NAJMTX5/ref=ppx_yo_dt_b_asin_title_o02_s00?ie=UTF8&psc=1
We draw from Rolfsen as well as Adams' introductory 'The Knot Book' and some other sources. They're all great, depending on the level and emphasis one is looking for.
Depending on some choices, the determinant can vary up to sign (positive, negative). So we typically just take the absolute value and call that positive value the determinant. Note that for the divisibility test (3-colorable if divisible by 3, etc.), it won't matter if it is positive or negative.
The funny thing is that right after I commented, I got to the part when the lecturer started to talk about how RI changes the determinant by multiplying it by -1
A question: I was working on the Conway polynomials for 5_1 and 5_2 when I came across something I was slightly confused by. While calculating the polynomial for 5_1, I had to calculate the polynomial for a link of two unknots with 4 crossings, commonly called Solomon's Knot. I initially calculated this as z^3+2z, and when I used this in the calculation for the Conway polynomial of 5_1, I got the correct answer and the relation between Conway's Polynomial and Alexander's Polynomial worked just fine. However, I thought that perhaps I could reduce z^3+2z to just z^2+2 and use that. However, it very quickly got me a different polynomial and the transformation for the Conway polynomial to the Alexander polynomial no longer worked. I assume that my reduction isn't allowed. It also just now occurs to me that the simpler link is just z and if we could simplify it by dividing the z out to make it 1, the Conway polynomial would likely be incorrect as well. Is it generally true for Conway Polynomials that be cannot divide or multiply by z?
That's Rolfsen's "Knots and Links"--a classic! You can track down a free PDF online. There is a knot table at the end. Though, you can also access that table in digital form here: katlas.org/wiki/The_Rolfsen_Knot_Table
Hi, I'm a junior in high school who's investigating knot theory for my high school math project. These videos are incredibly helpful since they break down more complex concepts, but I was wondering if there is any way I can contact you through email for help regarding the project? It would be a lifesaver.
That's right--good catch! This correction is noted in the handout. Also, it and other corrections appear in the upper-righthand corner of the video during the lecture.
@@MathatAndrews Excellent lectures. I wish there was a talk on computational ways to determine knots. Say, I have a curve's coordinates (3D discrete polymer simulations). How to find the Alexander polynomial and knot type? Project the coordinates on different planes and find the crossings? Any projection would work up to a power of +-t^m?
I thought since this is a lecture video so you would at least give some proof or sketch of proof,but the reality is even worse, you simply give all those results even without any explanation.Can you at least tell me some reference?
