Problem Set: drive.google.com/file/d/1MKbt... Knot Theory: Lecture 3 Andrews University: Math 487 (Spring, 2019) Andrews Math Department: www.andrews.edu/cas/math/ Anthony Bosman: www.anthonybosman.com/
I was skeptical about the assertion that it wasn't possible to connect two knots to produce an unknot since you didn't try merging a trefoil with its mirror image. So I did that and found that the resulting knot was 3-colorable. So much for that idea haha.
Sir, I just bumped into your videos as I was watching a video two days ago on Lisa Piccirillo’s theorem about the Conway knot not being slice. Two days ago, I had no idea about this field of mathematics and I took a deep dive into this beautiful theory. I start to get a glimpse of understanding into what Ms. Piccirillo means by her proposition but it seemed rather esoteric at first sight... I really love your lectures: very well explained with infectious enthusiasm! I would love to have some more proofs for instance, why is the genus bounded by half the span of the Alexander polynomial or why is the tricolorability an invariant (why can you just specialise to the 3 Reidermeister moves), why is the Conway polynomial the same as the Alexander one by just substituting z with t-t^-1, why do you need to get rid of one column and one row to get to the right déterminant of a knot, why do you have p-Colorability in only p prime...) In any case, really great course with a lot of thought provoking ideas: the best so far for me (after only your 3 first videos) is the Seifert trick to make a knot into a surface after you put it in soapy water! Beautiful... That really reconciles me a lot with the human nature, like when I hear a great piano sonata or read a beautiful poem. Maths can be so poetic!
I am starting these lectures, I only watched the three first. These are super clear and interesting, thanks for that ! Quick question : Is the reason why you don't provide proof of results as 1/2Span(Delta_k)≤g(k) or ∆_k(t)=Nabla_k(t^(1/2)-t^(-1/2)) is that it takes more advanced tools than the students had at their stage of study ? If yes, would you recommend a book that provides these in an understandable way ?
Questions:1) we may imagine that using Reidemeister transformation we obtain a knot representation (maybe several) with a minimal number of intersections. If we apply the construction of the surface, could we imagine that in such case we obtain the exact genus of the knot? 2)0bvoiuosly there are two way to build a connected sum of knots. Are bot the same?
I just discovered your course and been binge watching it. I am certainly more advanced than the audience you were teaching to. But man, this is very well explained and illuminating! Thanks for all the effort you've put into this!
One must also be careful of so called "pinching" (by virtue of homeomorphisms being bijective). Because one could push two antipodal points of a circle together along a diameter and this would not tear or create a hole; however, the figure eight is not homeomorphic to the circle.
Do you have some solved problems on knot theory ....... I was solving problems of Richard H. Crowell ..... but some of them were hard to me. ...... thanks for awesome explanation ...... How do you learned knot theory in such a beautiful way?
In the video description for each lecture, there is a handout that contains a few problems. You may also enjoy Adams free, excellent "Knot Book": math.harvard.edu/~ctm/home/text/books/adams/knot_book/knot_book.pdf
Hold on! The Alexander Polynomial of 5_1 isn't right. The sign on the last term needs to be flipped in both cases. After all, Alexander Polynomial coefficients are palindromic.
Also, I don't get how you're counting the faces. Like when you're counting the three-legged, two-disked surface bounded by the trefoil, shouldn't there be two faces for each disk instead of one? Why not? Thanks so much for all your help, these videos are awesome. What a beautiful subject.
Good question! The idea is that each face has two sides. For instance, the cube has 6 faces, but each of those faces has a side facing inside and a side outside of the cube. Similarly, the disk we consider as just one face which has a top and bottom. Hope that helps!
i listen to stuff like this and i wonder if they ever plugged tose numbers into music, light, emf, etc and then..... what does it mean??? lol as i brood over a plate of mashed poatoes hahaha, but seriously, its fun
Can you imagine giving an engaging lecture, only to have someone walk into your classroom every three to five minutes trying to sell your students something? Great presentation series made annoying by very-frequent ad disruptions. The price of obtaining knowledge is 5 seconds every three minutes, I guess.
