This lecture is part of an online course on algebraic topology. We calculate the fundamental group of (the complement of) a knot, and give a couple of examples. For the other lectures in the course see • Algebraic topology
I would like to also express my appreciation for these lectures. I'm a mathematics student at a school with a not-superb mathematics department in terms of topics available for study, and having some extremely high quality lectures which I can use in conjunction with a text for self study is so helpful that I don't have words to convey it. My gratitude for the lectures you produce is unbounded
Just want to share some of the love. Thank you so much for all your effort to share math with us. I got to a uni with a fantastic math dep but i always keep your videos on my weekly to solidify my understanding. You are a very friendly face to my friend group. We all appreciate it!
I am happy, that English understatement is alive and kicking. (or whatever hue of understatement you are happy to subscribe to, but cultivation of healthy understatement it is)
Dear Professor Borcherds, I can assure that I am not the only person here deeply grateful to you for bringing us this precious knowledge. I would like also to ask a question, respectfully: Is there any university around the world offering an online undergraduate mathematics Programme? I am really interested in coming back to university. I am a 43 math teacher, engineer, and want to enjoy pure mathematics. For any help, thanks in advance. Thanks, again, for your collosal help making these videos for the public here. :-)
www.open.ac.uk/courses/maths/degrees/bsc-mathematics-q31 This might help. Good luck I'm in a similar position, but i'm opting to self study instead, at least for now.
Livingston's _Knot Theory_ has a good approach, though he doesn't really investigate that much beyond defining it. Rolfsen does more, but tbh I find Rolfsen a lot more difficult to read. Really the study of the knot complement belongs to 3-manifold theory, and you're more likely to find deeper results in books that deal directly with that subject than with knot theory proper these days.
I don't quite understand why adding a ball kills off the boundary. Besides, a generator is just a loop. So why we could use an edge of the ball to represent a generator?
Hello, I have been watching your videos for a while now and want to thank you for the awesome content that you consistently upload. I am sure that you don’t have a lot of free time and I want to help you with that. I am a video editor and I want to ask you if you need any editing for your videos. it's possible to have a delivery for contact you and speak about it?
In case anyone is wondering, Prof. Borcherds is sweeping a lot under the table here. Specifically, how do we know the group we get doesn't depend on the particular projection of the knot? This takes us into the Reidemeister moves that define equivalence of knot diagrams, and some fiddling around with algebra to show that the group presentations we get from two different diagrams connected by a Reidemeister move give isomorphic groups. Flipping this around, we also get to why the knot group is NOT actually a very "easy" invariant of a knot. It's easy to define, but it's VERY DIFFICULT to tell whether two group presentations give isomorphic groups. In effect, the knot group replaces a question we don't know how to answer (whether two knots are isotopic) with a question we basically can't answer (whether two groups are isomorphic). We can, if we're lucky, prove that two groups are NOT isomorphic (as Prof. Borcherds does in the examples), but it's fairly difficult to work with the knot group as an invariant you can actually calculate with. Nice example of the fundamental group, though.
@@drmathochist06 And for those that don't, it means that *in general* there does not exist an algorithm that can correctly determine if two groups are isomorphic from their presentations. Indeed, many simple properties of groups like this are undecidable from the group presentation. We cannot even (in general) tell if it is abelian, or even trivial. While presentations are a very convenient way of specifying groups, they can be difficult to work with.
Isn't it obvious, that the group (the isomorphism class of the group to be precise) does not depend on the projection? Given the things shown informally in the video it is nothing else but the fundamental group of the complement of a knot. And the homotopy type of the complement of a knot is invariant under equivalence of knots (orientation preserving homeomorphisms of R³ to R³ mapping one knot to the other). So at least the isomorphism class of this group is independent of the projection. And more is not possible because the fundamental groups are non-abelian and therefore the fundamental groups are not canonically isomorphic.
