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!
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.
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.
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)
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?
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.
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?
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
Professor Borcherds, hope you continue these lectures. They are great! Learning a lot from you. Many thanks!
suddenly not hearing from any of your lectures... missing you much, are you doing ok sir?
We are hungry for more!
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!
Hope all is well Richard. Love this series.
Yes, why is there such a long break this time? He said "Next lecture....". I hope all is well too.
These lectures are golden, they are so clear and on topic keep it up!!
I love this algebraic topology lecture series so much!!!
hope you continue the series!
Does anyone have a good reference for the fundamental group of knot complements? It is only briefly mentioned in Hatcher.
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.
Stillwell's Classical Topology and Combinatorial Group Theory includes some more information.
@@drmathochist06 Thank you!
@@Michael-sq5ju Thank you!
Completely unrelated to your question: Vegan 👍 Mathematician 👍.
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.
@@jeffrey8770 Thanks a lot for recommending this web site. I'll chek it out. Thanks a lot, Jeffrey!
Agree. A wonderful teacher
Thank you for these wonderfull Videos Professor, I really enjoy watching these!
I really hope he is doing ok :(
I keep checking back in. I can't find a twitter or anything for him. Incredible man... Incredible channel.
Appreciate your efforts a lot. Please keep going 👍🏻
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)
I hope everythings is well . thank you very much for your work... please consider continue this class.
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?
Your brother Mr Borcherds taught me maths at Queen Mary's, both incredible teachers !
Professor, hope you are doing fine... Looking forward to seeing your new lectures.
Sorry, what happened to the "half time does not work"?
Waiting patiently for the next lecture on algebraic topology….
Which whiteboard are you using?
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.
Technically: the "very difficult" group isomorphism problem is in general "undecidable", for those who know what that means.
@@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.
It seems you could match sections by parts of the sections by say a 296 or a 729
2187. Connects too to these No.s so that’s a way
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.
thank you professor!
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?
Why he stop making video?
Probably because it's the summer breeak
@@f5673-t1h Too long! I hope there is nothing wrong.
@@staj6236 same
Dag nabbit. I was looking to see if you posted
Şu konular üzerine muhabbet çevirebileceğimiz hiç Türk yok mu? Çok esaslı konularmış gibi geliyor bana.
idk know why but it reminds me Escher's paingtings... I have just finished lecture about SVK theo today XD
I have seen that knots directions are on ancient temples. Budda
first yeey
yeeeeeeeeeeeeeeeeeeee
@@migarsormrapophis2755 yeeeeeeeee
yeeeeeee