Re: [math-fun] infinite comic strip proof of knot non-cancellation?
I'm worried about the validity of that proof. The issue I have is that perhaps this would "prove" that unknot+unknot does not equal unknot? See, if we had a finite sum of unknots, then they ought to be untangleable in a finite number of steps to make it obvious that the result is the unknot. But it perhaps is not obvious that an infinite sum can thus be untangled.
My reservations are related. There are tame knots and there are wild knots, and there are different notions of deformation and cancellation in the two respective settings. Before I believe Dan's (admittedly cute) argument I'll need to see an analysis that pays attention to this nicety. Jim On Thu, Oct 8, 2015 at 11:53 AM, Warren D Smith <warren.wds@gmail.com> wrote:
I'm worried about the validity of that proof.
The issue I have is that perhaps this would "prove" that unknot+unknot does not equal unknot?
See, if we had a finite sum of unknots, then they ought to be untangleable in a finite number of steps to make it obvious that the result is the unknot.
But it perhaps is not obvious that an infinite sum can thus be untangled.
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Just to be clear, the argument is not my own, but I don't know whose it it. I first saw it in Topology of 3-Manifolds and Related Topics, M.K. Fort ed., 1961 — a wonderful book with lots of geometric topology that's fascinating to flip through. [It's been reissued by Dover: http://www.amazon.com/Topology-3-Manifolds-Related-Topics-Mathematics/dp/048... <http://www.amazon.com/Topology-3-Manifolds-Related-Topics-Mathematics/dp/0486477533>.] But the argument is sound, because it only appears to be analogous to dubious tricks with infinite series. The wild knot that is created is an actual topological subspace of R^3, and the homeomorphisms described are actual homeomorphisms of R^3 — which is all that has to be checked. (Amusingly, by complete coincidence I just happened to be reviewing a letter I wrote to Martin Gardner in 1969 that included the same proof, in response to a column he had written about knots. In his response he expressed virtually the same reservations as were expressed in math-fun. But after checking with some experts, his notes read: "The wild (or oo) knot. The proof is sound, axioms and defs are such that it is a valid proof." (John Conway also devised a simple proof that two knots cannot cancel, which avoids any wild knots.) I'm not sure who originated this proof, but a) it appears in an essay by Ralph Fox in that Dover book, and elsewhere is referred to as the "Mazur swindle" — which I suspect refers to the mathematician Barry Mazur. —Dan
On Oct 8, 2015, at 9:38 AM, James Propp <jamespropp@gmail.com> wrote:
My reservations are related. There are tame knots and there are wild knots, and there are different notions of deformation and cancellation in the two respective settings. Before I believe Dan's (admittedly cute) argument I'll need to see an analysis that pays attention to this nicety.
Jim
On Thu, Oct 8, 2015 at 11:53 AM, Warren D Smith <warren.wds@gmail.com> wrote:
I'm worried about the validity of that proof.
The issue I have is that perhaps this would "prove" that unknot+unknot does not equal unknot?
See, if we had a finite sum of unknots, then they ought to be untangleable in a finite number of steps to make it obvious that the result is the unknot.
But it perhaps is not obvious that an infinite sum can thus be untangled.
Yes indeed, this argument is due to Barry Mazur. It's contained in his PhD thesis from Princeton (in 1958 I believe). It's only about 6 or 7 pages long! The story is that there was a seminar being given at Princeton about the attempts to prove the generalized Schoenflies conjecture (that if you embed an n-sphere into an n+1-sphere, that the "inside" and "outside" are both homotopic to a point). After the introductory session Mazur went home to think about it, and came up with the above proof. Victor On Fri, Oct 9, 2015 at 2:19 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Just to be clear, the argument is not my own, but I don't know whose it it.
I first saw it in Topology of 3-Manifolds and Related Topics, M.K. Fort ed., 1961 — a wonderful book with lots of geometric topology that's fascinating to flip through. [It's been reissued by Dover: http://www.amazon.com/Topology-3-Manifolds-Related-Topics-Mathematics/dp/048... <http://www.amazon.com/Topology-3-Manifolds-Related-Topics-Mathematics/dp/0486477533>.]
But the argument is sound, because it only appears to be analogous to dubious tricks with infinite series. The wild knot that is created is an actual topological subspace of R^3, and the homeomorphisms described are actual homeomorphisms of R^3 — which is all that has to be checked.
(Amusingly, by complete coincidence I just happened to be reviewing a letter I wrote to Martin Gardner in 1969 that included the same proof, in response to a column he had written about knots. In his response he expressed virtually the same reservations as were expressed in math-fun. But after checking with some experts, his notes read:
"The wild (or oo) knot. The proof is sound, axioms and defs are such that it is a valid proof."
(John Conway also devised a simple proof that two knots cannot cancel, which avoids any wild knots.)
I'm not sure who originated this proof, but a) it appears in an essay by Ralph Fox in that Dover book, and elsewhere is referred to as the "Mazur swindle" — which I suspect refers to the mathematician Barry Mazur.
—Dan
On Oct 8, 2015, at 9:38 AM, James Propp <jamespropp@gmail.com> wrote:
My reservations are related. There are tame knots and there are wild knots, and there are different notions of deformation and cancellation in the two respective settings. Before I believe Dan's (admittedly cute) argument I'll need to see an analysis that pays attention to this nicety.
Jim
On Thu, Oct 8, 2015 at 11:53 AM, Warren D Smith <warren.wds@gmail.com> wrote:
I'm worried about the validity of that proof.
The issue I have is that perhaps this would "prove" that unknot+unknot does not equal unknot?
See, if we had a finite sum of unknots, then they ought to be untangleable in a finite number of steps to make it obvious that the result is the unknot.
But it perhaps is not obvious that an infinite sum can thus be untangled.
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On Oct 9, 2015, at 11:54 AM, Victor Miller <victorsmiller@gmail.com> wrote:
Yes indeed, this argument is due to Barry Mazur. It's contained in his PhD thesis from Princeton (in 1958 I believe). It's only about 6 or 7 pages long! The story is that there was a seminar being given at Princeton about the attempts to prove the generalized Schoenflies conjecture (that if you embed an n-sphere into an n+1-sphere, that the "inside" and "outside" are both homotopic to a point). After the introductory session Mazur went home to think about it, and came up with the above proof.
As I learned from the topology textbook by Hocking & Young, the "Alexander horned sphere" as in this illustration: https://www.pinterest.com/pin/330803535106498064/ <https://www.pinterest.com/pin/330803535106498064/> shows that an embedding of the n-sphere into an (n+1)-sphere better be differentiable for the Schoenflies theorem to have any chance of being true! —Dan
I think that Mazur's construction need needed the embedding to be C1 at one point! Victor On Fri, Oct 9, 2015 at 3:11 PM, Dan Asimov <asimov@msri.org> wrote:
On Oct 9, 2015, at 11:54 AM, Victor Miller <victorsmiller@gmail.com> wrote:
Yes indeed, this argument is due to Barry Mazur. It's contained in his PhD thesis from Princeton (in 1958 I believe). It's only about 6 or 7 pages long! The story is that there was a seminar being given at Princeton about the attempts to prove the generalized Schoenflies conjecture (that if you embed an n-sphere into an n+1-sphere, that the "inside" and "outside" are both homotopic to a point). After the introductory session Mazur went home to think about it, and came up with the above proof.
As I learned from the topology textbook by Hocking & Young, the "Alexander horned sphere" as in this illustration:
https://www.pinterest.com/pin/330803535106498064/ <https://www.pinterest.com/pin/330803535106498064/>
shows that an embedding of the n-sphere into an (n+1)-sphere better be differentiable for the Schoenflies theorem to have any chance of being true!
—Dan
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curious apocrypha for Barry Mazur: https://books.google.com.mx/books?id=8mBdvAjk_gQC&pg=PA38&lpg=PA38&dq=Barry+Mazur%27s+thesis&source=bl&ots=2tbOIm3m8U&sig=qtcFimc5WsdPQzhZJ45uTMmJEHc&hl=en&sa=X&redir_esc=y#v=onepage&q=Barry%20Mazur's%20thesis&f=false (I see there is a .mx in that link ... I hope you can still see it) On Fri, Oct 9, 2015 at 2:11 PM, Dan Asimov <asimov@msri.org> wrote:
On Oct 9, 2015, at 11:54 AM, Victor Miller <victorsmiller@gmail.com> wrote:
Yes indeed, this argument is due to Barry Mazur. It's contained in his PhD thesis from Princeton (in 1958 I believe). It's only about 6 or 7 pages long! The story is that there was a seminar being given at Princeton about the attempts to prove the generalized Schoenflies conjecture (that if you embed an n-sphere into an n+1-sphere, that the "inside" and "outside" are both homotopic to a point). After the introductory session Mazur went home to think about it, and came up with the above proof.
As I learned from the topology textbook by Hocking & Young, the "Alexander horned sphere" as in this illustration:
https://www.pinterest.com/pin/330803535106498064/ < https://www.pinterest.com/pin/330803535106498064/>
shows that an embedding of the n-sphere into an (n+1)-sphere better be differentiable for the Schoenflies theorem to have any chance of being true!
—Dan
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I'm responsible for that story. Peter Markstein is an old friend of mine, and he told me that story. I passed it along to Krantz. Victor On Fri, Oct 9, 2015 at 6:23 PM, James Buddenhagen <jbuddenh@gmail.com> wrote:
curious apocrypha for Barry Mazur:
(I see there is a .mx in that link ... I hope you can still see it)
On Fri, Oct 9, 2015 at 2:11 PM, Dan Asimov <asimov@msri.org> wrote:
On Oct 9, 2015, at 11:54 AM, Victor Miller <victorsmiller@gmail.com> wrote:
Yes indeed, this argument is due to Barry Mazur. It's contained in his PhD thesis from Princeton (in 1958 I believe). It's only about 6 or 7 pages long! The story is that there was a seminar being given at Princeton about the attempts to prove the generalized Schoenflies conjecture (that if you embed an n-sphere into an n+1-sphere, that the "inside" and "outside" are both homotopic to a point). After the introductory session Mazur went home to think about it, and came up with the above proof.
As I learned from the topology textbook by Hocking & Young, the "Alexander horned sphere" as in this illustration:
https://www.pinterest.com/pin/330803535106498064/ < https://www.pinterest.com/pin/330803535106498064/>
shows that an embedding of the n-sphere into an (n+1)-sphere better be differentiable for the Schoenflies theorem to have any chance of being true!
—Dan
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participants (6)
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Dan Asimov -
Dan Asimov -
James Buddenhagen -
James Propp -
Victor Miller -
Warren D Smith