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.