Re: [math-fun] [xbbn] Graduate Student Solves Decades-Old Conway Knot Problem | Quanta Magazine
Knotting happens in codimension 2 regardless of the dimension. So a 2-sphere can be knotted in 4-dimensional space. Here's an example: Think of R^4 as R^2 x R^2. Notice that R^4 - (R^2 x {0}) = R^2 x (R^2 - {0}) and that factor R^2 - {0} is the result of rotating a ray (0, oo) around a circle. Now embed a closed arc in R^4 with its endpoints meeting R^2 x {0} perpendicularly ... and with an overhand knot in its middle. Finally, spin that arc around the R^2 x {0} (using the circular symmetry) to sweep out a surface. This surface is homeomorphic to an ordinary 2-sphere S^2. But it is knotted! Similar methods create even higher-dimensional, codimension-2 knots. —Dan Cris Moore wrote: ----- This is great. But I have a question. I had the impression that d-dimensional surfaces can be “knotted” (homeomorphic but not homotopic to a sphere, if I’m remembering the correct uses of those lovely greek-derived words) in 2d+1 dimensions: 1-dimensional curves in 3 dimensions, 2-dimensional surfaces in 5, and so on. But this artlcle says that 2-dimensional surfaces can be knotted in 4 dimensions. I can see how two spheres (2-dimensional surfaces) can be linked in 5 dimensions: put them in 3 dimensions so that the overlap and intersect in a circle, and then “lift” each point on that circle to a pair of antipodal points on a circle in another 2 dimensions. Then the two spheres cannot be drawn apart without having them intersect in all 5 dimensions at some point. But I don’t see a similar construction in 4 dimensions. Can anyone give a simple example of a knot or link of 2-surfaces in 4 dimensions? -----
Thanks. I wonder where I got this 2d+1 thing… maybe I’m misremembering some other embedding theorem. I do like the 5-dimensional construction of two linked 2-spheres, though, assuming I got that right. Cris
On May 23, 2020, at 8:12 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Knotting happens in codimension 2 regardless of the dimension.
So a 2-sphere can be knotted in 4-dimensional space. Here's an example:
Think of R^4 as R^2 x R^2. Notice that
R^4 - (R^2 x {0}) = R^2 x (R^2 - {0})
and that factor R^2 - {0} is the result of rotating a ray (0, oo) around a circle.
Now embed a closed arc in R^4 with its endpoints meeting R^2 x {0} perpendicularly ... and with an overhand knot in its middle.
Finally, spin that arc around the R^2 x {0} (using the circular symmetry) to sweep out a surface. This surface is homeomorphic to an ordinary 2-sphere S^2. But it is knotted!
Similar methods create even higher-dimensional, codimension-2 knots.
—Dan
Cris Moore wrote: ----- This is great. But I have a question. I had the impression that d-dimensional surfaces can be “knotted” (homeomorphic but not homotopic to a sphere, if I’m remembering the correct uses of those lovely greek-derived words) in 2d+1 dimensions: 1-dimensional curves in 3 dimensions, 2-dimensional surfaces in 5, and so on. But this artlcle says that 2-dimensional surfaces can be knotted in 4 dimensions.
I can see how two spheres (2-dimensional surfaces) can be linked in 5 dimensions: put them in 3 dimensions so that the overlap and intersect in a circle, and then “lift” each point on that circle to a pair of antipodal points on a circle in another 2 dimensions. Then the two spheres cannot be drawn apart without having them intersect in all 5 dimensions at some point.
But I don’t see a similar construction in 4 dimensions. Can anyone give a simple example of a knot or link of 2-surfaces in 4 dimensions? -----
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