On Oct 8, 2015, at 12:04 PM, Cris Moore <moore@santafe.edu> wrote:
Athanasius Kircher said the world is bound with secret knots: and Kelvin thought atoms were knots in the ether (or rather the æther). In college I was hoping that elementary particles would turn out to be little 3-manifolds… why knot?
Can this æther be used for anæsthesia?
On Oct 8, 2015, at 12:01 PM, Dave Dyer <ddyer@real-me.net> wrote:
Knots are only possible in three dimensions. That's why space has three dimensions. Particles are knots in the fabric of spacetime. Anti-particles are anti-knots, which self untangle (and release energy) when brought together.
This is true for the usual kind of knots: simple closed curves. Indeed they can knot only in three dimensions. But interestingly, there are higher dimensional knots made of k-spheres S^k homeomorphically mapped into R^(k+2).* For example, just as Closure(R^2-R^0) = ([0,oo) x R^0) x S^1 (R^0 is just a single point), we also have Closure(R^3-R^1) = ([0,oo) x R^1) x S^1 and Closure(R^4-R^2) = ([0,oo) x R^2) x S^1. Note that each space above of the form [0,oo) x R^(k-1) is just the closed half-space H^k of R^k. Now let's make an overhand knot K from a *closed interval* in a closed half-space H^3 = ([0,oo) x R^2), so that the endpoints lie on bd(H^3) = R^2. Assume K is smoothly embedded such that its endpoints are arranged to be *perpendicular* to the R^2 = bd(H^3). Because Closure(R^4-R^2) = H^3 x S^1, we have a circle's worth of H^3's. So we can can take K and *spin* it around the R^2 = bd(H^3), keeping its endpoints fixed throughout. Just as the union of longitude lines on a sphere equals the sphere, the union of all these spun images of K forms a S^2 as well. Call the sphere Spun(K). It can be shown that Spun(K) is knotted because the fundamental group of its complement: pi_1(R^4 - Spun(K)) is not isomorphic to pi_1(R^4 - S^2) = Z where S^2 is a standard 2-sphere in R^4. —Dan ________________________________________ * Note: An easy but *uninteresting* way to knot a 2-sphere in R^4 is to just consider an ordinary knot K in R^3 x {0} \sub R^4, and then take the union of all line L(p,q) segments of the form L(p,(0,0,0,1)) and L(p,(0,0,0,-1)) where p ranges over all points p of K. Call this set S(K). This set S(K) will in fact be a topological 2-sphere, but its embedding in R^4 is not what is called "locally flat" at the poles. Instead of being truly knotted globally, it is knotted only because of weirdness at (0,0,0,1) and (0,0,0,-1). This is somewhat related to the reason that the top illustration at https://www.win.tue.nl/~aeb/at/algtop-5.html <https://www.win.tue.nl/~aeb/at/algtop-5.html> is not equivalent to a line segment via any homeomorphism of R^3 to itself.