I got pmy question answered at MathOverflow. Apparently no such cancellation is possible. See: http://mathoverflow.net/questions/220313/cancellation-of-2-component-links On Wednesday, October 7, 2015, James Propp <jamespropp@gmail.com> wrote:
Dan Asimov writes:
"The fisherman's knot is not a knot in the sense of topology, since it has two separate strands. That means it can be thought of as a link, but then I don't know how to define two "cancelling links"."
What I mean by a "cancellation" is a way to deform the knot into a trivial cable, i.e., one in which the x-coordinates of both ropes are monotone increasing as one travels along the rope from one end to the other. The two ropes can twist around one another, but not double back.
One could give a more stringent definition that forbids twisting, but I'm inclined to use the more lenient definition, since I think such a cancellation would make a cool magic trick. (Though maybe no such trick exists.)
I should mention that about twenty years ago I accidentally invented a trick in which one APPEARS to be able to cancel two opposite-orientation trefoil knots (thereby "disproving" the noncellation theorem). I've never been able to reconstruct the trick. This may have been an instance in which Murphy's Law trumped mathematical or physical law: I was trying to show someone that you can't cancel knots, and somehow I managed to cancel them! The fact that it was late at night may be relevant. (But no alcohol was consumed.)
Jim
On Wed, Oct 7, 2015 at 10:57 AM, Dan Asimov <dasimov@earthlink.net <javascript:_e(%7B%7D,'cvml','dasimov@earthlink.net');>> wrote:
A knot in topology is defined as a simple closed curve in R^3.
Two knots K and L are called equivalent if there is an orientation-preserving homeomorphism of R^3 to itself taking K onto L.
The cutest proof that two knots cannot cancel goes like this:
Imagine an infinite blank comic strip's panels, but tapering as it heads toward infinity so the top and bottom edges converge at some finite point (forcing the panels to have a series of widths that converge).
Use this template to concatenate infinitely many copies of K and L in this order:
S := K + L + K + L + ... + K + ...
to get a so-called wild knot S.
Knot concatenation is commutative, so K + L = L + K = 0 (the unknot), since we are assuming K and L cancel. So we get:
S = K + (L+K) + (L+K) + ... = K
but also
S = (K+L) + (K+L) + (K+L) + ... = 0.
So K is the unknot, and likewise so is L. So the only knots that can cancel are the trivial knot with itself. ---------------------------------
The fisherman's knot is not a knot in the sense of topology, since it has two separate strands. That means it can be thought of as a link, but then I don't know how to define two "cancelling links".
—Dan
On Oct 7, 2015, at 7:35 AM, James Propp <jamespropp@gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp@gmail.com');>> wrote:
There's an old result in knot theory that has a nice proof due to Conway (which I learned about from a Martin Gardner column ages ago): You can't tie nontrivial knots in different parts of a rope and pass them through each other in such a fashion that they cancel. (Can anyone provide a reference to the relevant column?) Proof idea: Create a tube that follows one knot and swallows the other.
It occurs to me that this proof does not apply to configurations like the fisherman's knot (see the top picture in https://en.wikipedia.org/wiki/File:Fisherman's_knot.png). Or at least, it's not obvious to me how to apply Conway's trick in this context. (Maybe use a tube that isn't cylindrical, but bifurcates like a pair of pants? I don't see how to make this work.)
So now I'm wondering: Is it possible to have something nontrivial like the fisherman's knot in one stretch of a two-rope cable, and something nontrivial in another stretch of the two-rope cable, and cancel them by appropriate manipulations of the cable?
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