[FOM] Formalization Thesis

Bill Taylor W.Taylor at math.canterbury.ac.nz
Sat Jan 19 22:34:57 EST 2008

->Bill Taylor wrote:
->> if the knot pair is somehow untied, the cellophane must continuously
->> deform, so that a line in it, originally from the north point of one circle
->> to the north point of the other, will still become a line in the vertical
->> plane joining them.  But what was knotted before, has now become unknotted!
->> Impossible, QED.

Vaughan Pratt responded:

->Yes, great out-of-the-box line of reasoning, even for an adult.

Well put.

->I have the following problem with it.  Consecutive knots commute, as can 
->be seen by leaving one loose while snugging the other up and sliding the 
->string along the shape formed by the loose knot until the snug knot has 
->passed all the way through the loose knot.  Now with the cellophane in 
->place it's clear that the cellophane need not be disturbed at all when 
->you snug the bagged knot and thread it around the sleeved knot.  However 
->if you try to snug the sleeved knot and thread it around the shape 
->formed by the bagged knot all topological hell breaks loose in the 
->cellophane.  How do you argue that the cellophane doesn't get snagged on 
->itself somewhere in this process and block further progress? 

I don't see how there's a problem here.  Isn't this simple enough?

Note the fiddling with the string, and record the closest any two
separate pieces of string ever get to one another.  (We must never let
the string touch itself, but that is trivial.)  Whatever this distance,
let the distance between string and sleeve remain always at most 1/3 of
this distance, then two bits of cellophane can never get closer to touching.

Am I overlooking something simple?

Bill Taylor

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