My understanding was just the opposite -- I've heard knot theorists say that they've never seen an unknot which was not 'obvious', and thus they expected the problem to be easy even though no good general methods are at hand. I have relatively little understanding of the field, personally, so I have no opinion of my own. Charles Greathouse Analyst/Programmer Case Western Reserve University On Mon, Dec 30, 2013 at 6:15 PM, Cris Moore <moore@santafe.edu> wrote:
beautiful movies!
I have heard that making the units of string repel each other, while keeping the arc length constant, seems to be a good heuristic for unknotting unknots. However, since telling whether a knot is an unknot or not is believed to be computationally hard (even showing it's in NP is pretty challenging) such a heuristic shouldn't work in all cases --- there should exist knots where it gets stuck.
A similar issue arises in numerical general relativity: it shouldn't be easy to tell whether two 4-manifolds are equivalent (it's undecidable because of the word problem!) but there are no known examples where numerical algorithms that try to tell whether a 4-manifold is just a sphere get stuck for a long time.
Does anyone know of references on either of these?
Cris
On Dec 30, 2013, at 1:58 PM, Warren D Smith <warren.wds@gmail.com> wrote:
I've posted before about knots are really machines and should be analysed as such. This idea turns out not to be new. Here is a paper
http://arxiv.org/abs/1002.1723
with computer simulations of knots trying to find the "tightest" form of each knot type. You can see movies of the tightening process for many knots & links here:
http://www.jasoncantarella.com/movs/
Many questions suggest themselves...
-- Warren D. Smith http://RangeVoting.org
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