Problems & Puzzles: Puzzles

Puzzle 748. A third follow up to Puzzle 747 (about 3D-golygons) Here we ask you to make a special 3D-golygon: one that produces a "knot" satisfying all the conditions to be a valid 3D-golygon (See puzzle 747) Perhaps you already know that the simplest known knot is the so-called "trefoil-knot". Here an illustration of it: I suppose that these kind of 3D-golygons (knotted-golygons) not only exists but are a kind of easy to produce. But I must tell you frankly that I have never produced any one of these creatures... Q. Send your minimal 3D-knotted-golygon (data and drawing, please) a) using consecutive natural numbers as sides. b) using consecutive prime numbers as sides.

Contributions came from Jan van Delden

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Jan wrote:

The simplest knot on a cube lattice is the trefoil or “klaverbladknoop” in Dutch.
(picture from www.shapeways.com):

Let’s give the lengths of the edges a name: n[i].

 

We must satisfy the following relations, to close the curve:

n[1+i]+n[7+i]=n[4+i]+n[10+i] with i in the set {0,1,2}.

Since n[i]<n[j] for all i<j is requested, we see that we can’t have a solution even if we use strictly increasing real numbers.

In order to construct a trefoil we must add some slack to some one or more loops.
Therefore the number of edges must be larger than 12.

Instead of trying other types of knots, I decided to look for a golygon in 3D, using the following steps:
Find a 3D golgyon.
Identify edges that are part of a loop. (*) [Repeat if necessary].

Reduce the golygon by replacing the marked edges by a shorter xyz-loop if possible.

Use parallel projection for each main direction on the reduced golygon. 
Check if there is at least one line which is crossed twice, once from above, once from below.
Output the original golygon if everything checks out o.k. (**)

(*)   An edge is part of a loop if the two rectangles spanned by this edge combined with its neighbours don’t cross another (unmarked) edge. 
(**) This is not a sufficient condition, but it is a necessary condition, so I used visual inspection to select correct solutions.

Unlucky enough the identifying step combined with the reduction step don’t proof to be sufficient. So one should inspect solutions.
Since this procedure takes considerable time (especially the visual inspection), I’m not sure my solutions are minimal.

 

Natural solution, n=48:  1 through 48.

 

Prime solution, n=36: 3 through 157:

See attachment for input files for Gygols.jar.

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