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I have a few questions about nonplanar "stick circuits" (or hexagons and higher n-gons) $n$-gons) that you might be able to help with:

(I know that n=6 $n=6$ is the minimum number of points to form a stick knot.)

(1) Given n=6 $n=6$ points in R3 $\mathbb{R}^3$ in general position connected by a specific "stick circuit" (nonplanar hexagon), what test can be done to see if it forms a stick knot vs. an unknot?

(2) Given n=6 $n=6$ points in R3 $\mathbb{R}^3$ in general position, there are 60 different stick circuits connecting them. True or false, at least one forms a knot?

(3) Given n=6 $n=6$ points in R3 $\mathbb{R}^3$ in general position, does the minimum-length stick circuit on these n $n$ points ever form a knot? ("knotted 6-point traveling salesman problem with return)

All these can be generalized to $n > 66$.

These questions occurred to me over the last few days. I suspect (1) has a known answer but I have no idea about (2) or (3).

Any info will be eagerly read! Thank you.

Steve Gray

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# Stick knot questions: simple but may not be easy

I have a few questions about nonplanar "stick circuits" (or hexagons and higher n-gons) that you might be able to help with:

(I know that n=6 is the minimum number of points to form a stick knot.)

(1) Given n=6 points in R3 in general position connected by a specific "stick circuit" (nonplanar hexagon), what test can be done to see if it forms a stick knot vs. an unknot?

(2) Given n=6 points in R3 in general position, there are 60 different stick circuits connecting them. True or false, at least one forms a knot?

(3) Given n=6 points in R3 in general position, does the minimum-length stick circuit on these n points ever form a knot? ("knotted 6-point traveling salesman problem with return)

All these can be generalized to n > 6.

These questions occurred to me over the last few days. I suspect (1) has a known answer but I have no idea about (2) or (3).

Any info will be eagerly read! Thank you.

Steve Gray