I'm looking for an answer to the following question:

Given a knot in $\mathbb{R^{3}}$ can we find a piecewise-linear diagram of it wich is minimal (has a minimal number of verticies)?


closed as unclear what you're asking by Benoît Kloeckner, Stefan Kohl, Ricardo Andrade, Daniel Moskovich, Theo Johnson-Freyd Mar 2 '14 at 2:13

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    $\begingroup$ What do you mean by "can we find"? Are you asking an algorithmic question? $\endgroup$ – Lee Mosher Jun 23 '12 at 16:41
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    $\begingroup$ If by "diagram" you mean a representative in $\mathbb{R}^3$, this is called the stick number of a knot. en.wikipedia.org/wiki/Stick_number However, if by diagram you mean a projection to the plane, I'm not sure if it has been named. There are some patterns of crossings which can't come from a projection of straight line segments, so perhaps a knot could have a projection with fewer vertices than its stick number. $\endgroup$ – Douglas Zare Jun 23 '12 at 17:10
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    $\begingroup$ You might refer to these earlier MO questions on the stick number of a knot: "Which knots' stick numbers are twice their crossing numbers?" mathoverflow.net/questions/39870 , "Lattice Stick Number vs. Stick Number of Knot" mathoverflow.net/questions/28241 , or check out the "stick-knots" MO tag. $\endgroup$ – Joseph O'Rourke Jun 23 '12 at 17:24
  • $\begingroup$ Yes, it seems that the notion of a "stick number" is what I'm looking for. Yes, by "can we find" I mean an algorithm. It is obvious that this stick number is an invariant, so it seems interesting to look for an invariant, depending on this number, or think about knots with the same stick number. Thank you all for answer. It is sometimes difficult to find something, because of terminology. $\endgroup$ – Andrew Jun 24 '12 at 14:56
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    $\begingroup$ I feel your question is not precise enough to get an interesting answer. Do you ask for an explicit algorithm, for complexity bounds, for NP-hardness? $\endgroup$ – Benoît Kloeckner Mar 1 '14 at 16:25