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Can the lattice stick number of a knot be bounded by the stick number of the knot?

The stick number $S(K)$ of a knot $K$ is the fewest number of segments needed to realize it by a simple 3D polygon. The lattice stick number $S_L(K)$ is the fewest segments in a realization in the cubic lattice, with all segments parallel to a coordinate axis. For example, the stick number of the trefoil knot $K=3_1$ is 6, and its lattice stick number is 12 (the latter a result of Huh and Oh from 2005).

My question is whether it is possible to bound $S_L(K)$ by $m S(K)$, where $m$ is some multiplier factor. Ideally $m$ would be a constant, but perhaps it is more realistic to expect it to depend on the complexity of the knot (e.g., on its crossing number $cr(K)$). What I have in mind is replacing each stick in a stick realization by a bounded lattice path.

Addendum. Tracy Hall's clever example below indicates that it is unlikely that $m$ could be a constant.

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Clarification: for lattice stick number, can a "segment" by arbitrarily long, or are you counting the total number of unit segment lengths? (Both questions look equally interesting a priori.) – Cap Khoury Jun 15 '10 at 12:50
@Cap Khoury: The former. In general, the length of the sticks is not part of the definition of $S_L(K)$. One stick can consist of several collinear unit-length segments. – Joseph O'Rourke Jun 15 '10 at 12:57
up vote 7 down vote accepted

I wouldn't be surprised by something like a quadratic bound, or possibly something reasonable in terms of another complexity measure for the knot, but I see no hope for making $m$ constant. Consider the following construction: given $m$, choose some large number like $N=(10m)^6$ of points uniformly at random in the unit sphere, and connect them sequentially in a cycle with straight line segments to define a knot $K$. By construction $K$ has stick number no more than $N$, but each stick has a long narrow tunnel that it must traverse in a very precise direction, which is difficult to do with only $m$ lattice sticks. Of course any one tunnel can be made shorter and wider with an affine transformation (or any small collection of tunnels, with a piecewise affine transformation) but I am convinced (without attempting a rigorous proof) that with probability approaching $1$ a knot so constructed has a lattice stick number much higher than $mS_L(K)$.

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This is a convincing example, Tracy! Thanks! – Joseph O'Rourke Aug 4 '10 at 13:12

The lattice stick number is obviously bounded by some function of the stick number: there are finitely many graphs with stick number at most k (because there are finitely many line segment arrangements in the plane and choices of over-under relationships on each crossing) so the lattice stick number of stick-number-k graphs is just the max of the lattice stick number of this finite set of graphs. This argument doesn't give an explicit or good bound on the number, though.

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Thanks, David, for that clear argument! – Joseph O'Rourke Aug 5 '10 at 16:22

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