Stick number explained

In the mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight "sticks" stuck end to end needed to form a knot. Specifically, given any knot

K , the stick number of K , denoted by \operatorname{stick}(K) , is the smallest number of edges of a polygonal path equivalent

Known values

Six is the lowest stick number for any nontrivial knot. There are few knots whose stick number can be determined exactly. Gyo Taek Jin determined the stick number of a

(p,q) -torus knot T(p,q) in case the parameters p and q are not too far from each other:

The same result was found independently around the same time by a research group around Colin Adams, but for a smaller range of parameters.

Bounds

The stick number of a knot sum can be upper bounded by the stick numbers of the summands:^[1] $$\backslash text(K\_1\backslash \#K\_2)\backslash le\; \backslash text(K\_1)+\; \backslash text(K\_2)-3\; \backslash ,$$

Related invariants

The stick number of a knot

K is related to its crossing number c(K) by the following inequalities:^[2] $$\backslash frac12(7+\backslash sqrt)\; \backslash le\; \backslash text(K)\backslash le\; \backslash frac32\; (c(K)+1).$$

These inequalities are both tight for the trefoil knot, which has a crossing number of 3 and a stick number of 6.

References

Introductory material

• . An accessible introduction into the topic, also for readers with little mathematical background.
• .

Research articles

External links

• • "Stick numbers for minimal stick knots", KnotPlot Research and Development Site.

Notes and References

1. ,
2. ,,