Stick number

File:Trefoil valknut cropped.png

2,3 torus (or trefoil) knot has a stick number of six.

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 $\mathrm{stick}(K)$, is the smallest number of edges of a polygonal path equivalent to $K$.

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: Template:Bi

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

File:Square knot sum of trefoils stick number.png

Square knot = trefoil + trefoil reflection.

The stick number of a knot sum can be upper bounded by the stick numbers of the summands: $$\text{stick}(K_1\mathrm{\#}{K}_2)\le \text{stick}(K_1)+\text{stick}(K_2)-3$$

Related invariants

The stick number of a knot $K$ is related to its crossing number $c(K)$ by the following inequalities: $$\frac{1}{2}(7+\sqrt{8\text{c}(K)+1})\le \text{stick}(K)\le \frac{3}{2}(c(K)+1).$$

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.
• .

External links

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