Stick number

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

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 ${\displaystyle (p,q)}$-torus knot ${\displaystyle T(p,q)}$ in case the parameters ${\displaystyle p}$ and ${\displaystyle q}$ are not too far from each other:^[1]

${\displaystyle \operatorname {stick} (T(p,q))=2q}$, if ${\displaystyle 2\leq p<q\leq 2p.}$

The same result was found independently around the same time by a research group around Colin Adams, but for a smaller range of parameters.^[2]

The stick number of a knot sum can be upper bounded by the stick numbers of the summands:^[3] $${\displaystyle {\text{stick}}(K_{1}\#K_{2})\leq {\text{stick}}(K_{1})+{\text{stick}}(K_{2})-3\,}$$

The stick number of a knot ${\displaystyle K}$ is related to its crossing number ${\displaystyle c(K)}$ by the following inequalities:^[4] $${\displaystyle {\frac {1}{2}}(7+{\sqrt {8\,{\text{c}}(K)+1}})\leq {\text{stick}}(K)\leq {\frac {3}{2}}(c(K)+1).}$$

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

• Adams, C. C. (May 2001), "Why knot: knots, molecules and stick numbers", Plus Magazine. An accessible introduction into the topic, also for readers with little mathematical background.
• Adams, C. C. (2004), The Knot Book: An elementary introduction to the mathematical theory of knots, Providence, RI: American Mathematical Society, ISBN 0-8218-3678-1.

• Adams, Colin C.; Brennan, Bevin M.; Greilsheimer, Deborah L.; Woo, Alexander K. (1997), "Stick numbers and composition of knots and links", Journal of Knot Theory and its Ramifications, 6 (2): 149–161, doi:10.1142/S0218216597000121, MR 1452436
• Calvo, Jorge Alberto (2001), "Geometric knot spaces and polygonal isotopy", Journal of Knot Theory and its Ramifications, 10 (2): 245–267, arXiv:math/9904037, doi:10.1142/S0218216501000834, MR 1822491
• Eddy, Thomas D.; Shonkwiler, Clayton (2019), New stick number bounds from random sampling of confined polygons, arXiv:1909.00917
• Jin, Gyo Taek (1997), "Polygon indices and superbridge indices of torus knots and links", Journal of Knot Theory and its Ramifications, 6 (2): 281–289, doi:10.1142/S0218216597000170, MR 1452441
• Negami, Seiya (1991), "Ramsey theorems for knots, links and spatial graphs", Transactions of the American Mathematical Society, 324 (2): 527–541, doi:10.2307/2001731, MR 1069741
• Huh, Youngsik; Oh, Seungsang (2011), "An upper bound on stick number of knots", Journal of Knot Theory and its Ramifications, 20 (5): 741–747, arXiv:1512.03592, doi:10.1142/S0218216511008966, MR 2806342

• Weisstein, Eric W., "Stick number", MathWorld
• "Stick numbers for minimal stick knots", KnotPlot Research and Development Site.