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 ${\displaystyle K}$. A related quantity is the equilateral stick number, the smallest number of edges of the same length that are required to form a knot. It is not currently known whether the equilateral stick number is the same as the stick number for every knot.

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]

There are certain knots for which the upper bound and lower bound of the stick number are the same, such that the stick number is known exactly. These include 3₁ with a stick number of 6, 4₁ (7), all 5 and 6 crossing knots (8), and all 7 crossing knots (9). The 8 crossing knots 16 through 21 in Alexander-Briggs notation (8 or 9), and 9-crossing knots 29, 34, 35, and 39 through 49 (9), and 10₁₂₄ (10, a torus knot) have known crossing numbers. There are 19 additional non-alternating 11- and 13-crossing knots with a stick number of exactly 10.^[3]

The stick number of a knot sum can be upper bounded by the stick numbers of the summands:^[5] $${\displaystyle {\text{stick}}(K_{1}\#K_{2})\leq {\text{stick}}(K_{1})+{\text{stick}}(K_{2})-3\,}$$

The stick number of a knot ${\displaystyle K}$ is related to its crossing number ${\displaystyle c(K)}$ by the following inequalities:^[6] $${\displaystyle {\frac {1}{2}}(7+{\sqrt {8\,{\text{c}}(K)+1}})\leq {\text{stick}}(K)\leq {\frac {3}{2}}(c(K)+1).}$$

These inequalities are both tight for the trefoil knot, which has a crossing number of 3 and a stick number of 6. The upper bound on the stick number does not apply to the unknot, which has crossing number 0 but stick number 3.

• 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", Experimental Mathematics, 31 (4): 1373–1395, arXiv:1909.00917, doi:10.1080/10586458.2021.1926000
• 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, JSTOR 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.