Hellenica World

# .

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 stick(K), is the smallest number of edges of a polygonal path equivalent to K.

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 (Jin 1997):

$$\text{stick}(T(p,q)) = 2q\text{, if } 2 \le p < q \le 2p. \,$$

The same result was found independently around the same time by a research group around Colin Adams, but for a smaller range of parameters (Adams et al. 1997). They also found the following upper bound for the behavior of stick number under knot sum (Adams et al. 1997, Jin 1997):

$$\text{stick}(K_1\#K_2)\le \text{stick}(K_1)+ \text{stick}(K_2)-3 \,$$

The stick number of a knot K is related to its crossing number c(K) by the following inequalities (Negami 1991, Calvo 2001):

$$\frac12(7+\sqrt{8\,\text{cr}(K)+1}) \le \text{stick}(K)\le 2 c(K).$$

References
Introductory material

C. C. Adams. Why knot: knots, molecules and stick numbers. Plus Magazine, May 2001. An accessible introduction into the topic, also for readers with little mathematical background.
C. C. Adams, The Knot Book: An elementary introduction to the mathematical theory of knots. American Mathematical Society, Providence, RI, 2004. xiv+307 pp. ISBN 0-8218-3678-1

Research articles

C. C. Adams, B. M. Brennan, D. L. Greilsheimer, A. K. Woo. Stick numbers and composition of knots and links. J. Knot Theory Ramifications 6(2):149–161, 1997.

J. A. Calvo. Geometric knot spaces and polygonal isotopy. J. Knot Theory Ramifications 10(2):245–267, 2001.

G. T. Jin. Polygon indices and superbridge indices of torus knots and links. J. Knot Theory Ramifications 6(2):281–289, 1997.

S. Negami. Ramsey theorems for knots, links and spatial graphs. Trans. Amer. Math. Soc. 324(2):527–541, 1991.