Stick number

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2,3 torus (or trefoil) knot has a stick number of six. q = 3 and 2 × 3 = 6.

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.

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 (Jin 1997):

{\text{stick}}(T(p,q))=2q{\text{, if }}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 (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})\leq {\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, Huh & Oh 2011):

{\frac 12}(7+{\sqrt {8\,{\text{cr}}(K)+1}})\leq {\text{stick}}(K)\leq {\frac 32}(c(K)+1).

Further reading

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.
  • Y. Huh, S. Oh. An upper bound on stick number of knots, J. Knot Theory Ramifications 20(5):741–747, 2011.

External links

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