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.

Known values[edit]

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 (pq)-torus knot T(pq) in case the parameters p and q are not too far from each other (Jin 1997):

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).


Square knot = trefoil + trefoil reflection.

The stick number of a knot sum can be upper bounded by the stick numbers of the summands (Adams et al. 1997, Jin 1997):

Related invariants[edit]

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):

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

Further reading[edit]

Introductory material[edit]

Research articles[edit]

  • 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, doi:10.1142/S0218216501000834, MR 1822491 .
  • 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, doi:10.1142/S0218216511008966, MR 2806342 .

External links[edit]