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Stick number

2,3 torus (or trefoil) knot has a stick number of six.

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 , the stick number of , denoted by , is the smallest number of edges of a polygonal path equivalent to .

Known values

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 -torus knot in case the parameters and are not too far from each other:[1]

, if

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]


Square knot = trefoil + trefoil reflection.

The stick number of a knot sum can be upper bounded by the stick numbers of the summands:[3]

Related invariants

The stick number of a knot is related to its crossing number by the following inequalities:[4]

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



Introductory material

  • 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.

Research articles

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Stick number
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