{"document_url": "https://en.wikipedia.org/wiki/Stick_number"}
Smallest number of edges of an equivalent polygonal path for a knot
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:
, 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.

Bounds
The stick number of a knot sum can be upper bounded by the stick numbers of the summands:
Related invariants

The stick number of a knot
is related to its crossing number
by the following inequalities:
These inequalities are both tight for the trefoil knot, which has a crossing number of 3 and a stick number of 6.