In the mathematical theory of knots, the stick number is a

knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some i ...

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

\operatorname(K)

, is the smallest number of edges of a

polygonal path In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...

equivalent

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

(p,q)

torus knot In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of cop ...

T(p,q)

in case the parameters

p

and

q

are not too far from each other: 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

Square knot sum of trefoils stick number

The stick number of a

knot sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifica ...

can be upper bounded by the stick numbers of the summands:

\text(K_1\#K_2)\le \text(K_1)+ \text(K_2)-3 \,

Related invariants

The stick number of a knot

K

is related to its crossing number

c(K)

by the following inequalities:, ,

\frac12(7+\sqrt) \le \text(K)\le \frac32 (c(K)+1).

These inequalities are both tight for the

trefoil knot In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest kno ...

, which has a crossing number of 3 and a stick number of 6.

References

Notes

Introductory material

*. An accessible introduction into the topic, also for readers with little mathematical background. *.

Research articles

* * * * * *

External links

*
Stick numbers for minimal stick knots
, ''KnotPlot Research and Development Site''. {{Knot theory, state=collapsed Knot invariants