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Stick Number

Let the stick number $s(K)$ of a Knot $K$ be the least number of straight sticks needed to make a Knot $K$. The smallest stick number of any Knot is $s(T)=6$, where $T$ is the Trefoil Knot. If $J$ and $K$ are Knots, then

s(J+K)\leq s(J)+s(K)+1.

For a nontrivial Knot $K$, let $c(K)$ be the Crossing Number (i.e., the least number of crossings in any projection of $K$). Then

{\textstyle{1\over 2}}[5+\sqrt{25+8(c(K)-2)}]\leq s(K)\leq 2c(K).

The following table gives the stick number for some common knots.

Trefoil Knot 6
Whitehead Link 8

See also Crossing Number (Link), Triangle Counting


Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 27-30, 1994.

© 1996-9 Eric W. Weisstein