Stick number

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 $T(p,q)$ in case the parameters $p$ and $q$ are not too far from each other:{{harvnb|Jin|1997}}

{{bi|left=1.6|$\backslash operatorname\{stick\}(T(p,q))\; =\; 2q$, 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.{{harvnb|Adams|Brennan|Greilsheimer|Woo|1997}}

File:Square knot sum of trefoils stick number.png

The stick number of a knot sum can be upper bounded by the stick numbers of the summands:{{harvnb|Adams|Brennan|Greilsheimer|Woo|1997}}, {{harvnb|Jin|1997}}

$$\backslash text\{stick\}(K\_1\backslash \#K\_2)\backslash le\; \backslash text\{stick\}(K\_1)+\; \backslash text\{stick\}(K\_2)-3\; \backslash ,$$

The stick number of a knot $K$ is related to its crossing number $c(K)$ by the following inequalities:{{harvnb|Negami|1991}}, {{harvnb|Calvo|2001}}, {{harvnb|Huh|Oh|2011}}

$$\backslash frac12(7+\backslash sqrt\{8\backslash ,\backslash text\{c\}(K)+1\})\; \backslash le\; \backslash text\{stick\}(K)\backslash le\; \backslash frac32\; (c(K)+1).$$

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

{{reflist}}

• {{citation|first=C. C.|last=Adams|authorlink=Colin Adams (mathematician)|url=http://plus.maths.org/issue15/features/knots/index.html|title=Why knot: knots, molecules and stick numbers|journal=Plus Magazine|date=May 2001}}. An accessible introduction into the topic, also for readers with little mathematical background.
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}}

• {{MathWorld|title=Stick number|urlname=StickNumber|mode=cs2}}
• "[http://www.colab.sfu.ca/KnotPlot/sticknumbers/ Stick numbers for minimal stick knots]", KnotPlot Research and Development Site.

{{Knot theory|state=collapsed}}

Category:Knot invariants