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
[edit]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]
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]
Bounds
[edit]
The stick number of a knot sum can be upper bounded by the stick numbers of the summands:[3]
Related invariants
[edit]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.
References
[edit]Notes
[edit]Introductory material
[edit]- 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
[edit]- Adams, Colin C.; Brennan, Bevin M.; Greilsheimer, Deborah L.; Woo, Alexander K. (1997), "Stick numbers and composition of knots and links", Journal of Knot Theory and its Ramifications, 6 (2): 149–161, doi:10.1142/S0218216597000121, MR 1452436
- Calvo, Jorge Alberto (2001), "Geometric knot spaces and polygonal isotopy", Journal of Knot Theory and its Ramifications, 10 (2): 245–267, arXiv:math/9904037, doi:10.1142/S0218216501000834, MR 1822491
- Eddy, Thomas D.; Shonkwiler, Clayton (2019), New stick number bounds from random sampling of confined polygons, arXiv:1909.00917
- Jin, Gyo Taek (1997), "Polygon indices and superbridge indices of torus knots and links", Journal of Knot Theory and its Ramifications, 6 (2): 281–289, doi:10.1142/S0218216597000170, MR 1452441
- Negami, Seiya (1991), "Ramsey theorems for knots, links and spatial graphs", Transactions of the American Mathematical Society, 324 (2): 527–541, doi:10.2307/2001731, MR 1069741
- Huh, Youngsik; Oh, Seungsang (2011), "An upper bound on stick number of knots", Journal of Knot Theory and its Ramifications, 20 (5): 741–747, arXiv:1512.03592, doi:10.1142/S0218216511008966, MR 2806342
External links
[edit]- Weisstein, Eric W., "Stick number", MathWorld
- "Stick numbers for minimal stick knots", KnotPlot Research and Development Site.
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