THC Science

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 $K$ , the stick number of $K$ , denoted by $\operatorname {stick} (K)$ , is the smallest number of edges of a polygonal path equivalent to $K$ .

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 $(p,q)$ -torus knot $T(p,q)$ in case the parameters $p$ and $q$ are not too far from each other:^[1]

\operatorname {stick} (T(p,q))=2q

, if

2\leq p<q\leq 2p.

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]

{\text{stick}}(K_{1}\#K_{2})\leq {\text{stick}}(K_{1})+{\text{stick}}(K_{2})-3\,

Related invariants[edit]

The stick number of a knot $K$ is related to its crossing number $c(K)$ by the following inequalities:^[4]

{\frac {1}{2}}(7+{\sqrt {8\,{\text{c}}(K)+1}})\leq {\text{stick}}(K)\leq {\frac {3}{2}}(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.

References[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.

[1] Jin 1997

[2] Adams et al. 1997

[3] Adams et al. 1997, Jin 1997

[4] Negami 1991, Calvo 2001, Huh & Oh 2011

[1]

[2]

[3]

[4]

Knot theory (knots and links)
Hyperbolic	Figure-eight (4₁) Three-twist (5₂) Stevedore (6₁) 6₂ 6₃ Endless (7₄) Carrick mat (8₁₈) Perko pair (10₁₆₁) Conway knot (11n34) Kinoshita–Terasaka knot (11n42) (−2,3,7) pretzel (12n242) Whitehead (5² ₁) Borromean rings (6³ ₂) L10a140
Satellite	Composite knots Granny Square Knot sum
Torus	Unknot (0₁) Trefoil (3₁) Cinquefoil (5₁) Septafoil (7₁) Unlink (0² ₁) Hopf (2² ₁) Solomon's (4² ₁)
Invariants	Alternating Arf invariant Bridge no. 2-bridge Brunnian Chirality Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability Unknotting no. and problem
Notation and operations	Alexander–Briggs notation Conway notation Dowker–Thistlethwaite notation Flype Mutation Reidemeister move Skein relation Tabulation
Other	Alexander's theorem Berge Braid theory Conway sphere Complement Double torus Fibered Knot List of knots and links Ribbon Slice Sum Tait conjectures Twist Wild Writhe Surgery theory
Category Commons