Number of segments (knot theory)

2,3 toric knot (trefoil) has a number of segments equal to six. q = 3 and 2 × 3 = 6.

In knot theory, the number of segments is an invariant of a node that defines the smallest number of straight “segments” that, connecting end to end, form a knot. More specifically, for any node K, the number of segments of K , denoted by stick ( K ), is the smallest number of links in a polyline equivalent to K.

Content

Known Values

The smallest number of segments for nontrivial nodes is six. There are a small number of nodes for which the number of segments can be determined accurately. Gyo Taek Jin determined the number of segments ( p , q ) - toric nodes T ( p , q ) for cases when the parameters p and q are not very different ^[1] :

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

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

if a

2\leq p<q\leq 2p.

{\ displaystyle 2 \ leq p <q \ leq 2p.}

At about the same time, the same result was independently obtained by a research group led by , but for a smaller range of parameters ^[2] .

Borders

The number of segments of the composition of nodes from above is limited by the total number of segments of the source nodes ^[2] ^[1] :

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

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

Related Invariants

The number of segments of the knot K is related to its number of intersections c (K) by the following inequality ^[3] ^[4] ^[5] :

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

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

Notes

↑ ¹ ² Jin, 1997 .
↑ ¹ ² Adams, Brennan, Greilsheimer, Woo, 1997 .
↑ Negami, 1991 .
↑ Calvo, 2001 .
↑ Huh, Oh, 2011 .

Literature

Introductory materials

CC Adams. Why knot: knots, molecules and stick numbers // Plus Magazine. - 2001. - Vol. May . An introduction for readers with little knowledge of mathematics
CC Adams. The Knot Book: An elementary introduction to the mathematical theory of knots. - Providence, RI: American Mathematical Society, 2004 .-- ISBN 0-8218-3678-1 .

Research Articles

Colin C. Adams, Bevin M. Brennan, Deborah L. Greilsheimer, Alexander K. Woo. Stick numbers and composition of knots and links // Journal of Knot Theory and its Ramifications. - 1997. - T. 6 , no. 2 . - S. 149—161 . - DOI : 10.1142 / S0218216597000121 .
Jorge Alberto Calvo. Geometric knot spaces and polygonal isotopy // Journal of Knot Theory and its Ramifications. - 2001. - T. 10 , no. 2 . - S. 245-267 . - DOI : 10.1142 / S0218216501000834 .
Gyo Taek Jin. Polygon indices and superbridge indices of torus knots and links // Journal of Knot Theory and its Ramifications. - 1997. - T. 6 , no. 2 . - S. 281-289 . - DOI : 10.1142 / S0218216597000170 .
Seiya Negami. Ramsey theorems for knots, links and spatial graphs // Transactions of the American Mathematical Society. - 1991.- T. 324 , no. 2 . - S. 527-541 . - DOI : 10.2307 / 2001731 .
Youngsik Huh, Seungsang Oh. An upper bound on stick number of knots // Journal of Knot Theory and its Ramifications. - 2011 .-- T. 20 , no. 5 . - S. 741-747 . - DOI : 10.1142 / S0218216511008966 .

Links

Weisstein, Eric W. Stick number on the Wolfram MathWorld website.
" Stick numbers for minimal stick knots ", KnotPlot Research and Development Site .

[_6573d405e0a408e4-1] ¹ ² Jin, 1997 .

[_f13925b0f6b94ffe-2] ¹ ² Adams, Brennan, Greilsheimer, Woo, 1997 .

[_bd05fb2bf415735a-3] Negami, 1991 .

[_439f0184cea350ef-4] Calvo, 2001 .

[_3ef03916cdf48ca5-5] Huh, Oh, 2011 .