Narrow Results

Lattice stick number s_L(K) is defined to be the minimal number of sticks required to construct a polygonal representation of the knot K in the cubic lattice. In this paper, we give lattice stick numbers of small knots such as 3₁ and 4₁. More precisely we prove that s_L(3₁) = 12 and s_L(K) ≥ 14 for any other non-trivial knot K.

For a nontrivial knot $K$, Negami found an upper bound on the stick number $s(K)$ in terms of its crossing number $c(K)$ which is $s(K)\le 2c(K)$. Later, Huh and Oh utilized the arc index $\alpha (K)$ to present a more precise upper bound $s(K)\le \frac{3}{2}c(K)+\frac{3}{2}$. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number $s_{=}(K)$ as follows; $s_{=}(K)\le 2c(K)+2$. As a sequel to this research program, we similarly define the stick number $s(G)$ and the equilateral stick number $s_{=}(G)$ of a spatial graph $G$, and present their upper bounds as follows;

In 1991, Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of crossing number c(K) which is s(K) ≤ 2c(K). In this paper we give a new upper bound in terms of arc index, and improve Negami's upper bound to . Moreover if K is a nonalternating prime knot, then .

We prove that the knots $13n_{592}$ and $15n_{41,127}$ both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known.

This work is motivated by a paper of Huh and Oh, in which the authors prove that the minimum number of sticks required to form a knot in ℤ³ is 12. In this article the authors prove that the stick number in the simple hexagonal lattice is 11. Moreover, the stick number of the trefoil in the simple hexagonal lattice is 11.

An algorithm that produces polygonal cable links is described, and applications discussed. In particular, the stick numbers of T_p,q torus links are shown to be 4p for 2p < q ≤ 3p, and it is shown that, in general, . Further, it is shown that the Ramsey number of a link is at least the sum of its arc index and bridge number. Using these results, we relate the Ramsey, stick and crossing numbers of torus links, showing .

This paper gives new upper bounds on the stick numbers of the knots $9_{18}$, $10_{18}$, $10_{58}$, $10_{66}$, $10_{68}$, $10_{80}$, $10_{82}$, $10_{84}$, $10_{93}$, $10_{100}$, and $10_{152}$, as well as on the equilateral stick number of $10_{79}$. These bounds imply that the knots $10_{58}$, $10_{66}$, and $10_{80}$ have superbridge index $\le 5$, completing the project of showing that no prime knots through 10 crossings can have superbridge index larger than 5. The current best bounds on stick number and superbridge index for prime knots through 10 crossings are given in Appendix A.

We address the concept of stick number for knots and links under various restrictions concerning the length of the sticks, the angles between sticks, and placements of the vertices. In particular, we focus on the effect of composition on the various stick numbers. Ultimately, we determine the traditional stick number for an infinite class of knots, which are the (n,n-1)-torus knots together with all of the possible compositions of such knots. The exact stick number was previously known for only seven knots.