Narrow Results

A method is given for economically constructing any algebraic knot or link K. This construction, which involves tree diagrams, gives a new upper bound for the edge number of K that is proven to be at most twice the crossing number of K. Furthermore, it realizes a minimal-crossing projection.

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.

We introduce a geometric construction, called a clasp move, which is used to prove several results on stick numbers of links. In particular, we show that all two-component six-crossing links have stick number 8, produce small crossing number knots that can be constructed with fewer sticks than crossings, and improve known upper bounds for the stick number of certain classes of links.

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 .

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.

The conformation space of α-regular knots, that is, of spatial polygons which are both equilateral and equiangular, has been investigated by both chemists and mathematicians for quite some time. More recently, the study of α-regular stick numbers, the minimum number of sticks needed to create an α-regular polygon representing a given knot, has also gained some traction. In most of these studies, the value of the angle α has been a fixed constant, playing a secondary role to the number of edges in the polygon or its knot type. In this paper, we consider the entire spectrum of minimal α-regular stick numbers for a knot as a function of α. In particular, we compute this spectrum for the special case of the unknot.

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 .

An equilateral stick number s₌(K) of a knot K is defined to be the minimal number of sticks required to construct a polygonal knot of K which consists of equal length sticks. Rawdon and Scharein [Upper bounds for equilateral stick numbers, in Physical Knots: Knotting, Linking, and Folding Geometric Objects in ℝ³, Contemporary Mathematics, Vol. 304 (American Mathematical Society, Providence, RI, 2002), pp. 55–76] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithms in the software KnotPlot. In this paper, we find an upper bound on the equilateral stick number of a non-trivial knot K in terms of the minimal crossing number c(K) which is s₌(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s₌(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s₌(K₁♯K₂) ≤ 2c(K₁) + 2c(K₂).

Utilizing both twisting and writhing, we construct integral tangles with few sticks, leading to an efficient method for constructing polygonal 2-bridge links. Let L be a two-bridge link with crossing number c, stick number s, and n tangles. It is shown that . We also show that if c > 12n + 3, then minimal stick representatives do not admit minimal crossing projections.

An $n$-string tangle is a pair $(B,A)$ such that $A$ is a disjoint union of properly embedded $n$ arcs in a topological $3$-ball $B$. And an $n$-string tangle is said to be trivial (or rational)${}^{\mathrm{\text{a}}}$, if it is homeomorphic to $(D\times I,\{x_1,\dots ,x_n\}\times I)$ as a pair, where $D$ is a 2-disk, $I$ is the unit interval and each $x_i$ is a point in the interior of $D$. A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an $n$-string stick tangle its stick-order is defined to be a nonincreasing sequence $(s_1,s_2,\dots ,s_n)$ of natural numbers such that, under an ordering of the arcs of the tangle, each $s_i$ denotes the number of sticks constituting the $i$th arc of the tangle. And a stick-order $S$ is said to be trivial, if every stick tangle of the order $S$ is trivial.

In this paper, restricting the $3$-ball $B$ to be the standard 3-ball, we give the complete list of trivial stick-orders.

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;

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.

Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$: $s(K)\le 2c(K)$. Huh and Oh found an improved upper bound: $s(K)\le \frac{3}{2}(c(K)+1)$. Huh, No and Oh proved that $s(K)\le c(K)+2$ for a $2$-bridge knot or link $K$ with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let $K$ be a knot or link which admits a reduced Montesinos diagram with $c(K)$ crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then $s(K)\le c(K)+3$. Furthermore, if $K$ is alternating, then we can additionally reduce the upper bound by $2$.

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.

Previous work [C. Shonkwiler, New computations of the superbridge index, J. Knot Theory Ramifications 29(14) (2020) 2050096] used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. This work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygonal realizations do not minimize the stick number of the knot; these seem to be the first such examples. Appendix A gives a complete summary of what is currently known about superbridge indices of prime knots through 10 crossings and Appendix B gives all knots through 16 crossings for which the superbridge index is known.

Consider two parallel lines ${\ell }_1$ and ${\ell }_2$ in ${ℝ}^3$. A rail arc is an embedding of an arc in ${ℝ}^3$ such that one endpoint is on ${\ell }_1$, the other is on ${\ell }_2$, and its interior is disjoint from ${\ell }_1\cup {\ell }_2$. Rail arcs are considered up to rail isotopies, ambient isotopies of ${ℝ}^3$ with each self-homeomorphism mapping ${\ell }_1$ and ${\ell }_2$ onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper calculates the stick number of rail arcs classes with a crossing number at most 2 and uses a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for 29 knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial $11$-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$).

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.