Journal of Knot Theory and Its Ramifications

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.

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S³. In this paper, we define the Gordian complex of virtual knots which is a simplicial complex whose vertices consist of all virtual knots by using the local move which makes a real crossing a virtual crossing. We show that for any virtual knot K₀ and for any given natural number n, there exists a family of virtual knots {K₀, K₁,…,K_n} such that for any pair (K_i, K_j) of distinct elements of the family, the Gordian distance of virtual knots d_v(K_i, K_j) = 1. And we also give a formula of the f-polynomial for the sum of tangles of virtual knots.

If a knot has the Alexander polynomial not equivalent to 1, then it is linearly n-colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e. the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in detail.

We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, and we give an inequality between the span and the dealternating number.

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knot of type (2, p), modulo p. This expression is t + 2l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is the minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.

We exhibit a finite set of local moves that connect any two surgery presentations of the same 3-manifold via framed links in S³. The moves are handle-slides and blow-downs/ups of a particular simple kind.

We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

In this paper, we associate a plane graph G with an oriented link by replacing each vertex of G with a special oriented n-tangle diagram. It is shown that such an oriented link has the minimum genus over all orientations of its unoriented version if its associated plane graph G is 2-connected. As a result, the genera of a large family of unoriented links are determined by an explicit formula in terms of their component numbers and the degree sum of their associated plane graphs.

In this note, we show that a homomorphism from the braid group B_n into an arbitrary group is injective if and only if its image can be left ordered in a particular way. It follows from this criterion that the image of B_n (n ≥ 2) under any homomorphism is bi-orderable if and only if it is infinite cyclic. As a second application, this criterion also provides a new proof that the Artin representation of B_n into Aut(F_n) is injective.

In this paper, we introduce the notion of a virtual surface-knot as a 2-dimensional generalization of a virtual knot. We define the upper and lower groups of a virtual surface-knot to prove that there are infinitely many non-classical virtual surface-knots.

In 1983 Conway and Gordon proved that any embedding of the complete graph K₇ into ℝ³ contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recently. In this paper, we are interested in knotted Hamiltonian cycles in linear embedding of K₇. Concretely it is shown that any linear embedding of K₇ contains at most three figure-8 knots.