Journal of Knot Theory and Its Ramifications

We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N = 1 are biquandles, which we call bikei or 双圭. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, and we give an example of a non-involutory birack whose counting invariant detects the non-invertibility of a virtual knot.

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, the nullification process may be the pathway for a knotted circular DNA to unknot itself (through recombination of its DNA strands). The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. It turns out that there are several different ways to define such a number, since different conditions may be applied in the nullification process. We show that these definitions are not equivalent, thus they lead to different nullification numbers for a knot/link in general, not just one single nullification number. Our aim is to explore some mathematical properties of these nullification numbers. First, we give specific examples to show that the nullification numbers we defined are different. We provide a detailed analysis of the nullification numbers for the well known 2-bridge knots and links. We also explore the relationships among the three nullification numbers, as well as their relationships with other knot invariants. Finally, we study a special class of links, namely those links whose general nullification number equals one. We show that such links exist in abundance. In fact, the number of such links with crossing number less than or equal to n grows exponentially with respect to n.

We determine prime amphicheiral links with at least two components and up to 11 crossings. There are 27 such links. We check also special amphicheiralities. Most of prime links with up to 11 crossings are detected not to be amphicheiral by a condition on the Jones polynomial. For the rest links, we applied conditions from the Alexander polynomial. We added new necessary conditions for a special case.

We provide necessary conditions for the Alexander polynomials of algebraically split component-preservingly amphicheiral links. We raise a conjecture that the Alexander polynomial of an algebraically split component-preservingly amphicheiral link with even components is zero. Our necessary conditions and some examples support the conjecture.

In [The planar algebra of a bipartite graph, Knots in Hellas '98 (World Scientific, 2000), pp. 94–117, MR1865703], Jones found two copies of the cyclic category cΔ in the annular Temperley–Lieb category Atl. We give an abstract presentation of Atl to discuss how these two copies of cΔ generate Atl together with the coupling constants and the coupling relations. We then discuss modules over the annular category and homologies of such modules, the latter of which arises from the cyclic viewpoint.

We show that there is no non-trivial 3-adjacency relation between two-bridge knots or links. As an application of our proof argument, we also show that if a knot K is 3-adjacent to a two-bridge knot or fibered knot K′, then K and K′ have the same determinant.

In this paper, we investigate the behaviors of the index polynomial and the arrow polynomial for periodic virtual links. As applications, we give necessary conditions for a given virtual link to have period n in terms of those polynomials, which can be used to estimate the possible periods for a given virtual link with non-trivial index polynomial or non-trivial arrow polynomial.

Suppose P_n is a regular n-gon in ℝ². An embedding f : P_n ↪ ℝ³ is called an α-regular stick knot provided the image of each side of P_n under f is a line segment of length 1 and any two consecutive line segments meet at an angle of α. The main result of this paper proves the existence of α-regular stick unknots for odd n ≥ 7 with α in the range . All knots constructed will have trivial knot type, and we will show that any non-trivial α-regular stick knot must have .

We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.

This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte's activity letters that arise because the Jones polynomial is a specialization of the signed version of the Tutte polynomial. The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above. Evaluations of these activity words are related to the chain complex for the Champanerkar–Kofman spanning tree model of reduced Khovanov homology.