Journal of Knot Theory and Its Ramifications

We classify the isotopy types of negatively curved closed strips on an arbitrarily given knot which have prescribed negative Gaussian curvature at each point on the knot.

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 .

Using the Fourier expansion of Markov traces for Ariki–Koike algebras over ℚ(q, u₁, …, u_e), we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite a simple form. As a consequence, we show that the Alexander polynomial of a mixed link is essentially equal to the Alexander polynomial of the link obtained by resolving the twisted parts.

Classical knots in ℝ³ can be represented by diagrams in the plane. These diagrams are formed by curves with a finite number of transverse crossings, where each crossing is decorated to indicate which strand of the knot passes over at that point. A pseudodiagram is a knot diagram that may be missing crossing information at some of its crossings. At these crossings, it is undetermined which strand passes over. Pseudodiagrams were first introduced by Ryo Hanaki in 2010. Here, we introduce the notion of a pseudoknot, i.e. an equivalence class of pseudodiagrams under an appropriate choice of Reidemeister moves. In order to begin a classification of pseudoknots, we introduce the concept of a weighted resolution set, or WeRe-set, an invariant of pseudoknots. We compute the WeRe-set for several pseudoknot families and discuss extensions of crossing number, homotopy, and chirality for pseudoknots.

A virtual link comes with a variety of link complements. This paper is concerned with the Dehn space, a pseudo-manifold with boundary, and the Dehn complex, a two-dimensional spine of the Dehn space. In the classical case where the link is planar, the Dehn space is the link complement in the 3-sphere. We study topological and geometric properties of the Dehn complex of a virtual link. Among other things, we show that every finitely presented group is the fundamental group of a Dehn complex, and that the Dehn complex of any alternating triple of an alternating virtual link is a non-positively curved squared complex.

We describe the space of arrow diagram formulas for virtual knot diagrams in the annulus ℝ × 𝕊¹ as the kernel of a linear map, inspired from a conjecture due to Polyak. As a main application, we slightly improve Grishanov–Vassiliev's theorem for planar chain invariants.

Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of bands required to present the link as boundaries of an unknotted (flat) banded surface. Banded links of small (flat, respectively) band index are considered here. Some upper bounds are provided for these band indices of a link using braid representatives and canonical Seifert surfaces of the link. The relation between the band indices and genera of links is studied and the band indices of pretzel knots are calculated.

Lomonaco and Kauffman [Quantum knots and mosaics, Quantum Inf. Process. 7(2–3) (2008) 85–115] introduced the notion of knot mosaics in their work on quantum knots. It is conjectured that knot mosaic type is a complete invariant of tame knots. In this paper, we answer a question of C. Adams by constructing an infinite family of knots whose mosaic number can be realized only when the crossing number is not. That is, there is an infinite family of non-minimal knots whose mosaic numbers are known.