Journal of Knot Theory and Its Ramifications

By considering unknotting operations, we obtain ways of measuring how knotted a knot is. Unknotting phenomena can be seen not only in knot theory, but also in various settings such as DNA knots, mind knots and so on ([C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004); A. Kawauchi, K. Kishimoto and A. Shimizu, Knot theory and game (in Japanese) (Asakura Publishing, Tokyo, 2013); L. Rudolph, Qualitative Mathematics for the Social Sciences (Routledge, London, 2013); K. Murasugi, Knot Theory and Its Applications, Translated from the 1993 Japanese original by Bohdan Kupita (Birkhauser, Boston, MA, 1996)], etc.). In this paper, we see how knots can be unknotted (and therefore how they are knotted) by considering various unknotting operations and their associated unknotting numbers.

Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the absolute values of the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa–Murasugi conjecture for two-bridge knots.

In this paper, we consider local moves on classical and welded diagrams of string links, and the notion of welded extension of a classical move. Such extensions being non-unique in general, the idea is to find a topological criterion which could isolate one extension from the others. To that end, we turn to the relation between welded string links and knotted surfaces in ${ℝ}^4$, and the ribbon subclass of these surfaces. This provides a topological interpretation of classical local moves as surgeries on surfaces, and of virtual local moves as surgeries on ribbon surfaces. Comparing these surgeries leads to the notion of ribbon residue of a classical local move, and we show that up to some broad conditions there can be at most one welded extension which is a ribbon residue. We provide three examples of ribbon residues, for the self-crossing change, the Delta and the band-pass moves. However, for the latter, we note that the given residue is actually not an extension of the band-pass move, showing that a classical move may have a ribbon residue and a welded extension, but no ribbon residue which is an extension.

We improve the upper bound on superbridge index $\mathrm{\text{sb}}[K]$ in terms of bridge index $b[K]$ from $\mathrm{\text{sb}}[K]\le 5b[K]-3$ to $\mathrm{\text{sb}}[K]\le 3b[K]-1$.

A flat plumbing basket is a Seifert surface consisting of a disk and bands contained in distinct pages of the disk open book decomposition of the 3-sphere. In this paper, we examine close connections between flat plumbing baskets and the contact structure supported by the open book. As an application we give lower bounds for the flat plumbing basket numbers and determine the flat plumbing basket numbers for various knots and links, including the torus links.

Cho and Nelson introduced the notion of a quandle coloring quiver, which is a quiver-valued link invariant. This invariant is in general a stronger link invariant than the quandle coloring number. In this paper, we study quandle coloring quiver using dihedral quandle. We show that when we use a dihedral quandle of prime order, the quandle coloring quivers are equivalent to the quandle coloring numbers. We also discuss shadow versions of their invariants.