Journal of Knot Theory and Its Ramifications

A (1,1)-knot in the 3-sphere is a knot that admits a 1-bridge presentation with respect to a Heegaard torus in ${𝕊}^3$. A new parameterization of $(1,1)$-knots distinct from the classical ones is introduced. This parameterization is obtained from minimum length representatives of homotopy classes of arcs in the multipunctured plane. In the particular case of satellite (1,1)-knots, it is proven that the introduced parameterization is essentially unique. A generalization of this parameterization to the family of $(g,1)$-knots for any $g\ge 1$ is proposed.

A $4$-move is a local change for knots and links which changes 4 half twists to 0 half twists or vice versa. In 1979, Yasutaka Nakanishi conjectured that every knot can be changed by $4$-moves to the unknot. This is still an open problem. In this paper, we consider the $4$-move distance of knots, which is the minimal number of $4$-moves needed to deform one into the other. In particular, the $4$-move unknotting number of a knot is the $4$-move distance to the unknot. We give a table of the $4$-move unknotting number of knots with up to nine crossings.

Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang–Baxter equation. A homology theory for the set-theoretic Yang–Baxter equation was developed by Carter et al. in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang–Baxter equation. We obtain some concrete examples of nontrivial $n$-cocycles for Alexander biquandles. For a biquandle $X,$ its geometric realization $BX$ is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of $BX$ is finitely generated if the biquandle $X$ is finite.

Taking the Levine–Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017, Cimasoni and Conway generalized this formula to the multivariable signature of the closure of colored tangles. In this paper, we extend even further their result by using a different 4-dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the isotropic functor ${\mathcal{ℱ}}_{\omega }$. We also show that in the case of colored braids this defect can be rewritten in terms of the Meyer cocycle and the colored Gassner representation, making it a direct generalization of the formula of Gambaudo and Ghys.

Given a biquandle $(X,S)$, a function $\tau$ with certain compatibility and a pair of non commutative cocyles $f,h:X\times X\to G$ with values in a non necessarily commutative group $G$, we give an invariant for singular knots/links. Given $(X,S,\tau )$, we also define a universal group $U_{nc}^{fh}(X)$ and universal functions governing all 2-cocycles in $X$, and exhibit examples of computations. When the target group is abelian, a notion of abelian cocycle pair is given and the “state sum” is defined for singular knots/links. Computations generalizing linking number for singular knots are given. As for virtual knots, a “self-linking number” may be defined for singular knots.

In this paper we prove that the Rohlin invariant is the unique invariant inducing a homomorphism on the Torelli group. Using this result we generalize the construction of invariants of homology 3-spheres from families of trivial 2-cocycles on the Torelli group given by Pitsch to include invariants with values on an abelian group with 2-torsion.

A quandle coloring quiver is a quiver structure, introduced by Cho and Nelson, which is defined on the set of quandle colorings of an oriented knot or link by a finite quandle. We study quandle coloring quivers of $(p,2)$-torus knots and links with respect to dihedral quandles.

We discuss meridians and longitudes in reduced Alexander modules of classical and virtual links. When these elements are suitably defined, each link component will have many meridians, but only one longitude. Enhancing the reduced Alexander module by singling out these peripheral elements provides a significantly stronger link invariant. In particular, the enhanced module determines all linking numbers in a link; in contrast, the module alone does not even detect how many linking numbers are 0.