Related papers: Reorienting quandle orbits
Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions…
We work with a generalization of knot theory, in which one diagram is reachable from another via a finite sequence of moves if a fixed condition, regarding the existence of certain morphisms in an associated category, is satisfied for every…
The concept of turnaround surface in an accelerating universe is generalized to arbitrarily large deviations from spherical symmetry, to close the gap between the idealized theoretical literature and the real world observed by astronomers.…
Quandles with good involutions, which are called symmetric quandles, can be used to define invariants of unoriented knots and links. In this paper, we determine the necessary and sufficient condition for good involutions of a generalized…
We give a negative answer to a question by J.M. Landsberg on the nature of normalizations of orbit closures. A counterexample originates from the study of complex, ternary, cubic forms.
We introduce the warping matrix which is a new description of oriented knots from a viewpoint of warping degree.
The transverse beam pattern, usually observed in experiment, is a result of averaging the optical-frequency oscillations of the electromagnetic field distributed over the beam cross section. An analytical criterion is derived that these…
A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle…
In this paper, we aim to provide a notion of "relative objects", i.e. objects equipped with some sort of subobjects, in differential topology. In spite of active researches relating them, e.g. knot theory or the theory of manifolds with…
A quandle is an algebraic system which excels at describing limited symmetries of a space. We introduce the concept of Schl\"{a}fli quandles which are defined relating to chosen rotational symmetries of regular tessellations. On the other…
The reassignment method for the wavelet transform is investigated. Particularly good results are obtained if the wavelet is an extremal for the uncertainty relation of the affine group.
We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves…
Let $R$ be a unital ring with involution.In this paper, several new necessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in a ring $R$ are given.In addition, the formulae of the Moore-Penrose…
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular…
In quantum field theory, physicists routinely use "normal ordering" of operators, which just amounts to shuffling all creation operators to the left. Potentially confusing, then, is the occurrence in the literature of normal-ordered…
We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory.
Introducing a radially dependent magnetic field into Newton's off-center circular orbits potential so as to preserve the $E=0$ dynamical symmetry leads to a unique choice of field that can be identified as the inclusion of a magnetic…
To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for the tables of knots and tangles, to investigate which tangles may or may not embed in knots in the tables.
We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-classical virtual knots.
This paper is a survey of several papers in quandle homology theory and cocycle knot invariants that have been published recently. Here we describe cocycle knot invariants that are defined in a state-sum form, quandle homology, and methods…