On colorability of knots by rotations, torus knot and PL trochoid (Q2253720)

A quandle is an algebraic system introduced by \textit{D. Joyce} [J. Pure Appl. Algebra 23, 37--65 (1982; Zbl 0474.57003)] and \textit{S. V. Matveev} [Math. USSR, Sb. 47, 73--83 (1984); translation from Mat. Sb., Nov. Ser. 119(161), No. 1, 78--88 (1982; Zbl 0523.57006)]. They showed that quandles derived from knots are isomorphic if and only if the knots are equivalent. A pair \((Q, *)\) of a set \(Q\) and a binary operation \(*\) is called a quandle if is satisfies the three axioms: (1) \(x*x=x\), (2) the map \(*y : Q \ni x \mapsto x*y \in Q\) is a bijection, and (3) \((x*y)*z=(x*z)*(y*z)\) for all \(x,y,z \in Q\). A coloring of an oriented knot diagram \(D\) by a quandle \(Q\) is a map \(\{ \text{arcs of }D \} \rightarrow Q\) such that \(x*y=z\) is satisfied at each crossing, where \(y,x,z\) are the elements assigned to the over arc, and the under arcs on the right-hand side and on the left-hand side of it, respectively. It is non-trivial if it is not a constant map. It is known that, for a fixed quandle, the number of all colorings of an oriented knot diagram gives a knot invariant. Let Rot\({\mathbb E}^2\) be the set consisting of all rotations of the Euclidean plane \({\mathbb E}^2\), and \((z,e^{\theta\sqrt{-1}})\;(\in {\mathbb C} \times \text{U(1)})\) the rotation about \(z\) by angle \(\theta\), where we identify the complex plane \({\mathbb C}\) with \({\mathbb E}^2\). Then Rot\({\mathbb E}^2\) forms a quandle under the conjugation \((z,e^{\theta\sqrt{-1}})*(w,e^{\eta\sqrt{-1}})=((z-w)e^{\eta\sqrt{-1}}+w,e^{\theta\sqrt{-1}})\). It is shown that for any knot diagram colored by Rot\({\mathbb E}^2\), the rotation angles of elements assigned to the arcs are the same, and that a knot has a non-trivial coloring by Rot\({\mathbb E}^2\), whose rotation angles are \(\theta\), if and only if its Alexander polynomial has a root \(e^{\theta\sqrt{-1}}\) on the unit circle in \({\mathbb C}\). Note that under the same condition on the Alexander polynomial, it is known that there is a non-trivial representation of the knot group to the group of all orientation preserving similarities of \({\mathbb C}\) mapping a positive meridian to a rotation (Proposition 14.5 in [\textit{G. Burde} and \textit{H. Zieschang}, Knots. 2nd revised and extended ed. 2nd revised and extended ed. Berlin: Walter de Gruyter (2003; Zbl 1009.57003)]). A non-trivial Rot\({\mathbb E}^2\) coloring remains to be a non-trivial coloring if the centers of rotations are transformed by an orientation perserving similarity. The paper under review enumerates all non-trivial colorings of the standard diagram of a torus knot by Rot\({\mathbb E}^2\). For any pair of integers \(k, \ell\) with \(1 \leq k \leq |p|-1, 1 \leq \ell \leq |q|-1\), there is a non-trivial coloring of the standard diagram of the \((p,q)\)-torus knot with the rotation angle \(\theta(|p|,k;|q|,\ell)=((|p|-2k)/|p| - (|q|-2\ell)/|q|)\pi\). The author considered the \(\lq\lq (|p|,k;|q|,\ell)\)-trochoid'', to describe the centers of rotations. Moreover, the Alexander polynomial of the \((p,q)\)-torus knot is completely factrized into prime factors \(t-e^{\theta(|p|,k;|q|,\ell)\sqrt{-1}}\).