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R-equivalence classes of $\mathrm{Rot} \mathbb{E}^{2}$-colorings of torus knots

Mai Sato

TL;DR

This work addresses the problem of classifying R-equivalence classes of non-trivial $Rot\mathbb{E}^{2}$-colorings on torus-knot diagrams $D(p,q)$. It builds on Inoue's trochoid parametrization, connecting colorings with $(|p|,k;|q|,l)$-trochoids and analyzing their behavior under Reidemeister moves via two geometric deformations: shifts and switches. The main result shows that, when the parity condition $p'q'$ is even, R-equivalence is generated by shifts and even counts of switches, effectively reducing classification to a combinatorial action; the paper also establishes two necessary conditions for R-equivalence using quandle 2-cocycles and weights. An appendix relates cyclotomic unit theory to the algebraic structure underlying the colorings, highlighting deeper number-theoretic aspects of the problem.

Abstract

We introduce a new equivalence relation, named R-equivalence relation, on the set of colorings of an oriented knot diagram by a quandle. We determine the R-equivalence classes of colorings of a diagram of a torus knot by a quandle, called $\mathrm{Rot} \mathbb{E}^{2}$, under a certain condition.

R-equivalence classes of $\mathrm{Rot} \mathbb{E}^{2}$-colorings of torus knots

TL;DR

This work addresses the problem of classifying R-equivalence classes of non-trivial -colorings on torus-knot diagrams . It builds on Inoue's trochoid parametrization, connecting colorings with -trochoids and analyzing their behavior under Reidemeister moves via two geometric deformations: shifts and switches. The main result shows that, when the parity condition is even, R-equivalence is generated by shifts and even counts of switches, effectively reducing classification to a combinatorial action; the paper also establishes two necessary conditions for R-equivalence using quandle 2-cocycles and weights. An appendix relates cyclotomic unit theory to the algebraic structure underlying the colorings, highlighting deeper number-theoretic aspects of the problem.

Abstract

We introduce a new equivalence relation, named R-equivalence relation, on the set of colorings of an oriented knot diagram by a quandle. We determine the R-equivalence classes of colorings of a diagram of a torus knot by a quandle, called , under a certain condition.
Paper Structure (11 sections, 19 theorems, 42 equations, 13 figures)

This paper contains 11 sections, 19 theorems, 42 equations, 13 figures.

Table of Contents

  1. Introduction
  2. R-equivalence for quandle colorings
  3. $\mathrm{Rot} \space \mathbb{E}^{2}$-colorings of a diagram of a torus knot
  4. Main theorem
  5. Two necessary conditions for R-equivalence
  6. Proof of Theorem \ref{['main_thm']}
  7. On cyclotomic integers of absolute value $1$ by Takashi Hara (Department of Mathematics, Tsuda University)
  8. Materials from algebraic number theory
  9. Cyclotomic fields and its maximal real subfields
  10. Restatement via the norm map
  11. Proof of Proposition \ref{['prop:main2']}

Key Result

Theorem 3.1

Set $m = |p|$ and $n = |q|$ in the above. Let $a_{ij}$ denote the arcs of $D(p, q)$ as depicted in Figure a_diagram_of_(p,q)-torus_knot, although $a_{i0}$ and $a_{[i + 1]_{|q|}, |p| - 1}$ mark the same arc for each $i$($0 \leq i \leq |q| - 1$). Then the map $\mathscr{C}(Q, P_{0})$ from the set of al is well-defined and a non-trivial $\mathrm{Rot} \space \mathbb{E}^{2}$-coloring of $D(p,q)$. Here,

Figures (13)

  • Figure 1: The condition for an $X$-coloring.
  • Figure 2: The colorings of a diagram of the trefoil knot by the dihedral quandle of order $3$.
  • Figure 3: Reidemeister moves relate $X$-colorings of the diagrams uniquely.
  • Figure 4: The diagram $D(p, q)$ of the $(p, q)$-torus knot.
  • Figure 5: Regular polygons of type $(4, k)$, $(5, k)$ and $(6, k)$.
  • ...and 8 more figures

Theorems & Definitions (39)

  • Remark 2.1
  • Theorem 3.1: Theorem 4.1 of I2015
  • proof
  • Remark 3.2
  • Theorem 4.1
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • Lemma 5.3
  • proof
  • ...and 29 more