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Orderability of link quandles

Hitesh Raundal, Mahender Singh, Manpreet Singh

TL;DR

It is proved that knot quandle of many fibred prime knots are right-orderable, whereas link quandles of most non-trivial torus links are not right- orderable, and it is deduce that the knot qu andle of the trefoil is neither left nor right-ordered.

Abstract

The paper develops a general theory of orderability of quandles with a focus on link quandles of tame links and gives some general constructions of orderable quandles. We prove that knot quandles of many fibered prime knots are right-orderable, whereas link quandles of most non-trivial torus links are not right-orderable. As a consequence, we deduce that the knot quandle of the trefoil is neither left nor right orderable. Further, it is proved that link quandles of certain non-trivial positive (or negative) links are not bi-orderable, which includes some alternating knots of prime determinant and alternating Montesinos links. The paper also explores interconnections between orderability of quandles and that of their enveloping groups. The results establish that orderability of link quandles behave quite differently than that of corresponding link groups.

Orderability of link quandles

TL;DR

It is proved that knot quandle of many fibred prime knots are right-orderable, whereas link quandles of most non-trivial torus links are not right- orderable, and it is deduce that the knot qu andle of the trefoil is neither left nor right-ordered.

Abstract

The paper develops a general theory of orderability of quandles with a focus on link quandles of tame links and gives some general constructions of orderable quandles. We prove that knot quandles of many fibered prime knots are right-orderable, whereas link quandles of most non-trivial torus links are not right-orderable. As a consequence, we deduce that the knot quandle of the trefoil is neither left nor right orderable. Further, it is proved that link quandles of certain non-trivial positive (or negative) links are not bi-orderable, which includes some alternating knots of prime determinant and alternating Montesinos links. The paper also explores interconnections between orderability of quandles and that of their enveloping groups. The results establish that orderability of link quandles behave quite differently than that of corresponding link groups.

Paper Structure

This paper contains 8 sections, 39 theorems, 55 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of linear orderings on quandles
  4. Constructions of orderable quandles and order-preserving automorphisms
  5. Orderability of some general quandles
  6. Orderability of some link quandles
  7. Orderability of link quandles of torus links
  8. Orderability of involutory quandles of alternating links

Key Result

Theorem 2.2

Let $Q$ be a quandle with a presentation $Q= \langle X~~|~~R \rangle$. Then its enveloping group has a presentation $\operatorname{Env}(Q)\cong\langle \tilde{x}, ~~x \in X~~|~~\tilde{R} \rangle$, where $\tilde{R}$ consists of relations obtained from relations in $R$ with an expression $x*y$ replaced

Figures (7)

  • Figure 2: If the component $K_2$ of $L_2$ has an exterior arc in $D_2$
  • Figure 3: If the component $K_2$ of $L_2$ has no exterior arc in $D_2$
  • Figure 6: Montesinos link $M(r_1,r_2,\ldots,r_k)$
  • Figure 7: Toric braid $\tau(m,n)$
  • Figure 8: Toric braid $\tau(m,1)$
  • ...and 2 more figures

Theorems & Definitions (78)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • Corollary 2.8
  • Definition 3.1
  • ...and 68 more