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Fundamental $n$-quandles of links are residually finite

Neeraj Kumar Dhanwani, Deepanshi Saraf, Mahender Singh

TL;DR

The paper proves that the fundamental $n$-quandle $Q_n(L)$ of every oriented link $L$ in $S^3$ is residually finite for all $n\ge2$, enriching the link-quandle correspondence via $n$-fold cyclic branched covers and Thurston geometry. It develops a general framework for subquandle separability, showing how separability transfers from group-theoretic contexts to quandle constructions and proving residually finite behavior for broad quandle families, including free and twisted-conjugation quandle variants, as well as finitely generated abelian quandles. The results imply solvability of the word problem for finitely presented residually finite quandles and provide practical criteria for identifying subquandle separable structures. Together, these findings extend the landscape of residually finite quandles and connect quandle finiteness properties to 3-manifold topology and group separability phenomena.

Abstract

In this paper, we investigate the residual finiteness and subquandle separability of quandles, properties that respectively imply the solvability of the word problem and the generalized word problem for quandles. From Winker's work, we know that fundamental $n$-quandles of oriented links, which are canonical quotients of their fundamental quandles, are closely associated with $n$-fold cyclic branched covers of the 3-sphere branched over these links. We prove that the fundamental $n$-quandle of any oriented link in the 3-sphere is residually finite for each $n\ge 2$. This supplements the recent result by Bardakov, Singh and the third author on residual finiteness of fundamental quandles of oriented links, and the classification by Hoste and Shanahan of links whose fundamental $n$-quandles are finite for some $n$. We also establish several general results on these finiteness properties and identify many families of quandles admitting them.

Fundamental $n$-quandles of links are residually finite

TL;DR

The paper proves that the fundamental -quandle of every oriented link in is residually finite for all , enriching the link-quandle correspondence via -fold cyclic branched covers and Thurston geometry. It develops a general framework for subquandle separability, showing how separability transfers from group-theoretic contexts to quandle constructions and proving residually finite behavior for broad quandle families, including free and twisted-conjugation quandle variants, as well as finitely generated abelian quandles. The results imply solvability of the word problem for finitely presented residually finite quandles and provide practical criteria for identifying subquandle separable structures. Together, these findings extend the landscape of residually finite quandles and connect quandle finiteness properties to 3-manifold topology and group separability phenomena.

Abstract

In this paper, we investigate the residual finiteness and subquandle separability of quandles, properties that respectively imply the solvability of the word problem and the generalized word problem for quandles. From Winker's work, we know that fundamental -quandles of oriented links, which are canonical quotients of their fundamental quandles, are closely associated with -fold cyclic branched covers of the 3-sphere branched over these links. We prove that the fundamental -quandle of any oriented link in the 3-sphere is residually finite for each . This supplements the recent result by Bardakov, Singh and the third author on residual finiteness of fundamental quandles of oriented links, and the classification by Hoste and Shanahan of links whose fundamental -quandles are finite for some . We also establish several general results on these finiteness properties and identify many families of quandles admitting them.
Paper Structure (10 sections, 26 theorems, 43 equations, 1 figure)

This paper contains 10 sections, 26 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Enveloping group
  4. Homogeneous representation of $n$-quandles
  5. Residual finiteness and fundamental $n$-quandles of links
  6. Residual finiteness and subquandle separability of quandles
  7. Residual finiteness of fundamental $n$-quandles of oriented links
  8. Residual finiteness of general quandles
  9. Subquandle separability of general quandles
  10. Declaration

Key Result

Proposition 2.3

Let $X$ be an $n$-quandle and $\{x_i \mid i \in I\}$ a set of representatives of orbits of $X$ under the action of $\operatorname{Env}(X)$. Let $H_i$ be the stabiliser of $x_{i}$ in $\operatorname{Env}_n(X)$ under the above action. Then $H_i$ lies in the centraliser of $\bold{e}_{x_i}$ in $\operator

Figures (1)

  • Figure 1: Quandle relations at crossings.

Theorems & Definitions (54)

  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 44 more