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Finiteness of canonical quotients of Dehn quandles of surfaces

Neeraj K. Dhanwani, Mahender Singh

TL;DR

This work analyzes finiteness phenomena for canonical quotients (n-quandles) of the Dehn quandle $\mathcal{D}_g^{ns}$ of a closed orientable surface. It develops a general finiteness criterion linking finite $n$-quandles to finite quotients of underlying groups, and proves that the involutory (2-)quandle of Artin quandles coincides with the corresponding Coxeter quandle. The principal results determine, for each genus, when $(\mathcal{D}_g^{ns})_n$ is finite, giving precise descriptions such as $(\mathcal{D}_g^{ns})_2$ is isomorphic to the projective primitive homological quandle $\mathcal{P}_{g,2}$ and, for low genus, explicit finiteness sets: $n\in\{2,3,4,5\}$ for $g=1$, $n\in\{2,3\}$ for $g=2$, and $g\ge3$ cases with finiteness only for $n=2$ (and a known infinite status for many other $n$). The results extend the Hoste–Shanahan program to Dehn quandles and identify sharp lower bounds on the size of smallest non-trivial quotients via homological constructions. Overall, the paper clarifies the structure of Dehn quandles of surfaces and connects quandle finiteness to surface mapping class group and symplectic representations.

Abstract

The Dehn quandle of a closed orientable surface is the set of isotopy classes of non-separating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider finiteness of some canonical quotients of these quandles. For a surface of positive genus, we give a precise description of the 2-quandle of its Dehn quandle. Further, with some exceptions for genus more than two, we determine all values of $n$ for which the $n$-quandle of its Dehn quandle is finite. The result can be thought of as the Dehn quandle analogue of a similar result of Hoste and Shanahan for link quandles. We also compute the size of the smallest non-trivial quandle quotient of the Dehn quandle of a surface. Along the way, we prove that the involutory quotient of an Artin quandle is precisely the corresponding Coxeter quandle and also determine the smallest non-trivial quotient of a braid quandle.

Finiteness of canonical quotients of Dehn quandles of surfaces

TL;DR

This work analyzes finiteness phenomena for canonical quotients (n-quandles) of the Dehn quandle of a closed orientable surface. It develops a general finiteness criterion linking finite -quandles to finite quotients of underlying groups, and proves that the involutory (2-)quandle of Artin quandles coincides with the corresponding Coxeter quandle. The principal results determine, for each genus, when is finite, giving precise descriptions such as is isomorphic to the projective primitive homological quandle and, for low genus, explicit finiteness sets: for , for , and cases with finiteness only for (and a known infinite status for many other ). The results extend the Hoste–Shanahan program to Dehn quandles and identify sharp lower bounds on the size of smallest non-trivial quotients via homological constructions. Overall, the paper clarifies the structure of Dehn quandles of surfaces and connects quandle finiteness to surface mapping class group and symplectic representations.

Abstract

The Dehn quandle of a closed orientable surface is the set of isotopy classes of non-separating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider finiteness of some canonical quotients of these quandles. For a surface of positive genus, we give a precise description of the 2-quandle of its Dehn quandle. Further, with some exceptions for genus more than two, we determine all values of for which the -quandle of its Dehn quandle is finite. The result can be thought of as the Dehn quandle analogue of a similar result of Hoste and Shanahan for link quandles. We also compute the size of the smallest non-trivial quandle quotient of the Dehn quandle of a surface. Along the way, we prove that the involutory quotient of an Artin quandle is precisely the corresponding Coxeter quandle and also determine the smallest non-trivial quotient of a braid quandle.
Paper Structure (7 sections, 15 theorems, 44 equations, 3 figures)

This paper contains 7 sections, 15 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finiteness of $n$-quandles of Dehn quandles
  4. Involutory quotients of Artin quandles
  5. Finiteness of $n$-quandles of Dehn quandles of surfaces
  6. 2 and 3-quandles of Dehn quandles of surfaces
  7. $n$-quandles of Dehn quandles of surfaces

Key Result

Proposition 3.1

Let $Q$ be a quandle with a presentation $\langle S \mid R \rangle$. Then, for each $n \ge 2$, $(Q)_n$ has a presentation $\langle S \mid R \cup T \rangle$, where $T=\{x*^n y=x \mid x,y\in S\}$.

Figures (3)

  • Figure 1: Arcs $c_{i, j}$ on the disc for $n=6$.
  • Figure 2: Surface of genus $g$ with curves $a_i$, $b_i$ and $c_i$.
  • Figure 3: Surface of genus $g$ with curves $d_i$.

Theorems & Definitions (28)

  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 18 more