Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A preorder on the set of links defined via orbifolds

Michel Boileau, Teruaki Kitano, Yuta Nozaki

TL;DR

This work introduces the π-orbifold preorder on prime links with at least three bridges, defined by epimorphisms between π-orbifold groups $G^{orb}$. It provides a detailed classification of potential dominated links when the dominator is a Montesinos or Seifert-type link, shows finiteness results for small links, and analyzes determinant-zero cases that yield broad domination phenomena for 2-bridge links. It further extends the framework to arborescent and symmetric-union contexts, yielding structural constraints on dominated links and strong interactions with 2-fold branched covers and orbifold geometry. The paper also develops the notion of $\pi$-minimal links and connects π-dominance to other preorders on knots, culminating in several open questions about bridge numbers, genus, and volume that guide future research. Overall, the results expose rigid patterns in how orbifold-group epimorphisms constrain link types and offer tools for studying symmetric unions and related knot families through the lens of 3-manifold topology.

Abstract

For a link $L$ in the $3$-sphere, the $π$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$, we denote this by $L\succeq L'$ and explore the relationships between the two links. Specifically, we prove that if $L\succeq L'$ and $L$ is a Montesinos link with $r$ rational tangles $(r\geq 3)$, then $L'$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L'$ satisfying $L\succeq L'$. In contrast, if $L$ has determinant zero, then $L\succeq L'$ for every $2$-bridge link $L'$. Additionally, we discuss applications to symmetric unions of knots and connections to other preorders on the set of knots. Finally, we raise open questions on bridge number and volume.

A preorder on the set of links defined via orbifolds

TL;DR

This work introduces the π-orbifold preorder on prime links with at least three bridges, defined by epimorphisms between π-orbifold groups . It provides a detailed classification of potential dominated links when the dominator is a Montesinos or Seifert-type link, shows finiteness results for small links, and analyzes determinant-zero cases that yield broad domination phenomena for 2-bridge links. It further extends the framework to arborescent and symmetric-union contexts, yielding structural constraints on dominated links and strong interactions with 2-fold branched covers and orbifold geometry. The paper also develops the notion of -minimal links and connects π-dominance to other preorders on knots, culminating in several open questions about bridge numbers, genus, and volume that guide future research. Overall, the results expose rigid patterns in how orbifold-group epimorphisms constrain link types and offer tools for studying symmetric unions and related knot families through the lens of 3-manifold topology.

Abstract

For a link in the -sphere, the -orbifold group is defined as a quotient of the link group of . When there exists an epimorphism , we denote this by and explore the relationships between the two links. Specifically, we prove that if and is a Montesinos link with rational tangles , then is either a Montesinos link with at most rational tangles or a certain connected sum. We further show that if is a small link, then there are only finitely many links satisfying . In contrast, if has determinant zero, then for every -bridge link . Additionally, we discuss applications to symmetric unions of knots and connections to other preorders on the set of knots. Finally, we raise open questions on bridge number and volume.
Paper Structure (19 sections, 36 theorems, 27 equations, 1 figure)

This paper contains 19 sections, 36 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A preorder defined via orbifolds
  3. Arborescent links
  4. Small links and finiteness
  5. Preliminaries
  6. Montesinos links
  7. Orbifolds
  8. Proof of Theorem \ref{['thm:order']} concerning the preorder $\succeq$
  9. Proof of Theorem \ref{['thm:arb']} concerning arborescent links
  10. Small links and finiteness
  11. Proof of Theorem \ref{['thm:small']}(1)
  12. Small links and determinants
  13. Proof of Theorem \ref{['thm:small']}(2)
  14. $\pi$-minimal links
  15. Symmetric unions of knots
  16. ...and 4 more sections

Key Result

Proposition 1.2

Let $L$ be a link in $S^3$.

Figures (1)

  • Figure 1: Symmetric union $(D \cup D^*)(\infty, n_1, \dots, n_k)$ and its partial knot $K_D$.

Theorems & Definitions (96)

  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.11
  • Definition 2.1: Kaw96
  • ...and 86 more