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Characterizations of knot groups and knot symmetric quandles of surface-links

Jumpei Yasuda

TL;DR

This work provides a unified, presentation-theoretic framework to characterize knot groups and knot symmetric quandles of surface-links (including non-orientable ones) via plat closures of adequate braided surfaces. It introduces $(2m,n)$-presentations with inverses and a weak $\partial$-condition, together with Euler-characteristic and abelianization constraints, to completely determine when a group is a knot group of a surface-link and when a symmetric quandle is its knot symmetric quandle. The authors extend these characterizations to symmetric-quandle presentations, and show that every dihedral quandle with a good involution can be realized as a knot symmetric quandle of some surface-link. They also explore implications for $P^2$-irreducibility, providing a mechanism to distinguish surface-links via their knot symmetric quandles and constructing infinite families of $P^2$-irreducible examples.

Abstract

The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method. González-Acuña and Kamada independently extended this characterization to the knot groups of orientable surface-links. Kamada applied the closed 2-dimensional braid method to show this result. In this paper, we generalize these results to characterize the knot groups of surface-links, including non-orientable ones. We use a plat presentation for surface-links to prove it. Furthermore, we show a similar characterization for the knot symmetric quandles of surface-links. As an application, we show that every dihedral quandle with an arbitrarily good involution can be realized as the knot symmetric quandle of a surface-link.

Characterizations of knot groups and knot symmetric quandles of surface-links

TL;DR

This work provides a unified, presentation-theoretic framework to characterize knot groups and knot symmetric quandles of surface-links (including non-orientable ones) via plat closures of adequate braided surfaces. It introduces -presentations with inverses and a weak -condition, together with Euler-characteristic and abelianization constraints, to completely determine when a group is a knot group of a surface-link and when a symmetric quandle is its knot symmetric quandle. The authors extend these characterizations to symmetric-quandle presentations, and show that every dihedral quandle with a good involution can be realized as a knot symmetric quandle of some surface-link. They also explore implications for -irreducibility, providing a mechanism to distinguish surface-links via their knot symmetric quandles and constructing infinite families of -irreducible examples.

Abstract

The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method. González-Acuña and Kamada independently extended this characterization to the knot groups of orientable surface-links. Kamada applied the closed 2-dimensional braid method to show this result. In this paper, we generalize these results to characterize the knot groups of surface-links, including non-orientable ones. We use a plat presentation for surface-links to prove it. Furthermore, we show a similar characterization for the knot symmetric quandles of surface-links. As an application, we show that every dihedral quandle with an arbitrarily good involution can be realized as the knot symmetric quandle of a surface-link.
Paper Structure (14 sections, 19 theorems, 38 equations, 1 figure)

This paper contains 14 sections, 19 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Braids and Braided surfaces
  3. Braids and Braided surfaces
  4. Braid systems of braided surfaces
  5. The plat closure of braided surfaces
  6. Proof of Theorem \ref{['Main theorem']}
  7. Quandles and symmetric quandles
  8. Quandles
  9. Symmetric quandles
  10. Symmetric quandle presentation
  11. Proof of Theorem \ref{['Main Theorem2']}
  12. Knot symmetric quandles and dihedral quandles
  13. Proof of Thoerem \ref{['Theorem: dihedral quandle and knot symmetric quandle']}
  14. Remarks on $P^2$-irreduciblity of non-orientable surface-links

Key Result

Theorem 1.1

A group $G$ is the knot group of a link if and only if there exists an $m$-braid $b$ for some $m \geq 1$ such that $G$ has a presentation

Figures (1)

  • Figure 1: Nooses of $K$ (left) and an operation result of the knot quandle (right)

Theorems & Definitions (32)

  • Theorem 1.1: Alexander1923Artin1925
  • Theorem 1.2: Kamada1994-01
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2: Rudolph1983
  • Definition 2.3
  • Proposition 2.4: Yasuda21
  • Proposition 2.5: Yasuda24
  • ...and 22 more