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Knot quandle decomposition along a torus

Marco Bonatto, Alessia Cattabriga, Eva Horvat

TL;DR

The paper develops a comprehensive framework for augmented fundamental quandles of knots in 3-manifolds with incompressible tori, notably satellites, by relating the quandle of a satellite to the quandles and fundamental groups of its pattern and companion knots. It provides general presentations for the fundamental quandle of links in the solid torus and in lens spaces, and derives a precise augmented-quandle decomposition for satellites using a amalgamated-group construction and a concrete presentation. Building on the connection to affine quandles, it defines the Alexander module $ ext{M}(Q)$ of a connected quandle, constructs the Alexander quandle $ ext{Alex}(Q)$, and shows $ ext{Alex}(Q)igsqcup Q/oldsymbol extgamma_Q$, with explicit block-form matrices for satellites. The work yields practical tools for computing knot-quandle invariants in nontrivial 3-manifolds, establishes colorability criteria via affine quandles, and clarifies the relationship between quandle theory and the classical Alexander module in satellite and lens-space contexts.

Abstract

We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot and the fundamental quandles/groups of its companion and pattern knots. General presentations of the fundamental quandles of a link in a solid torus, a link in a lens space and a satellite knot are described. In the last part of the paper, an algebraic approach to the study of affine quandles is presented and some known results about the Alexander module and quandle colorings are obtained.

Knot quandle decomposition along a torus

TL;DR

The paper develops a comprehensive framework for augmented fundamental quandles of knots in 3-manifolds with incompressible tori, notably satellites, by relating the quandle of a satellite to the quandles and fundamental groups of its pattern and companion knots. It provides general presentations for the fundamental quandle of links in the solid torus and in lens spaces, and derives a precise augmented-quandle decomposition for satellites using a amalgamated-group construction and a concrete presentation. Building on the connection to affine quandles, it defines the Alexander module of a connected quandle, constructs the Alexander quandle , and shows , with explicit block-form matrices for satellites. The work yields practical tools for computing knot-quandle invariants in nontrivial 3-manifolds, establishes colorability criteria via affine quandles, and clarifies the relationship between quandle theory and the classical Alexander module in satellite and lens-space contexts.

Abstract

We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot and the fundamental quandles/groups of its companion and pattern knots. General presentations of the fundamental quandles of a link in a solid torus, a link in a lens space and a satellite knot are described. In the last part of the paper, an algebraic approach to the study of affine quandles is presented and some known results about the Alexander module and quandle colorings are obtained.

Paper Structure

This paper contains 12 sections, 16 theorems, 58 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quandles as algebraic structures
  4. Quandle presentations
  5. The fundamental quandle of a link
  6. Fundamental quandle of a link in the solid torus
  7. Fundamental quandle of a link in a lens space
  8. Fundamental quandle of satellite knots
  9. The Alexander module of a connected quandle
  10. Abelian and affine quandles
  11. Presentation of the Alexander module
  12. Knot colorings by affine quandles

Key Result

Theorem 2

FR Given a diagram $D_{L}$ of a link $L$ in $S^{3}$, let $A(D_L)$ be the set of arcs and let $C(D_L)$ be the set of crossing relations at the crossings of $D_L$. Then the fundamental quandle $Q(L)$ of the link $L$ is given by the quandle presentation $[A(D_L)\colon C(D_L)]$.

Figures (8)

  • Figure 1: The crossing relation at a positive crossing (left) and at a negative crossing (right)
  • Figure 2: The four types of crossing relations in a diagram of a link in a 3-manifold
  • Figure 3: A diagram of a link in the solid torus
  • Figure 4: Surgery diagram of a link $L$ in the solid torus
  • Figure 5: Surgery diagram of a link $L$ in the lens space
  • ...and 3 more figures

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Definition 5
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Proposition 5
  • ...and 27 more