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The fundamental quandle of ribbon concordances

Eva Horvat, Luka Marčič

Abstract

We describe the fundamental quandle of a properly embedded surface $F$ (possibly with boundary) in $\mathbb{R} ^{3}\times I$, and derive its presentation in terms of a motion picture diagram or a CH-diagram of $F$. Our study is based on the topological definition of the fundamental quandle. We prove that a ribbon concordance $C$ from a classical knot $K_1$ to $K_0$ gives rise to an injective quandle homomorphism $Q(K_0)\to Q(C)$ and a surjective quandle homomorphism $Q(K_1)\to Q(C)$.

The fundamental quandle of ribbon concordances

Abstract

We describe the fundamental quandle of a properly embedded surface (possibly with boundary) in , and derive its presentation in terms of a motion picture diagram or a CH-diagram of . Our study is based on the topological definition of the fundamental quandle. We prove that a ribbon concordance from a classical knot to gives rise to an injective quandle homomorphism and a surjective quandle homomorphism .
Paper Structure (7 sections, 6 theorems, 14 equations, 6 figures)

This paper contains 7 sections, 6 theorems, 14 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quandles
  4. Ribbon concordances
  5. The concordance quandle
  6. Fundamental quandle of an embedded surface
  7. Fundamental quandle of a ribbon concordance

Key Result

Lemma 1.1

If $C$ is a ribbon concordance from $K_1$ to $K_0$, then $\pi _{1}(K_1)\to \pi _{1}(C)$ is surjective and $\pi _{1}(K_0)\to \pi _{1}(C)$ is injective.

Figures (6)

  • Figure 1: The quandle crossing relations
  • Figure 2: Motion picture transformations
  • Figure 3: The resolutions of a marker below (left) and above the critical point (right)
  • Figure 4: Example of a ribbon concordance with one critical point of index 0 and 1 each.
  • Figure 5: Fixing $H(0,s)$ so that it doesn't pass $N_C((0,1)_D \times \{1\}_D)$.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Lemma 1.1: G
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2: Conjugation quandle
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7: FR
  • ...and 10 more