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Core groups

Daniel S. Silver, Lorenzo Traldi, Susan G. Williams

TL;DR

This work introduces three core-group invariants for virtual links—arc core group $AC(D)$ and two region-based groups $RC(D)$ and $RRC(D)$—and shows how they extend the classical Wirtinger/Dehn perspectives to virtual diagrams and thickened-surface links. It proves structural decompositions tying $RC(D)$ to $AC(D)$ (for classical diagrams, $RC(D) \cong \mathbb{Z} * AC(D)$) and establishes a second theorem describing how $RC(D)$ decomposes into region-subgroups plus free factors, with clear behavior under splitting. The paper also develops abelianizations, alternative presentations of $AC(D)$, and a core-group functor unifying Wirtinger and Dehn frameworks; it connects these core groups to Coxeter groups and, in the Dehn setting, to Dehn groups via checkerboard colorability. Overall, the results illuminate how region- and arc-based core groups capture invariant, topology-linked information for virtual links and their embeddings, with concrete algebraic decompositions and functorial structure. The methods blend combinatorial group presentations, covering-space arguments, and classical Goeritz/Goeritz-like indices to relate different presentations of link groups.

Abstract

The core group of a classical link was introduced independently by A.J. Kelly in 1991 and M. Wada in 1992. It is a link invariant defined by a presentation involving the arcs and crossings of a diagram, related to Wirtinger's presentation of the fundamental group of a link complement. Two close relatives of the core group are defined by presentations involving regions rather than arcs; one of them is related to Dehn's presentation of a link group. The definitions are extended to virtual link diagrams and properties of the resulting invariants are discussed.

Core groups

TL;DR

This work introduces three core-group invariants for virtual links—arc core group and two region-based groups and —and shows how they extend the classical Wirtinger/Dehn perspectives to virtual diagrams and thickened-surface links. It proves structural decompositions tying to (for classical diagrams, ) and establishes a second theorem describing how decomposes into region-subgroups plus free factors, with clear behavior under splitting. The paper also develops abelianizations, alternative presentations of , and a core-group functor unifying Wirtinger and Dehn frameworks; it connects these core groups to Coxeter groups and, in the Dehn setting, to Dehn groups via checkerboard colorability. Overall, the results illuminate how region- and arc-based core groups capture invariant, topology-linked information for virtual links and their embeddings, with concrete algebraic decompositions and functorial structure. The methods blend combinatorial group presentations, covering-space arguments, and classical Goeritz/Goeritz-like indices to relate different presentations of link groups.

Abstract

The core group of a classical link was introduced independently by A.J. Kelly in 1991 and M. Wada in 1992. It is a link invariant defined by a presentation involving the arcs and crossings of a diagram, related to Wirtinger's presentation of the fundamental group of a link complement. Two close relatives of the core group are defined by presentations involving regions rather than arcs; one of them is related to Dehn's presentation of a link group. The definitions are extended to virtual link diagrams and properties of the resulting invariants are discussed.
Paper Structure (9 sections, 26 theorems, 39 equations, 7 figures)

This paper contains 9 sections, 26 theorems, 39 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main1']}
  3. Proof of Theorem \ref{['main2']}
  4. Reidemeister moves
  5. Abelianized core groups
  6. Another description of $AC(D)$
  7. Core group functor
  8. Coxeter groups
  9. Dehn core groups from Dehn groups

Key Result

Theorem 1.4

If $D$ is a classical link diagram, then $RC(D)$ is isomorphic to the free product $\mathbb Z * AC(D)$.

Figures (7)

  • Figure 1: The regions incident at a classical crossing in a classical diagram (on the left) or an abstract link diagram (on the right).
  • Figure 2: The group triples $(AC,RC,RRC)$ for these four diagrams are $(\mathbb Z, \mathbb Z,\mathbb Z*\mathbb Z)$, $(\mathbb Z, \mathbb Z*\mathbb Z,\mathbb Z*\mathbb Z)$, $(\mathbb Z, \mathbb Z*\mathbb Z,\mathbb Z*\mathbb Z*\mathbb Z_2)$ and $(\mathbb Z * \mathbb Z_3, \mathbb Z*\mathbb Z,\mathbb Z*\mathbb Z*\mathbb Z_2)$, from left to right.
  • Figure 3: $\Omega.2$ moves that change $RC$ or $RRC$.
  • Figure 4: An $\Omega.2$ move changes $D$ into $D'$.
  • Figure 5: This diagram of a $(2,m)$ torus link has $m$ crossings.
  • ...and 2 more figures

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 53 more