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Framed Braid Equivalences

Anastasios Kokkinakis

TL;DR

This work extends the classical Alexander–Markov framework to framed knots and links by developing framed analogues of $L$-moves and a one-move Markov theorem, enabling a closed-form, diagrammatic bridge between framed braids and framed links. It introduces framed versions of the Hilden groups and constructs explicit presentations for both the framed Hilden and pure framed Hilden groups, revealing their semidirect product and direct-sum decompositions with framing components. A framed version of Birman’s plat-theorem is formulated, showing that framed plat closures are controlled by double cosets in framed Hilden groups together with framed stabilizations, with framing adjustments managed by new generators. The results solidify a comprehensive, algebraic framework for Kirby-calculus-style moves within the framed setting, enhancing tractable representations of framed links via plat and standard closures and enabling systematic manipulations of framed 3-manifold presentations.

Abstract

We introduce framed versions of the $L$-moves and prove a one move theorem for the extension of the Markov theorem for framed braids. We further introduce framed versions of the Hilden and Pure Hilden groups, we give presentations and we use them to state and prove a framed version of the Birman theorem for framed links in plat representation.

Framed Braid Equivalences

TL;DR

This work extends the classical Alexander–Markov framework to framed knots and links by developing framed analogues of -moves and a one-move Markov theorem, enabling a closed-form, diagrammatic bridge between framed braids and framed links. It introduces framed versions of the Hilden groups and constructs explicit presentations for both the framed Hilden and pure framed Hilden groups, revealing their semidirect product and direct-sum decompositions with framing components. A framed version of Birman’s plat-theorem is formulated, showing that framed plat closures are controlled by double cosets in framed Hilden groups together with framed stabilizations, with framing adjustments managed by new generators. The results solidify a comprehensive, algebraic framework for Kirby-calculus-style moves within the framed setting, enhancing tractable representations of framed links via plat and standard closures and enabling systematic manipulations of framed 3-manifold presentations.

Abstract

We introduce framed versions of the -moves and prove a one move theorem for the extension of the Markov theorem for framed braids. We further introduce framed versions of the Hilden and Pure Hilden groups, we give presentations and we use them to state and prove a framed version of the Birman theorem for framed links in plat representation.

Paper Structure

This paper contains 18 sections, 17 theorems, 34 equations, 35 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: Classical Braids and Classical Links
  3. Classical Braids
  4. Classical Knots and Links
  5. Framed links and framed braids
  6. Representations of framed links
  7. Framed braids
  8. Relationship between classical links and braids via the standard closure
  9. Framed links and framed braids via the standard closure
  10. The Alexander theorem for framed links
  11. The Markov theorem for framed braids
  12. Ribbon or framed $L$-moves
  13. Framed $L$-equivalence
  14. The plat closure, the Hilden groups $H_{2n}$ and the plat equivalence
  15. Framed Hilden Groups and Pure Framed Hilden Groups
  16. ...and 3 more sections

Key Result

Theorem 2.1

The group $B_n$ is characterized algebraically to be the group with presentation:

Figures (35)

  • Figure 1: Braid isotopy moves.
  • Figure 2: The Artin generators.
  • Figure 3: The Reidemeister moves.
  • Figure 4: The standard closure of a braid.
  • Figure 5: The two types of crossing in a link diagram.
  • ...and 30 more figures

Theorems & Definitions (38)

  • Theorem 2.1: Artin
  • Definition 3.1
  • Definition 3.2: Framing
  • Definition 3.3
  • Definition 3.4
  • Corollary 3.0.1
  • Theorem 4.1: Alexander
  • Theorem 4.2: Markov
  • Definition 4.1: $L$-moves
  • Definition 4.2: $L$-moves revised
  • ...and 28 more