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Satellite fully positive braid links are braided satellite of fully positive braid links

Tetsuya Ito

TL;DR

The paper addresses when fully positive braid links arise as satellites, proving a precise equivalence: a fully positive braid link is a satellite iff it is a braided satellite of a fully positive companion with a braided pattern containing sufficiently many full twists. The approach combines a regular-form decomposition for braided satellites, Nielsen–Thurston reducibility analysis, and Garside theory to connect positivity across the satellite data, with braid foliation/FDTC methods ensuring the existence of compatible representations. Key contributions include explicit twist-bounds ensuring positivity, a modulo-structure characterization linking companion and pattern positivity, and a corollary-style unknot/trefoil test via braided satellites. The results advance understanding of how positivity properties propagate through satellite constructions and provide tools for identifying fully positive satellite structures in knot theory, with implications for Lorenz/Lorenz-like knots and related invariants.

Abstract

A link in $S^{3}$ is a fully positive braid link if it is the closure of a positive braid that contains at least one full-twist. We show that a fully positive braid link is a satellite link if and only if it is the satellite of a fully positive braid link $C$ such that the pattern is a positive braid that contains sufficiently many full twists, where the number of necessary full twists only depends on $C$. As an application, we give a characterization of the unknot by the property that certain braided satellite is a (fully) positive braid knot.

Satellite fully positive braid links are braided satellite of fully positive braid links

TL;DR

The paper addresses when fully positive braid links arise as satellites, proving a precise equivalence: a fully positive braid link is a satellite iff it is a braided satellite of a fully positive companion with a braided pattern containing sufficiently many full twists. The approach combines a regular-form decomposition for braided satellites, Nielsen–Thurston reducibility analysis, and Garside theory to connect positivity across the satellite data, with braid foliation/FDTC methods ensuring the existence of compatible representations. Key contributions include explicit twist-bounds ensuring positivity, a modulo-structure characterization linking companion and pattern positivity, and a corollary-style unknot/trefoil test via braided satellites. The results advance understanding of how positivity properties propagate through satellite constructions and provide tools for identifying fully positive satellite structures in knot theory, with implications for Lorenz/Lorenz-like knots and related invariants.

Abstract

A link in is a fully positive braid link if it is the closure of a positive braid that contains at least one full-twist. We show that a fully positive braid link is a satellite link if and only if it is the satellite of a fully positive braid link such that the pattern is a positive braid that contains sufficiently many full twists, where the number of necessary full twists only depends on . As an application, we give a characterization of the unknot by the property that certain braided satellite is a (fully) positive braid knot.
Paper Structure (13 sections, 21 theorems, 14 equations, 3 figures)

This paper contains 13 sections, 21 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Braided satellites and its regular form
  3. Satellite links
  4. Regular form of braided satellites
  5. Braided satellite that gives a (fully) positive braid link
  6. Reducible braids and regular forms
  7. Regular forms of reducible braids
  8. Compatible braid representatives
  9. Companion of braided satellite positive braid link
  10. Garside theory
  11. Reducibility and Garside theory
  12. Positivity criterion of companion and patterns
  13. Existence criterion of compatible braid representatives

Key Result

Theorem 1.2

Let $L$ be the satellite with companion $C=C_1 \cup C_2\cup \cdots \cup C_{m}$ and pattern $P=(P_1,\ldots,P_m)$. Then (a) and (b) are equivalent. Here $g(C_i)$ and $b(C_i)$ are the genus and the braid index of the knot $C_i$, respectively.

Figures (3)

  • Figure 1: Regular form. The box represents the braid $\Delta_{n_i}^{-2w(C_i,\beta_{\rm ext})}$
  • Figure 2: Reducible braid preserving round essential multicurve is close to a regular form. The difference is that there may be additional braiding inside tubes (gray box)
  • Figure 3: Taking conjugate to get a regular form

Theorems & Definitions (40)

  • Definition 1.1: Fully positive braid link
  • Theorem 1.2: Characterization of satellite fully positive braid link
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1: Braided pattern
  • Definition 2.2: Satellite link
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • ...and 30 more