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On Positive Braids, Monodromy Groups and Framings

Livio Ferretti

TL;DR

The paper defines the braid monodromy group MG($\beta$) for positive braids as a natural extension of the geometric monodromy group of plane curve singularities, embedding it as a subgroup of the mapping class group of the braid's Milnor fiber. It establishes invariance under fundamental braid moves and shows that for prime positive braids whose closures are knots and are not of type $A_n$, MG($\beta$) coincides with a framed mapping class group Mod$(\Sigma_{\beta},\phi_{\beta})$ for all but finitely many cases, yielding a knot-invariant from framing data. Consequently, the geometric monodromy group of an irreducible singularity is determined by the genus of the Milnor fiber and the Arf invariant of the associated knot, linking singularity topology to classical knot invariants. The work leverages A'Campo’s divide theory and framed mapping class group structure to connect braid combinatorics, monodromy, and singularity theory, while noting that certain infinite families fall outside the established framework.

Abstract

We associate to every positive braid a braid monodromy group, generalizing the geometric monodromy group of an isolated plane curve singularity. If the closure of the braid is a knot, we identify the corresponding group with a framed mapping class group. In particular, this gives a well defined knot invariant. As an application, we obtain that the geometric monodromy group of an irreducible singularity is determined by the genus and the Arf invariant of the associated knot.

On Positive Braids, Monodromy Groups and Framings

TL;DR

The paper defines the braid monodromy group MG() for positive braids as a natural extension of the geometric monodromy group of plane curve singularities, embedding it as a subgroup of the mapping class group of the braid's Milnor fiber. It establishes invariance under fundamental braid moves and shows that for prime positive braids whose closures are knots and are not of type , MG() coincides with a framed mapping class group Mod for all but finitely many cases, yielding a knot-invariant from framing data. Consequently, the geometric monodromy group of an irreducible singularity is determined by the genus of the Milnor fiber and the Arf invariant of the associated knot, linking singularity topology to classical knot invariants. The work leverages A'Campo’s divide theory and framed mapping class group structure to connect braid combinatorics, monodromy, and singularity theory, while noting that certain infinite families fall outside the established framework.

Abstract

We associate to every positive braid a braid monodromy group, generalizing the geometric monodromy group of an isolated plane curve singularity. If the closure of the braid is a knot, we identify the corresponding group with a framed mapping class group. In particular, this gives a well defined knot invariant. As an application, we obtain that the geometric monodromy group of an irreducible singularity is determined by the genus and the Arf invariant of the associated knot.
Paper Structure (8 sections, 18 theorems, 17 equations, 13 figures)

This paper contains 8 sections, 18 theorems, 17 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper:
  3. Acknowledgements:
  4. The monodromy group of a positive braid
  5. Divides and monodromy of singularities
  6. Framings
  7. A framing for positive braids
  8. Proof of the main theorem

Key Result

Theorem 1

Let $f:\mathbb{C}^2 \rightarrow \mathbb{C}$ define an isolated plane curve singularity and $L(f)$ be the link of $f$. Then there exists a positive braid $\beta$ representing $L(f)$ such that the geometric monodromy group of $f$ is equal to $\mathit{MG}(\beta)$.

Figures (13)

  • Figure 1: The fibre surface of $\sigma_3\sigma_1\sigma_2\sigma_1^2\sigma_3\sigma_2$, its brick diagram and the corresponding linking graph.
  • Figure 2: The isotopy between $\Sigma_{\beta\sigma_i}$ and $\Sigma_{\sigma_i\beta}$
  • Figure 3: The isotopy between $\Sigma_{\alpha}$ and $\Sigma_{\beta}$
  • Figure 5: A divide and the associated surface with some of the vanishing cycles. The corresponding link is the torus knot $T_{3,4}$.
  • Figure 6: The divides on the left are ordered Morse, the divides on the right are not.
  • ...and 8 more figures

Theorems & Definitions (46)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Example 2.1
  • Proposition 2.1: Elementary conjugation invariance
  • proof
  • ...and 36 more