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The crossing matrix and the extended first Johnson homomorphism of a braid group

Yusuke Kuno, Yoshiro Yaguchi

TL;DR

The paper relates two crossed homomorphisms on the braid group, the diagrammatic crossing matrix $C$ and Kawazumi's extended first Johnson homomorphism $\tau_1^{\theta}$, by proving $\tau_1^{\theta} = \delta \circ C$ with an injective $S_m$-equivariant map $\delta: {\rm Mat}_m^0 \to {\rm Hom}(H, \wedge^2 H)$. It then provides explicit computations for simple braids, expressing crossing data through cord homology via a bijection between cord configurations and the Johnson image. The results unify diagrammatic and algebraic invariants of braids and point to applications in Hurwitz-type problems and higher Johnson invariants. The work thus offers a concrete, computable bridge between combinatorial braid crossings and automorphism-induced Johnson data, with potential to inform surface-braid invariants and related actions.

Abstract

We compare two crossed homomorphisms on a braid group, one defined diagrammatically and the other defined algebraically. We show that these crossed homomorphisms are essentially the same, and compute them in detail for simple braids, namely elements conjugate to the standard generators of the braid group or to their inverses.

The crossing matrix and the extended first Johnson homomorphism of a braid group

TL;DR

The paper relates two crossed homomorphisms on the braid group, the diagrammatic crossing matrix and Kawazumi's extended first Johnson homomorphism , by proving with an injective -equivariant map . It then provides explicit computations for simple braids, expressing crossing data through cord homology via a bijection between cord configurations and the Johnson image. The results unify diagrammatic and algebraic invariants of braids and point to applications in Hurwitz-type problems and higher Johnson invariants. The work thus offers a concrete, computable bridge between combinatorial braid crossings and automorphism-induced Johnson data, with potential to inform surface-braid invariants and related actions.

Abstract

We compare two crossed homomorphisms on a braid group, one defined diagrammatically and the other defined algebraically. We show that these crossed homomorphisms are essentially the same, and compute them in detail for simple braids, namely elements conjugate to the standard generators of the braid group or to their inverses.

Paper Structure

This paper contains 5 sections, 11 theorems, 42 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The crossing matrix of a braid group
  3. The extended first Johnson homomorphism of the automorphism group of a free group
  4. The extended first Johnson homomorphism of a braid group
  5. Simple braids and cords in a disk

Key Result

Lemma 2.1

The map $C$ is a crossed homomorphism. Namely, for any $\beta_1, \beta_2\in B_m$, $C(\beta_1\beta_2)=C(\beta_1)+|\beta_1|({C(\beta_2)})$. Thus we have a group homomorphism $\widetilde{C}:B_m\to {\rm Mat}_m^0\rtimes S_m$ defined by $\widetilde{C}(\beta)=(C(\beta), |\beta|)$.

Figures (7)

  • Figure 1: the standard generator $\sigma_i \in B_m$
  • Figure 2: the crossing matrix for $\beta = \sigma_2^{-1} \sigma_1^2 \sigma_2^3 \sigma_1^{-1} \sigma_2$
  • Figure 3: the action of $\sigma_i$ on generators $x_1,\ldots,x_m$
  • Figure 4: the positive half-twist about the cord $\gamma_{\beta}$
  • Figure 5: the $5$-braid $\beta=\sigma_1 * (\sigma_2 \sigma_3^{-1} \sigma_4^{-2} \sigma_1^{-2})$ and its cord $\gamma_{\beta}$
  • ...and 2 more figures

Theorems & Definitions (21)

  • Lemma 2.1: BGKN02
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2: kawazumi, Lemma 2.1
  • Remark 3.3
  • ...and 11 more