The coloured reduced Burau matrix

Let $n\ge 2$ and let ${\sigma }_1,\dots ,{\sigma }_{n-1}$ be the standard generators of $B_n$ (The braid group by Artin presentation). Let $\beta ={\prod }_{r=1}^l{\sigma }_{i_r}^{{\epsilon }_r}\in B_n$ be a braid word. Labelling of the strings. Put the label $t_j$ on the string of $\beta$ which starts at the point $j$ at the bottom of the braid diagram, so that the labels are $t_1,\dots ,t_n$ read from the bottom left. Reading the letters of the word from left to right as the crossings from the top of the diagram, let $a_r$ be the label of the undercrossing string at crossing $r$, where for the positive generator ${\sigma }_i$ the undercrossing string is the one entering the crossing at position $i$ and for the negative generator ${\sigma }_i^{-1}$ the one entering at position $i+1$; this is the convention of Morton §2.1, checked against his example $\beta ={\sigma }_1{\sigma }_2^{-1}{\sigma }_1{\sigma }_2^{-1}{\sigma }_1{\sigma }_2^{-1}{\sigma }_3\in B_4$, where $a_1,\dots ,a_7=t_1,t_4,t_2,t_1,t_4,t_2,t_4$.

The matrices. For $1\le i\le n-1$ and a label $a$ let ${\bar{C}}_i(a)$ be the $(n-1)\times (n-1)$ matrix over the Laurent ring $\mathbb{Z}[a^{\pm 1}]$ which agrees with the identity matrix (Invertible square matrices and similarity over a commutative ring) except that its $i$-th row has the three entries $({\bar{C}}_i(a){)}_{i,i-1}=a,({\bar{C}}_i(a){)}_{i,i}=-a,({\bar{C}}_i(a){)}_{i,i+1}=1,$ where an entry is omitted when its column index lies outside $\{1,\dots ,n-1\}$: for $i=1$ the entry $a$ in column $0$ is omitted and for $i=n-1$ the entry $1$ in column $n$ is omitted, so that for $n>2$ each boundary row has exactly two non-zero entries, while for $n=2$ the sole row is the single entry $-a$. Each ${\bar{C}}_i(a)$ is upper triangular except for the single entry $a$ in position $(i,i-1)$, so $\det {\bar{C}}_i(a)=-a$ is a unit of $\mathbb{Z}[a^{\pm 1}]$ and the matrix is invertible (For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix); its inverse is the matrix whose $i$-th row has entries $({\bar{C}}_i(a{)}^{-1}{)}_{i,i-1}=1$, $({\bar{C}}_i(a{)}^{-1}{)}_{i,i}=-a^{-1}$ and $({\bar{C}}_i(a{)}^{-1}{)}_{i,i+1}=a^{-1}$, again truncated at the boundary columns. This is Morton's matrix ${\bar{C}}_i(a)$ (§2.1); the three places are the entries on row $i$ produced by the Fox derivatives of the elementary braid, and the truncation rule is his.

The coloured reduced Burau matrix. The coloured reduced Burau matrix of the braid word is ${\bar{B}}_{\beta }(t_1,\dots ,t_n):={\prod }_{r=1}^l({\bar{C}}_{i_r}(a_r){)}^{{\epsilon }_r},$ the product taken in the order of the word, an element of ${\mathrm{GL}}_{n-1}(\mathbb{Z}[t_1^{\pm 1},\dots ,t_n^{\pm 1}])$. It is a direct matrix product, so no well-definedness issue beyond matrix multiplication arises; the chosen word enters only through the labels $a_r$.

Equal labels and conventions. Specialising $t_1=\cdots =t_n=t$ gives Morton's equal-label matrix ${\bar{B}}_{\beta }(t,\dots ,t)$ over ${\mathrm{\Lambda }}_1=\mathbb{Z}[t^{\pm 1}]$. For comparison with the topological representation, assume AC, as in The reduced Burau representation and The Axiom of Choice. Put $D=\mathrm{diag}(t,t^2,\dots ,t^{n-1})$. The fixed adjacent weighted basis of that representation is $b_i=t^i(h_i-h_{i+1})$ with $h_n=0$. Its generator matrix has row $i$ entries $1,-t,t$, truncated at the boundary, by The topological and matrix Burau representations agree. Diagonal conjugation of the displayed row $t,-t,1$ gives these entries, so ${\bar{\rho }}_n({\sigma }_i)=D^{-1}{\bar{C}}_i(t)D,{\bar{\rho }}_n(\beta )=D^{-1}{\bar{B}}_{\beta }(t,\dots ,t)D\mathrm{.}$ Consequently $\det (I-{\bar{B}}_{\beta }(t,\dots ,t))=\det (I-{\bar{\rho }}_n(\beta ))$. These identities concern equal labels; the labelled matrix remains defined algebraically for the chosen word without a Choice assumption.

• The determinant of each ${\bar{C}}_i(a)$ is $-a$, so $\det {\bar{B}}_{\beta }=(-1{)}^l{\prod }_r{a}_r^{{\epsilon }_r}$ for a word of length $l$, a monomial in the labels; in particular the coloured matrix is invertible over $\mathbb{Z}[t_1^{\pm 1},\dots ,t_n^{\pm 1}]$.
• Morton's matrices act on column vectors with the product ordered as the word, exactly as displayed; no inverse order is taken. At equal labels the generator matrix ${\bar{C}}_i(t)$ has the same characteristic polynomial $(\lambda -1{)}^{n-2}(\lambda +t)$ as the reduced Burau generator. The determinant comparison for arbitrary braid words follows from simultaneous conjugacy by the fixed matrix $D$, rather than from the characteristic polynomials of individual generators alone.