Compute the twisted Burau determinant for a colored braid closure

Inputs. An integer $n\ge 2$; a finite Artin word $\beta \in B_n$ in the positive-half-twist convention where ${\sigma }_i$ sends the $i$th meridian $x_i$ to $x_i{x}_{i+1}{x}_i^{-1}$ and $x_{i+1}$ to $x_i$; a surjective color map $c$ from the cycles of the endpoint permutation to $\{1,\dots ,\mu \}$, written on strands as $c_1,\dots ,c_n$ and constant around each cycle; a field $\mathbb{F}$ with encoded exact arithmetic and equality; a positive integer $d$ and invertible matrices $\rho (x_i)\in {\mathrm{G}\mathrm{L}}_d(\mathbb{F})$; and compatible finite certificates of based deficiency-one $2$-spines for the closure exterior $X$ and the closure-with-axis exterior $Y$. The $X$ spine has generators $g_i=x_1\cdots x_i$ and relators $r_i=g_i^{-1}{\beta }_{\ast }(g_i)$ for $1\le i<n$. The $Y$ spine has generators $g_1,\dots ,g_n,x$ and relators $s_i={\beta }_{\ast }(g_i)(x^{-1}{g}_{i}x{)}^{-1}$ for $1\le i\le n$; $x$ is the meridian of the axis. The certificates include a based cell structure on $T$ with one vertex, $1$-cells $x,g_n$, and the commutator $2$-cell; the one-vertex, one-core-edge structure on $V=D^2\times S^1$; and compatible cell bases for $X=Y{\cup }_{T}V$. These spine and gluing certificates are geometric input and are not recognized by the procedure. Put $H=⟨t_1,\dots ,t_{\mu }⟩\cong {\mathbb{Z}}^{\mu }$, $\psi (x_i)=t_{c_i}$, $g_n=x_1\cdots x_n$, $\mathrm{\Lambda }=\mathbb{F}[H]$, and $Q=\mathrm{Frac}(\mathrm{\Lambda })$.

Instructions.

1. Read the supplied $X$ and $Y$ spine/presentation correspondences and compatible gluing as certified geometric input; check their finite cell counts and relator lists, but do not attempt to recognize whether the data describe exterior spines. Return not-applicable if either certificate is absent. Compute the induced free-group automorphism ${\beta }_{\ast }$ by substituting the displayed Artin rules in the declared word order. For the mapping-torus checks, set the axis meridian's representation matrix to $\rho (x)=I_d$. Return not-applicable if the coloring is not preserved on endpoints, if the representation fails any closure relator, or if it fails any mapping-torus relator $s_i={\beta }_{\ast }(g_i)(x^{-1}{g}_{i}x{)}^{-1}$. Equivalently the braid invariance checks $\rho ({\beta }_{\ast }(x_i))=\rho (x_i)$ must hold for all meridians.

2. Set $\psi (x)=1$ on the axis meridian. Verify the deck map on the relators $r_i$ and the axis relators $s_i$, and verify the colors are surjective, so $\psi$ descends to both spine groups and is an epimorphism. Compute the positive-meridian full abelianization basis first, then use the explicit color map $H_L\to H$; no unannounced equal-variable or total-linking specialization is performed.

3. Rewrite ${\beta }_{\ast }(g_i)$ in the free basis $g_1,\dots ,g_n$, using $x_i=g_{i-1}^{-1}{g}_i$ and $g_0=1$. Compute the block Fox matrix $B_{\beta }$ with $(i,j)$ block $\mathrm{\Phi }(\mathrm{\partial }{\beta }_{\ast }(g_i)\mathrm{/}\mathrm{\partial }{g}_j)$, using the product and inverse recursions of the Wada definition and $\mathrm{\Phi }(g)=\rho (g)\psi (g)$. The braid fixes the boundary word $g_n$, so its last block row is $(0,\dots ,0,I_d)$. Let ${\bar{B}}_{\rho }(\beta )$ be the upper-left $(n-1)d$-square block.

4. Compute $D=\det (\rho (g_n)\psi (g_n)-I_d)$. If $D=0$, return not-applicable: this designated deleted column is inadmissible, even if another column of the Fox matrix could be used.

5. Form the twisted cellular boundary matrices $d_1,d_2$ of the closure-exterior spine $X$ from the Fox lemma. Use exact Gaussian elimination over $Q$ to check the chain identity $d_1\circ d_2=0$ (the row-matrix product $J{A}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}=0$). If this identity fails, return not-applicable because the encoded data do not define a chain complex. If it holds but the ranks are not $\mathrm{rank}(d_1)=d$ and $\mathrm{rank}(d_2)=(n-1)d$, return $0$ under the declared torsion convention. Otherwise compute $N=\det ({\bar{B}}_{\rho }(\beta )-I_{(n-1)d})$ and return the rational function $N\mathrm{/}D$ as the twisted Burau determinant value, modulo the determinant/deck units of the based torsion convention.

The braid convention is shown for two adjacent downward-oriented strands. The left strand crosses over the right strand for ${\sigma }_i$; the gap records the underpass. The word, color map, and algebraic action above, rather than the diagram, determine the computation.

The displayed generator interchanges endpoint positions, with the original left strand passing over; the colors must follow the same permutation along each link component.

Twisted coefficient and torsion conventions

Admissible Fox quotient and denominator

Positive Artin action on punctured-disk meridians

Well-defined Artin representation of the braid group

Identification with the geometric braid action

Closed-braid closure and component cycles

Conway, §§2.1–2.4 and 3.1–3.3

Kirk and Livingston, §§2–4

Ranicki, Definitions 4 and 6