Compute dihedral Fox colorings by row reduction

Input a finite regular oriented link diagram $D$ with $a$ arcs and $c$ crossings, fixed finite orderings of its arcs and crossings, and an odd prime $p$. The algorithm returns the full set ${\mathrm{Col}}_p(D)$ of assignments of residues in ${\mathbb{F}}_p$ to the arcs satisfying the Fox crossing equations, together with $\mathrm{\mid }{\mathrm{Col}}_p(D)\mathrm{\mid }$.

1. Create one variable $x_j$ for each arc in the supplied order, and process crossings in the supplied order.
2. At each crossing with overarc $o$ and the two underarcs $i$ and $j$, append the homogeneous linear equation $2x_o-x_i-x_j=0$ over ${\mathbb{F}}_p$. At a crossing, add the coefficients of any repeated arc in that row.
3. Set the next pivot-row index to $r=1$ and process columns from left to right. If $r>c$, mark this and every remaining column free and stop. If $r\le c$, search rows $r,\dots ,c$ in order for the first nonzero entry in the current column. If none exists, mark this column free and continue without changing $r$. Otherwise swap the selected row into row $r$, divide row $r$ by its nonzero entry in this column, and add multiples of row $r$ to every other row to clear this column. Record this column as a pivot column with pivot row $r$, then increase $r$ by one. After the scan, every column not recorded as a pivot column is free.
4. List the free columns in increasing order and enumerate every tuple in ${\mathbb{F}}_p^f$ in lexicographic order using the residue representatives $0,\dots ,p-1$, where $f$ is the number of free columns. If $f=0$, process the empty tuple once. For each tuple, assign its coordinates to the free variables. For each pivot row, set its pivot variable equal to the negative of the sum of the row's free-column coefficients times those assigned free variables. Append the resulting assignment in the supplied arc order. Return all appended assignments and their number.

Quandles and knot-diagram colorings.

The dihedral operation is a quandle.

Regular oriented link diagrams.

Przytycki, 3-coloring and other elementary invariants of knots, §2.

Manturov, Knot Theory, Chapter 5.

Carter, A Survey of Quandle Ideas.