  Construction of a universal finite-type invariant can be reduced, under
suitable assumptions, to the solution of certain equations (the hexagon and
pentagon equations) in a particular graded associative algebra of chord
diagrams. An explicit, closed-form solution to these equations may, indirectly,
give information about various interesting properties of knots, such as which
knots are ribbon. However, while closed-form solutions (as opposed to solutions
which can only be approximated to successively higher degrees) are needed for
this purpose, such solutions have proven elusive, partly as a result of the
non-commutative nature of the algebra. To make the problem more tractable, we
restrict our attention to solutions of the equations in the subalgebra of
horizontal chord diagrams, viewed as a graded unital permutative algebra --
where `permutative' means that u[x,y]=0 whenever u has degree at least 1. We
show that this restriction leads in a straightforward and fairly short way to a
reduction of the hexagon and pentagon equations to a simpler equation taken
over the algebra of power series in two commuting variables. This equation had
been found and solved explicitly by Kurlin under a superficially different set
of reduction assumptions, which we show here are in fact equivalent to ours.
This paper thus provides an alternative (simpler and shorter) derivation of
Kurlin's equation.
