  The Kauffman Constraint Coefficients Kw and their corresponding Elementals Ew
are presented as solutions to the construction of the (beta)-derivative of
Kauffman's Theta-function. Additionally, a new recursion relation is provided
to construct the (beta)-derivative of Theta that requires only operational
substitutions and summations; this algorithmically simplifies Kauffman's
original technique. To demonstrate Kw, we generate the 30 Kw Coefficients from
the corresponding Elementals Ew for the (9)-derivative of Theta and find that
our results are in complete agreement with Kauffman's Mathematica\texttrademark
solutions. We further present a calculation of two coefficients for the
(12)-derivative of Theta and invite readers to use Mathematica\texttrademark or
any other means to calculate and verify our results. Finally, we present a
challenging calculation for a coefficient of the (40)-derivative of Theta;
owing to the vast numbers of permutations involved, a Mathematica\texttrademark
approach may require substantial computer resources to obtain the solution in a
reasonable time.
