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Let $\Sigma$ be a surface of negative Euler characteristic together with apants decomposition $\P$. Kra's plumbing construction endows $\Sigma$ with aprojective structure as follows. Replace each pair of pants by a triplypunctured sphere and glue, or `plumb', adjacent pants by gluing punctured diskneighbourhoods of the punctures. The gluing across the $i^{th}$ pants curve isdefined by a complex parameter $\tau_i \in \C$. The associated holonomyrepresentation $\rho: \pi_1(\Sigma) \to PSL(2,\C)$ gives a projective structureon $\Sigma$ which depends holomorphically on the $\tau_i$. In particular, thetraces of all elements $\rho(\gamma), \gamma \in \pi_1(\Sigma)$, arepolynomials in the $\tau_i$.  Generalising results proved in previous papers for the once and twicepunctured torus respectively, we prove a formula giving a simple linearrelationship between the coefficients of the top terms of $\rho(\gamma)$, aspolynomials in the $\tau_i$, and the Dehn-Thurston coordinates of $\gamma$relative to $\P$.  This will be applied elsewhere to give a formula for the asymptoticdirections of pleating rays in the Maskit embedding of $\Sigma$ as the bendingmeasure tends to zero.

Top terms of polynomial traces in Kra's plumbing construction