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We consider boundary conditions compatible with discrete holomorphicity forthe dilute O(n) and C_2^(1) loop models. In each model, for a general set ofboundary plaquettes, multiple types of loops can appear. A generalisation ofSmirnov's parafermionic observable is therefore required in order to maintainthe discrete holomorphicity property in the bulk. We show that there existnatural boundary conditions for this observable which are consistent withintegrability, that is to say that, by imposing certain boundary conditions, weobtain a set of linear equations whose solutions also satisfy the correspondingreflection equation. In both loop models, several new sets of integrableweights are found using this approach.

Discrete holomorphicity and integrability in loop models with open boundaries