Summary: We study the category \(\mathcal{F}_n\) of finite-dimensional integrable representations of the periplectic Lie superalgebra \(\mathfrak{p}(n)\). We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter 0 on the category \(\mathcal{F}_n\) by certain translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for \(\mathfrak{p}(n)\) resembling those for \(\mathfrak{gl}(m \vert n)\). We discover two natural highest weight structures. Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of irreducibles in standard and costandard modules and classify the blocks of \(\mathcal{F}_n\). We also prove the surprising fact that indecomposable projective modules in this category are multiplicity-free.