A finite element elasticity complex on tetrahedral meshes is devised. The $H^1$ conforming finite element is the smooth finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an $H(textrm{inc})$-conforming finite element for symmetric tensors is the main focus of this paper. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the $textrm{inc}$ operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition of polynomial tensor spaces. The trace of the $textrm{inc}$ operator is induced from a Greens identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Our construction appears to be the first $H(textrm{inc})$-conforming finite elements on tetrahedral meshes without further splits.
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