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The Tur\'an density of an $r$-uniform hypergraph $\mathcal{H}$, denoted$\pi(\mathcal{H})$, is the limit of the maximum density of an $n$-vertex$r$-uniform hypergraph not containing a copy of $\mathcal{H}$, as $n \to\infty$.  Denote by $\mathcal{C}_{\ell}$ the $3$-uniform tight cycle on $\ell$vertices. Mubayi and R\"odl gave an ``iterated blow-up'' construction showingthat the Tur\'an density of $\mathcal{C}_5$ is at least $2\sqrt{3} - 3 \approx0.464$, and this bound is conjectured to be tight. Their construction also doesnot contain $\mathcal{C}_{\ell}$ for larger $\ell$ not divisible by $3$, whichsuggests that it might be the extremal construction for these hypergraphs aswell. Here, we determine the Tur\'an density of $\mathcal{C}_{\ell}$ for alllarge $\ell$ not divisible by $3$, showing that indeed $\pi(\mathcal{C}_{\ell})= 2\sqrt{3} - 3$. To our knowledge, this is the first example of a Tur\'andensity being determined where the extremal construction is an iterated blow-upconstruction.  A key component in our proof, which may be of independent interest, is a$3$-uniform analogue of the statement ``a graph is bipartite if and only if itdoes not contain an odd cycle''.

The Tur\'an density of tight cycles in three-uniform hypergraphs