Three-periodic helices on elliptic curves and their associated regular algebras

Let $k$ denote an algebraically closed field of characteristic zero and let $X$ denote a smooth elliptic curve over $k$. Given a three-periodic elliptic helix $\underline{\mathcal{E}}$ of vector bundles over $X$ with endomorphism $\mathbb{Z}$-algebra $\operatorname{End} \underline{\mathcal{E}}$ and quadratic cover $\mathbb{S}^{nc}(\underline{\mathcal{E}})$, we prove that $\operatorname{End} \underline{\mathcal{E}}$ is the quotient of $\mathbb{S}^{nc}(\underline{\mathcal{E}})$ by a degree three family of normal elements, generalizing a result of the authors to the case in which $\operatorname{dim }(\operatorname{End} \underline{\mathcal{E}})_{i, i+1}$ isn't a constant function of $i$. We then show that $\operatorname{End} \underline{\mathcal{E}}$ is noetherian if and only if it has polynomial growth, and in this case, the ranks of any three consecutive bundles in the helix are a Markov triple. Furthermore, in this case $\mathbb{S}^{nc}(\underline{\mathcal{E}})$ is a noetherian GK-three $\mathbb{Z}$-algebra which is ${\sf Proj }$-equivalent to an elliptic algebra. We conclude the paper by constructing several new families of elliptic helices with exponential growth.