Triangular curves and cyclotomic Zariski tuples

The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any \(d>3\) we find Zariski tuples parametrized by the \(d\)-roots of unity up to complex conjugation. As a consequence, for any divisor \(m\) of \(d\), \(m\neq 1,2,3,4,6\), we find arithmetic Zariski \(\frac{\phi(m)}{2}\)-tuples with coefficients in the corresponding cyclotomic field. These curves have abelian fundamental group and they are distinguished using a linking invariant.