Cyclotomic exponent sequences of numerical semigroups

We study the cyclotomic exponent sequence of a numerical semigroup $S,$ and we compute its values at the gaps of $S,$ the elements of $S$ with unique representations in terms of minimal generators, and the Betti elements $b\in S$ for which the set $\{a \in \operatorname{Betti}(S) : a \le_{S}b\}$ is totally ordered with respect to $\le_S$ (we write $a \le_S b$ whenever $a - b \in S,$ with $a,b\in S$). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, Garc\'ia-S\'anchez and Moree (2016) stating that $S$ is cyclotomic if and only if it is a complete intersection.