are integers that can be expressed in terms of $(\bar{\beta}_0,\ldots, \bar{\beta}_g)$. The curve $Y$ is called the *monomial curve associated with ${\mathcal C}$*, but, to simplify the notation, we will refer to it as a *(space) monomial curve* $Y \subset {\mathbb C}^{g+1}$. Since such a monomial curve for $g = 1$ is just a cusp in the complex plane, of which the monodromy conjecture is already well-known, we are interested in the case where $g \geq 2$.

The first part in the monodromy conjecture for these curves (or their corresponding binomial ideals) has already been studied in [@MVV]: a complete list of poles of the motivic zeta function associated with a space monomial curve $Y \subset {\mathbb C}^{g+1}$ is given by $${\mathbb L}^g, \qquad {\mathbb L}^{\frac{\nu_k}{N_k}}, \qquad k = 1,\ldots, g,$$ where $$\frac{\nu_k}{N_k} = \frac{1}{n_k{\bar{\beta}}_k}\bigg(\sum_{l=0}^k {\bar{\beta}}_l - \sum_{l=1}^{k-1}n_l{\bar{\beta}}_l\bigg) + (k-1) + \sum_{l=k+1}^g\frac{1}{n_l}, \qquad k = 1,\ldots, g.$$Here, ${\mathbb L}$ denotes the class of the affine line in the *Grothendieck ring* of complex varieties.

It thus remains to investigate the monodromy eigenvalues of a space monomial curve $Y \subset {\mathbb C}^{g+1}$ and to show that every pole in the above list yields such an eigenvalue. To this end, we will make 