arXiv:2611.14816v1 Announce Type: new Abstract: We prove that any complete intersection numerical semigroup can be realized as the value semigroup of an end curve in a splice quotient singularity. Therefore, providing a converse to the semigroup-theoretic statement of the End Curve Theorem of Neumann and Wahl for integral homology sphere links. Moreover, for a complete-intersection numerical semigroup, we define a family of knots such that the Alexander polynomial of any knot of the family coincides (up to a factor) with the generating series of the semigroup. Let $(X,0)$ be a splice quotient singularity with $\mathbb{Z}HS^3$ link $\Sigma$ and let $(C,0)$ be a reduced curve-germ embedded in $(X,0)$ defined as the zero locus of the product of end curve functions. We prove, thorugh our topological characterization of complete intersection semigroups, the coincidence between the Alexander polynomial of the knot cut out by $C$ in $\Sigma$ and the Poincar\'e series of the embedded curve. Hence providing a generalization of the result due to Campillo, Delgado and Gusein-Zade for plane curve singularities. Furthermore, for any good resolution of $(X,0)$ compatible with the splice quotient analytic structure, we provide an explicit description of the semigroup of integer multiplicities of functions along any divisor of the resolution.
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