  Attached to a singular analytic curve germ in $d$-space is a numerical
semigroup: a subset $S$ of the non-negative integers which is closed under
addition and whose complement isfinite. Conversely, associated to any numerical
semigroup $S$ is a canonical mononial curve in $e$-space where $e$ is the
number of minimal generators of the semigroup. It may happen that $d < e =
e(S)$ where $S$ is the semigroup of the curve in $d$-space. Define the minimal
(or `honest') embedding of a numerical semigroup to be the smallest $d$ such
that $S$ is realized by a curve in $d$-space. Problem: characterize the
numerical semigroups having minimal embedding dimension $d$. The answer is
known for the case $d=2$ of planar curves and reviewed in an Appendix to this
paper. The case $d =3$ of the problem is open. Our main result is a
characterization of the multiplicity $4$ numerical semigroups whose minimal
embedding dimension is $3$. See figure 1. The motivation for this work came
from thinking about Legendrian curve singularities.
