A C-complex is a union of Seifert surfaces for the components of a link which intersect each other in clasps. The clasp number of a link is the minimal number of clasps amongst all C-complexes it bounds It gives a measure of complexity and can be used to provide bounds on other useful characteristics of a link. This paper provides a new lower bound for the number of clasps of all C-complexes bounded by a given 3-component link improving results of Amundsen-Anderson-D.-Guyer. Furthermore, we construct links that achieve these bounds. In order to do so, we express the triple linking numbers as the area bounded by three curves, called word curves, and then perform the geometry and discrete optimization needed to minimize the length of these curves.
