In this paper we construct smooth irreducible space curves $C$ which link geometrically by surfaces of minimal degree containing $C$ to curves $\Gamma$ of generic embedding dimension three. This produces interesting behavior with respect to both $C$ and $\Gamma$. The curves $\Gamma$ link to smooth connected curves by surfaces of low degree but cannot link to smooth connected curves by surfaces of high degree. The curves $C$ give counterexamples to a conjecture of Martin-Deschamps and Perrin.