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Let $C({\bf n})$ be a complete intersection monomial curve in the4-dimensional affine space. In this paper we study the complete intersectionproperty of the monomial curve $C({\bf n}+w{\bf v})$, where $w>0$ is an integerand ${\bf v} \in \mathbb{N}^{4}$. Also we investigate the Cohen-Macaulayness ofthe tangent cone of $C({\bf n}+w{\bf v})$.

Complete intersection monomial curves and the Cohen-Macaulayness of their tangent cones