We introduce a new type of slant curves in almost contact B-metric manifolds, called $\varphi $-slant curves, by an additional condition which is specific for these manifolds. In this paper we study $\varphi $-slant null curves in a class of 3-dimensional normal almost contact B-metric manifolds and prove that for non-geodesic of them there exists a unique Frenet frame for which the original parameter is distinguished. We investigate some of $\varphi $-slant null curves and with respect to the associated B-metric on the manifold and find relationships between the corresponding Frenet frames and curvatures. We construct the examined curves in a 3-dimensional Lie group and give their matrix representation.

Comment: 18 pages. arXiv admin note: text overlap with arXiv:1511.08037