We study a class of topological materials which in their momentum-space band structure exhibit three-fold degeneracies known as triple points. Focusing specifically on $mathcal{P}mathcal{T}$-symmetric crystalline solids with negligible spin-orbit coupling, we find that such triple points can be stabilized by little groups containing a three-, four- or six-fold rotation axis, and we develop a classification of all possible triple points as type-A vs. type-B according to the absence vs. presence of attached nodal-line arcs. Furthermore, by employing the recently discovered non-Abelian band topology, we argue that a rotation-symmetry-breaking strain transforms type-A triple points into multi-band nodal links. Although multi-band nodal-line compositions were previously theoretically conceived and related to topological monopole charges, a practical condensed-matter platform for their manipulation and inspection has hitherto been missing. By reviewing the known triple-point materials with weak spin-orbit coupling, and by performing first-principles calculations to predict new ones, we identify suitable candidates for the realization of multi-band nodal links in applied strain. In particular, we report that an ideal compound to study this phenomenon is Li$_2$NaN, in which the conversion of triple points to multi-band nodal links facilitates largely tunable density of states and optical conductivity with doping and strain, respectively.
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