Fig. 1. Topological acoustic triple point (TATP).

a TATP with $\mathfrak{q}\equiv \left({\mathfrak{n}}_{\mathrm{sk}},\mathfrak{e}\right)=\left(1,2\right)$ can appear with either a linear dispersion or a quadratic dispersion. Note that there is a gap between the L and the T modes away from the triply degenerate point. b TATP is distinct from spin-1 Weyl point, which is protected by Chern numbers. TATP is also distinct from the triple point formed by band inversion, where it is not possible to separate the highest energy band from two lower energy bands away from the triple point. c Skyrmion texture of longitudinal phonon on the sphere wrapping a TATP. The transverse modes span the tangent plane of the sphere. d Wilson loop spectrum for the transverse modes as a function of the polar angle θ computed on a sphere wrapping the TATP. $\mid \mathfrak{e}\mid$ is given by the number of times that one of the two branches of w_ϕ(θ) crosses π.