Chain of interacting $SU(2)_4$ anyons and quantum $SU(2)_k\times\overline{SU(2)_k}$ doubles

We consider a chain of $SU(2)_4$ anyons with transitions to a topologically ordered phase state. For half-integer and integer indices of the type of strongly correlated excitations, we find an effective low-energy Hamiltonian that is an analogue of the standard Heisenberg Hamiltonian for quantum magnets. We describe the properties of the Hilbert spaces of the system eigenstates. For the Drinfeld quantum $SU(2)_k \times\overline{SU(2)_k}$ doubles, we use numerical computations to show that the largest eigenvalues of the adjacency matrix for graphs that are extended Dynkin diagrams coincide with the total quantum dimensions for the levels $k=2,3,4,5$. We also formulate a hypothesis about the reason for the universal behavior of the system in the long-wave limit.