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×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.