  The braiding operations of quantum states have attracted substantial
attention due to their great potential for realizing topological quantum
computations. In this paper, we show that a three-fold degenerate eigen
subspace can be obtained in a four-level Hamiltonian which is the minimal
physical system. Braiding operations are proposed to apply to dressed states in
the subspace. The topology of the braiding diagram can be characterized through
physical methods once that the sequential braiding pulses are adopted. We
establish an equivalent relationship function between the permutation group and
the output states where different output states correspond to different values
of the function. The topological transition of the braiding happens when two
operations overlap, which is detectable through the measurement of the
function. Combined with the phase variation method, we can analyze the wringing
pattern of the braiding. Therefore, the experimentally-feasible system provides
a platform to investigate braiding dynamics, the SU(3) physics and the qutrit
gates.
