Polynomials without repelling periodic point of given period

Bergweiler proved that for any given integer k⩾2, every polynomial P of degree d⩾2 has at least one repelling periodic cycle of period k unless (k,d)∈{(2,2),(2,3),(2,4),(3,2)}. Here we classified these exceptional polynomials. We also showed that the Julia sets of these exceptional polynomials are connected.