A solution u of a Cauchy problem or a Cauchy-Dirichlet problem for a semilinear heat equation Let ϕ ∞ be the radially symmetric singular steady state of the Cauchy problem. Suppose that u 0 ∈ L ∞ is a radially symmetric function such that u 0 − ϕ ∞ and (u 0 ) t change sign at most finitely many times. By application of the braid group theory, we determine the exact blow-up rate of solution with initial data u 0 which undergoes type II blow-up in the case of p > p JL , where p JL is the exponent of Joseph and Lundgren.