For an edge-weighted graph G with n vertices and m edges, we present a new deterministic algorithm for computing a minimum k-way cut for k = 3, 4. The algorithm runs in O(nk-1 F(n, m)) = O(mnk log(n2/m)) time and O(n2) space for k = 3, 4, where F(n, m) denotes the time bound required to solve the maximum flow problem in G. The bound for k = 3 matches the current best deterministic bound Õ(mn3) for weighted graphs, but improves the bound Õ(mn3) to O(n2 F(n, m)) = O(min{mn8/3, m3/2n2}) for unweighted graphs. The bound Õ(mn4) for k = 4 improves the previous best randomized bound Õ(n6) (for m = o(n2)). The algorithm is then generalized to the problem of finding a minimum 3-way cut in a symmetric submodular system.