The granny and the square tangle and the unknotting number

We show that a knot with a diagram with n granny and square tangles has unknotting number at least n, bridge number >n, and braid index >n. As an application, we construct families of slice knots with arbitrarily high unknotting number, in which the number of knots of given crossing number c is exponential in c. We discuss a relation of our estimates to the homology group of the double branched covering of the knot complement, and derive from it new conditions on the values of the Jones and Brandt–Lickorish–Millett–Ho polynomial.