A quadratic lower bound for Topswops

A quadratic lower bound for the topswops function is exhibited. This provides a non-trivial lower bound for a problem posed by J.H. Conway, D.E. Knuth, M. Gardner and others. We describe an infinite family of permutations, each taking a linear number of steps for the topswops process to terminate, and a chaining process that creates from them an infinite family of permutations taking a quadratic number of steps to reach a fixed point with the identity permutation.