Enumerating Palindromes in Rank Two Free Groups

Conjugacy classes of primitive words in the free group of rank two can indexed by the rationals. A representative for each conjugacy class is given by a recursive enumeration scheme and the representative, usually called a Farey word, is denoted by Wp/q where p and q are relatively prime integers. Primitive pairs, that is, pairs of words that generate the free group, correspond to pairs of words (Wp/q , Wr/s) with |ps − rq| = 1. Here we give a new enumerative scheme for the conjugacy class representatives of primitive words in the free group of rank two. We denote these words by Ep/q and prove that Ep/q is a palindrome if pq is even and the product of two palindromes if pq is odd. We prove that the pairs (Ep/q , Er/s) again generate the group when |ps− rq| = 1.