Reducing quadratic forms by kneading sequences

Abstract

We introduce an invertible operation on finite sequences of positive integers and call it "kneading". Kneading preserves three invariants of sequences -- the parity of the length, the sum of the entries, and one we call the "alternant". We provide a bijection between the set of sequences with alternant $a$ and parity $s$ and the set of Zagier-reduced indefinite binary quadratic forms with discriminant $a^2 + (-1)^s \cdot 4$, and show that kneading corresponds to Zagier reduction of the corresponding forms. It follows that the sum of a sequence is a class invariant of the corresponding form. We conclude with some observations and conjectures concerning this new invariant.