Reducing Quadratic Forms by Kneading Sequences

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 + (−1)s · 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. 1 Kneading sequences We pinch the left end of a finite sequence of positive integers by transforming it through the rule: (x, y, z, . . .) 7→ { (1, x− 1, y, z, . . .), if x ≥ 2, (y + 1, z, . . .), if x = 1. We pinch the right end similarly. We will knead a finite sequence of positive integers by • Removing the leftmost entry, then