Generalized rational blow-down, torus knots, and Euclidean algorithm

We construct a Kirby diagram of the rational homology ball used in “generalized rational blow-down” developed by Jongil Park. The diagram consists of a dotted circle and a torus knot. The link is simpler, but the parameters are a little complicate. Euclidean Algorithm is used three times in the construction and the proof. 1 Main theorem For a coprime pair (m,n) of positive integers, we take a simple closed curve k(m,n) in the standardly embedded once-punctured torus F in S3 as in Figure 1. We study the Kirby diagram k(m,n)∪ u: the component k(m,n) is a torus knot T (m,n) with (mn)-framing, and u is a dotted unknoted circle (It is a 1-handle, see [A, AK] and [GS, p.168]) in the complement of F . This diagram defines a rational homology ball that has cyclic fundamental group of order (m+ n). It has a symmetry: k(n,m) ∪ u = k(m,n) ∪ u. In the next section, for a given coprime pair (p, q) of positive integers with 1 ≤ q < p, we will construct an involutive symmetric function A by Algorithm, to decide (another) coprime pair (m,n) = A(p − q, q) satisfying m + n = p. It holds that A(p − 1, 1) = (p − 1, 1), 2000 Mathematics Subject Classification: Primary 57M25, 57D65, Secondary 55A25.