A classification of (1,1)-positions

Abstract

In this paper, we describe the equivalence classes of simple arcs between the two punctures on a 2-punctured torus Sigma(1,2) up to isotopy by using the given four generators g(1), g(2), g(3) and g(4). Actually, we show that a class of simple arcs is represented by an ordered sequence of four integers. Also, we introduce an algorithm to check whether or not an ordered sequence of four integers represents a class of simple arcs in Sigma(1,2). We want to point out that this result classifies the (1, 1)-positions.