Word representation of cords on a punctured plane

In this paper a purely algebraic condition for a word in a free group to be representable by a simple curve on a punctured plane will be given. application, an algorithm for simple closed curves on a punctured plane will be obtained. Our solution is different from any algorithm due to Reinhart [Ann. of Math. 75 (1962) 209], Zieschang [Math. Scand. 17 (1965) 17] or Chillingworth [Bull. London Math. Soc. 1 (1969) 310]. Although the study here will be confined to the case of a plane, similar arguments could be carried out on the 2-sphere. This research was motivated by monodromy problems appearing in Lefschetz fibrations and surface braids. See [Math. Proc. Cambridge Philos. Soc. 120 (1996) 237; Kamada, Braid and Knots Theory in Dimension Four, American Mathematical Society, 2002; Kamada and Matsumoto, in: Proceedings of the International Conference on Knot Theory “Knots in Hellas ʹ98”, World Scientific, 2000, p. 118; Kamada and Matsumoto, Enveloping monoidal quandles, Preprint, 2002; Matsumoto, in: S. Kojima et al. (Eds.), Proc. the 37th Taniguchi Sympos., World Scientific, 1996, p. 123].