Classification of simple arcs with fixed ends in the punctured plane

Abstract

The simple arcs with fixed end in the pucntured plane is a special case of 1 lases. In order to classify the isotopy classes of 1 lace on the pucntured plane, we approch by two way, the geometric view point and algebraic view point. In the geometrical view point, the isotopy classes of 1 lace in the pucntured plane can be identified to the integer pair (r, s) which satisfy rs ≥ O, ｜r｜≠ (2m+1)n+m and ｜s｜≠ (2m+1) k for each n, k ≥ 0 and m ≥ 1. And the cobordism classes of 1 laces in the pucntured plane can be identified to the infinite cyclic group Z. In the algebraic view point, the isotopy classes can be identified to the coset ＜ A_{1,3},A_{2,3} ＞ / ＜ A_{1,3}A_{2.3}＞.