Relations of formal power series of one variable and coding of space curves by iterated path integrals

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The words consisting of two letters X,Y and their inverses constitute a group structure in algebra, where the concatenation of words is the product. An element of this group defines a polygonal path with vertices on the integral lattice. Plane curves are regarded as generalization of such linear paths, hence words of X,Y. From this point of view, the group of plane curves are investigated, especially curves are presented as generalized words of X,Y and the logarithms of those curves are investigated as elements of the free algebra generated by X,Y, and some results are obtained.