Analysis of Block Group Movement and Restoration Feasibility Using Laurent Series

In this paper, we investigate the feasibility of block group movement and restoration using Laurent series. The movement of block groups is represented as a function in the complex plane, and the restoration feasibility is analyzed based on the invertibility of transformations using Laurent series expansions. We discuss the conditions under which the block group can return to its original configuration and explore the mathematical complexity of such transformations.The movement of blocks in their original positions can be considered feasible if the set of movable permutations includes only the permutations that can be moved. In terms of the Laurent series, this corresponds to the set of coefficients \( \{a_n\} \). If the set of movable permutations includes all the required permutations, the transformation is feasible. However, if it does not, meaning the Laurent series introduces negative powers of \( z \) (i.e., \( z^{-n} \) terms), the transformation becomes infeasible.