What do you mean by "reshaping". From your reference to "topology" you might be allowing any continuous deformation- bending lines,etc.. In that case, any polygon can be reshaped to any other, or to a circle.

If you are referring only to changing the angles of a polygon, while leaving the sides of the same length, the the "SSS" congurence shows that you cannot "reshape" a triangle in that way- The lengths of the sides determines the angles.