To expand upon Douglas Zare's comment a bit, in the book
Farber, Michael. Invitation to topological robotics. European Mathematical Society, 2008,
Farber uses C&P (cut & paste) surgery and the Grothendiek ring $\mathfrak{C}$&$\mathfrak{P}$ to compute Euler characteristics of polyhedral configuration spaces (pp.55-56):
![C&P][1]