Simplicial approximation and refinement of monoidal topological complexity

This paper presents a combinatorial approximation of the monoidal topological complexity T C M \mathrm {TC}^M of a simplicial complex K K that controls reserved robot motions in K K . We introduce an upper bound S C r M \mathrm {SC}^M_r of T C M \mathrm {TC}^M using the r r -iterated barycentric subdivision of K × K K \times K modulo the diagonal and consider the refinement of the approximation. We show that T C M \mathrm {TC}^M can be described as S C r M \mathrm {SC}^M_r for sufficiently large r ≥ 0 r \geq 0 . As an example, we consider a simplicial model S n S_n of an n n -sphere and demonstrate that S C r M ( S n ) \mathrm {SC}^M_r(S_n) presents the best estimate of the monoidal topological complexity of an n n -sphere for r ≥ 1 r \geq 1 .