On Targeted Complexity of Discrete Motion
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909730977349632 |
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| author | Babaee, Ameneh Mirebrahimi, Hanieh Fahimi, Soheila |
| author_facet | Babaee, Ameneh Mirebrahimi, Hanieh Fahimi, Soheila |
| contents | In this paper, we investigate discrete topological complexity $TC(K)$ introduced for situations where the configuration space possesses a simplicial structure. %Simplicial complexes are well-known and commonly used in programming for robotic motion. Let $K$ be a complex and let $L$ be a subcomplex considered as the target of the motion. We introduce targeted simplicial complexity $TC(K,L)$, which yields smaller values than the discrete version $TC(K)$. We then demonstrate that targeted simplicial complexity is strongly homotopy invariant and it varies between simplicial LS-categories of $K$ and $K \prod K$. Utilizing this information, we calculate targeted simplicial complexity for scenarios such as strongly collapsible complexes. Finally, we compare targeted simplicial complexity with relative topological complexity and we show that $TC(|K|, |L|) \le TC (K,L)$ where $|\cdot|$ denotes the geometric realization functor. Although relative topological complexity is generally lower than targeted simplicial complexity, they are equal in certain cases, such as arbitrary wedges of triangulated circles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07052 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Targeted Complexity of Discrete Motion Babaee, Ameneh Mirebrahimi, Hanieh Fahimi, Soheila Algebraic Topology In this paper, we investigate discrete topological complexity $TC(K)$ introduced for situations where the configuration space possesses a simplicial structure. %Simplicial complexes are well-known and commonly used in programming for robotic motion. Let $K$ be a complex and let $L$ be a subcomplex considered as the target of the motion. We introduce targeted simplicial complexity $TC(K,L)$, which yields smaller values than the discrete version $TC(K)$. We then demonstrate that targeted simplicial complexity is strongly homotopy invariant and it varies between simplicial LS-categories of $K$ and $K \prod K$. Utilizing this information, we calculate targeted simplicial complexity for scenarios such as strongly collapsible complexes. Finally, we compare targeted simplicial complexity with relative topological complexity and we show that $TC(|K|, |L|) \le TC (K,L)$ where $|\cdot|$ denotes the geometric realization functor. Although relative topological complexity is generally lower than targeted simplicial complexity, they are equal in certain cases, such as arbitrary wedges of triangulated circles. |
| title | On Targeted Complexity of Discrete Motion |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2508.07052 |