Topological complexity of spatial polygon spaces
HTML articles powered by AMS MathViewer
- by Donald M. Davis PDF
- Proc. Amer. Math. Soc. 144 (2016), 3643-3645 Request permission
Abstract:
Let $\overline {\ell }=(\ell _1,\ldots ,\ell _n)$ be an $n$-tuple of positive real numbers, and let $N(\overline {\ell })$ denote the space of equivalence classes of oriented $n$-gons in $\mathbb {R}^3$ with consecutive sides of lengths $\ell _1,\ldots ,\ell _n$, identified under translation and rotation of $\mathbb {R}^3$. Using known results about the integral cohomology ring, we prove that its topological complexity satisfies $\operatorname {TC}(N(\overline {\ell }))= 2n-5$, provided that $N(\overline {\ell })$ is nonempty and contains no straight-line polygons.References
- D. M. Davis, Topological complexity of some planar polygon spaces, on arXiv.
- —, Topological complexity of planar polygon spaces with small genetic code, on arXiv.
- Michael Farber, Invitation to topological robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2455573, DOI 10.4171/054
- Jean-Claude Hausmann, Mod two homology and cohomology, Universitext, Springer, Cham, 2014. MR 3308717, DOI 10.1007/978-3-319-09354-3
- J.-C. Hausmann and A. Knutson, The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 281–321 (English, with English and French summaries). MR 1614965, DOI 10.5802/aif.1619
- Alexander A. Klyachko, Spatial polygons and stable configurations of points in the projective line, Algebraic geometry and its applications (Yaroslavl′, 1992) Aspects Math., E25, Friedr. Vieweg, Braunschweig, 1994, pp. 67–84. MR 1282021
- G. Panina and D. Siersma, Motion planning and control of a planar polygonal linkage, on arXiv.
Additional Information
- Donald M. Davis
- Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
- MR Author ID: 55085
- Email: dmd1@lehigh.edu
- Received by editor(s): July 9, 2015
- Received by editor(s) in revised form: October 8, 2015
- Published electronically: February 1, 2016
- Communicated by: Michael A. Mandell
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3643-3645
- MSC (2010): Primary 55M30, 58D29, 55R80
- DOI: https://doi.org/10.1090/proc/12998
- MathSciNet review: 3503733