Research Article | Open Access
The Configuration Space of -Tuples of Equiangular Unit Vectors for , , and
Let be the configuration space of -tuples of unit vectors in such that all interior angles are . The space is an -dimensional space. This paper determines the topological type of for , , and .
Recently, starting in , the topology of the configuration space of spatial polygons of arbitrary edge lengths has been considered by many authors. In the equilateral case, the definition is given as follows. For , we setHere denote the unit vectors in the directions of the edges of a polygon; the group acts diagonally on .
Many topological properties of are already known: First, it is clear that there is a homeomorphism
Second, it is proved in  that is homeomorphic to del Pezzo surface of degree 5.
In another direction, we consider the space of -tuples of equiangular unit vectors in . More precisely, we define the following: We fix and setwhere denotes the standard inner product on . Using (3), we define
It is expected that the space is much more difficult than . For example, the following trivial observation shows that does not admit a similar property to (2): when is odd, we have but .
We recall previous results on . First,  considered the case for . The main result is that, realizing as a homotopy colimit of a diagram involving and , we inductively computed . In particular, we obtained a homeomorphism , where denotes a connected closed orientable surface of genus 5.
Second, we set Note that is the configuration space of equilateral and equiangular -gons. Crippen  studied the topological type of for , and 5. The result is that is either , one point, or two points depending on . Later, O’Hara  studied the topological type of . The result is that is disjoint union of a certain number of ’s and points.
The purpose of this paper is to determine the topological type of for , and 5. In contrast to the fact that at most one-dimensional spaces appear in the results of [8, 9], surfaces appear in our results.
2. Main Results
Theorem A. The topological type of is given in Table 1.
Theorem C. (i) The topological type of is given in Table 3. Let be a connected closed orientable surface of genus .
(ii) (a) Let satisfy that . We study the situation where approaches . We identify the torus with the Dupin cyclide, which we denote by . (See Figure 2.)
Using this, we identify with , where the connected sum is formed by cutting a small circular hole away from the narrow part of . As approaches , the center of each narrow part pinches to a point. Thus the five singular points appear.
(b) We consider the situation where increases from . Then each pinched point of separates. Thus we obtain .
(c) Let satisfy that . We consider the situation where approaches . In contrast to (a), the center of exactly one narrow part pinches to a point. Thus one singular point appears.
Corollary D. As a subspace of , we define the spaceThen is a singular point of and has a neighborhood , where denotes the cone.
Remark 1. Cone-type singularities appear in Theorems B and C and Corollary D. We note that singularities of configuration spaces of mechanical linkages have been studied extensively by Blanc and Shvalb .
3. Proofs of the Main Results
We fix and set Normalizing and to be and , respectively, we have the following description:Hereafter we use (8).
In order to prove our main results, we use the following fact, whose proof is left to the reader.
Fact 2. Let satisfy thatThen, there exists such that
Now we first consider the case . Consider Fact 2 for , and . Then there exists such that
Next, we consider Fact 2 for , and . Then there exists such that
Now we define the function byWe can understand as a level set. More precisely, we define the functionby . Then we haveif .
Lemma 4. We set Then is given in Table 4.
Proof. The lemma is proved by direct computations.
Proof of Theorem C. We consider in (16) as a Morse function on . First, Table 4 and (17) show that .
Second, direct computation shows that Since this is nonzero, the space is smooth at . Actually, we can prove that the point is a nondegenerate critical point of the function . Hence Morse lemma shows that there is a homeomorphism for . But if we use [11, Corollary B], we need not check that is nondegenerate at . For our reference, we draw the figure of in Figure 3.
Third, the other parts of Table 3 follow from Table 4. This completes the proof of Theorem C.
Proof of Corollary D. The corollary is an immediate consequence of Theorem C.
Proof of Theorem B. We define as in (11). We also define to be the right-hand side of (12). We define the function by Similarly to (17), we have . Since is one-dimensional, it is easy to draw its figure. Thus Theorem B follows.
Proof of Theorem A. We define the function by . Since , Theorem A follows.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
- M. Kapovich and J. J. Millson, “The symplectic geometry of polygons in Euclidean space,” Journal of Differential Geometry, vol. 44, no. 3, pp. 479–513, 1996.
- A. A. Klyachko, “Spatial polygons and stable configurations of points in the projective line,” in Algebraic Geometry and Its Applications (Yaroslaevl' 1992), Vieweg, pp. 67–84, 1994.
- J.-C. Hausmann and A. Knutson, “The cohomology ring of polygon spaces,” Université de Grenoble. Annales de l'Institut Fourier, vol. 48, no. 1, pp. 281–321, 1998.
- M. Farber, Invitation to Topological Robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, Switzerland, 2008.
- D. H. Gottlieb, “Robots and fibre bundles,” Bulletin of the Belgian Mathematical Society, vol. 38, pp. 219–223, 1986.
- D. H. Gottlieb, “Topology and the robot arm,” Acta Applicandae Mathematicae, vol. 11, no. 2, pp. 117–121, 1988.
- Y. Kamiyama, “A hypersurface of the configuration space of a spatial robot arm,” JP Journal of Geometry and Topology, vol. 20, no. 1, pp. 27–38, 2017.
- G. M. Crippen, “Exploring the conformation space of cycloalkanes by linearized embedding,” Journal of Computational Chemistry, vol. 13, no. 3, pp. 351–361, 1992.
- J. O'Hara, “The configuration space of equilateral and equiangular hexagons,” Osaka Journal of Mathematics, vol. 50, no. 2, pp. 477–489, 2013.
- D. Blanc and N. Shvalb, “Generic singular configurations of linkages,” Topology and its Applications, vol. 159, no. 3, pp. 877–890, 2012.
- Y. Kamiyama, “On the level set of a function with degenerate minimum point,” International Journal of Mathematics and Mathematical Sciences, Art. ID 493217, 6 pages, 2015.
Copyright © 2018 Yasuhiko Kamiyama. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.