Hiroshima Mathematical Journal

The configuration space of a model for ringed hydrocarbon molecules

Abstract

We give a mathematical model of $n$-membered ringed hydrocarbon molecules, and study the topology of a configuration space $C_{n}$ of the model. Under the bond angle conditions required for ringed molecules, we prove that $C_{n}$ is homeomorphic to $(n-4)$-dimensional sphere $S^{n-4}$ when $n = 5, 6, 7$. This result gives an appropriate explanation of the configuration space of $n$-membered ringed hydrocarbon molecules when $n = 5, 6$.

Article information

Source
Hiroshima Math. J., Volume 42, Number 1 (2012), 115-126.

Dates
First available in Project Euclid: 30 March 2012

https://projecteuclid.org/euclid.hmj/1333113009

Digital Object Identifier
doi:10.32917/hmj/1333113009

Mathematical Reviews number (MathSciNet)
MR2952075

Zentralblatt MATH identifier
1238.92069

Citation

Goto, Satoru; Komatsu, Kazushi. The configuration space of a model for ringed hydrocarbon molecules. Hiroshima Math. J. 42 (2012), no. 1, 115--126. doi:10.32917/hmj/1333113009. https://projecteuclid.org/euclid.hmj/1333113009

References

• F. H. Allen and M. J. Doyle, Automated conformational analysis from crystallographic data. 6.Principal-component analysis for n-membered carbocyclic rings (n = 4,5,6): Symmetry considerations and correlations with ring-puckering parameters, Acta Cryst. B47 (1991), 412–424.
• G. M. Crippen and T. F. Havel, Distance Geometry and Molecular Conformation, Wiley, NewYork, 1988.
• S. Goto, The configuration space of a mathematical model for ringed hydrocarbon molecules (2) (in Japanese), (The 2008 autumn Conference):Society of Computer Chemistry, Japan(SCCJ)
• S. Goto, T. Munakata and K. Komatsu, Pharmacoinformatical and mathematical study for the diversity of three-dimensional structures and the conformational interconversion of macrocyclic compound, J. Pharm. Soc. Jpn. 126(2006), 266-269.
• T. F. Havel, Some examples of the use of distances as coordinates for Euclidean geometry, J. Symbolic Computation 11 (1991), 579–593.
• H. Kamiya, Weighted trace functions as examples of Morse functions, Jour. Fac. Sci. Shinshu Univ. 7 (1971), 85–96.
• M. Kapovich and J. Millson, On the moduli space of polygons in the Euclidean plane, J. Diff. Geom. 42 (1995), 430–464.
• K. Komatsu, The configuration space of a mathematical model for ringed hydrocarbon molecules (1) (in Japanese), (The 2008 autumn Conference):Society of Computer Chemistry, Japan(SCCJ)
• W. J. Lenhart and S. H. Whitesides, Reconfiguring closed polygonal chains in Euclidean $d$-space, Jour. Discrete Comput. Geom. 13 (1995), 123–140.
• R. J. Milgram and J.C. Trinkle, Complete path planning for closed kinematic chains with spherical joints, Internat. J. Robotics Res. 21 (2002), 773–789.
• R. J. Milgram and J.C. Trinkle, The geometry of configuration spaces for closed chains in two and three dimensions, Homology, Homot., Appl. 6 (2004), 237–267.
• J. Milnor, Morse Theory, Princeton University Press, Princeton, 1969.
• T. Munakata, S. Goto and K. Komatsu, Conformational flexibility effect of one configuration difference, J. Pharm. Soc. Jpn. 126, (2006), 270-273.
• R. Rosen, A weak form of the star conjecture for manifolds, Abstract 570-28, Notices Amer. Math. Soc. 7 (1960), 380.
• D. Shimamoto and C. Vanderwaart, Spaces of polygons in the plane and Morse theory, American Math. Month. 112 (2005), 289–310.