Topological symmetry groups of K_{4r+3}

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December 2011, 4(6): 1401-1411. doi: 10.3934/dcdss.2011.4.1401

Topological symmetry groups of $K_{4r+3}$

Dwayne Chambers ^1, , Erica Flapan ^2, and John D. O'Brien ^3,

1.	Department of Mathematics, Claremont Graduate University, Claremont, CA 91711, United States
2.	Department of Mathematics, Pomona College, Claremont, CA 91711, United States
3.	Centre for Genomics and Global Health, Oxford University, Oxford OX3 7BN, United Kingdom

Received February 2009 Revised October 2009 Published December 2010

Abstract
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We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of a complete graph of the form $K_{4r+3}$ in $S^3$.

Keywords: Topological symmetry groups, molecular symmetries, complete graphs..

Mathematics Subject Classification: 57M25, 05C1.

Citation: Dwayne Chambers, Erica Flapan, John D. O'Brien. Topological symmetry groups of $K_{4r+3}$. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1401-1411. doi: 10.3934/dcdss.2011.4.1401

References:

[1]	M. Boileau, B. Leeb and J. Porti, Geometrization of $3$-dimensional orbifolds, Ann. of Math., 162 (2005), 195-290. doi: 10.4007/annals.2005.162.195.
[2]	E. Flapan, Rigidity of graph symmetries in the $3$-sphere, Journal of Knot Theory and its Ramifications, 4 (1995), 373-388. doi: 10.1142/S0218216595000181.
[3]	E. Flapan, B. Mellor and R. Naimi, Spatial graphs with local knots,, \arXiv{1010.0479}., ().
[4]	E. Flapan, B. Mellor and R. Naimi, Complete graphs whose topological symmetry groups are polyhedral,, \arXiv{1008.1095}., ().
[5]	E. Flapan, R. Naimi, J. Pommersheim and H. Tamvakis, Topological symmetry groups of embedded graphs in the $3$-sphere, Commentarii Mathematici Helvetici, 80 (2005), 317-354. doi: 10.4171/CMH/16.
[6]	E. Flapan, R. Naimi and H. Tamvakis, Topological symmetry groups of complete graphs in the $3$-sphere, Journal of the London Mathematical Society, 73 (2006), 237-251. doi: 10.1112/S0024610705022490.
[7]	P. A. Smith, Transformations of finite period II, Annals of Math., 40 (1939), 690-711. doi: 10.2307/1968950.

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References:

[1]	M. Boileau, B. Leeb and J. Porti, Geometrization of $3$-dimensional orbifolds, Ann. of Math., 162 (2005), 195-290. doi: 10.4007/annals.2005.162.195.
[2]	E. Flapan, Rigidity of graph symmetries in the $3$-sphere, Journal of Knot Theory and its Ramifications, 4 (1995), 373-388. doi: 10.1142/S0218216595000181.
[3]	E. Flapan, B. Mellor and R. Naimi, Spatial graphs with local knots,, \arXiv{1010.0479}., ().
[4]	E. Flapan, B. Mellor and R. Naimi, Complete graphs whose topological symmetry groups are polyhedral,, \arXiv{1008.1095}., ().
[5]	E. Flapan, R. Naimi, J. Pommersheim and H. Tamvakis, Topological symmetry groups of embedded graphs in the $3$-sphere, Commentarii Mathematici Helvetici, 80 (2005), 317-354. doi: 10.4171/CMH/16.
[6]	E. Flapan, R. Naimi and H. Tamvakis, Topological symmetry groups of complete graphs in the $3$-sphere, Journal of the London Mathematical Society, 73 (2006), 237-251. doi: 10.1112/S0024610705022490.
[7]	P. A. Smith, Transformations of finite period II, Annals of Math., 40 (1939), 690-711. doi: 10.2307/1968950.

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