# American Institute of Mathematical Sciences

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

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

 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

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 $S^3$.
Citation: Dwayne Chambers, Erica Flapan, John D. O'Brien. Topological symmetry groups of $K_{4r+3}$. Discrete & 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.  Google Scholar [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.  Google Scholar [3] E. Flapan, B. Mellor and R. Naimi, Spatial graphs with local knots,, \arXiv{1010.0479}., ().   Google Scholar [4] E. Flapan, B. Mellor and R. Naimi, Complete graphs whose topological symmetry groups are polyhedral,, \arXiv{1008.1095}., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [7] P. A. Smith, Transformations of finite period II, Annals of Math., 40 (1939), 690-711. doi: 10.2307/1968950.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [3] E. Flapan, B. Mellor and R. Naimi, Spatial graphs with local knots,, \arXiv{1010.0479}., ().   Google Scholar [4] E. Flapan, B. Mellor and R. Naimi, Complete graphs whose topological symmetry groups are polyhedral,, \arXiv{1008.1095}., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [7] P. A. Smith, Transformations of finite period II, Annals of Math., 40 (1939), 690-711. doi: 10.2307/1968950.  Google Scholar
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