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تسجيل مستخدم جديدTopological Symmetry Groups of Complete Bipartite Graphs
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The symmetries of complex molecular structures can be modeled by the {em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to $A_4$, $S_4$ or $A_5$; in this paper we determine which complete bipartite graphs have an embedding in $S^3$ whose topological symmetry group is isomorphic to $mathbb{Z}_m$, $D_m$, $mathbb{Z}_r times mathbb{Z}_s$ or $(mathbb{Z}_r times mathbb{Z}_s) ltimes mathbb{Z}_2$.
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A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassilievs theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresp
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