Abstract. The symmetries of complex molecular structures can be modeled by the 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 [7]; 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 A4, S4 or A5 [12]; in this paper we determine which complete bipartite graphs have an embedding in S 3 whose topological symmetry group is isomorphic to Zm, Dm, Zr × Zs or (Zr × Zs) ⋉ Z2.

For a positive integer n ≥ 3, the collection of n-sided polygons embedded in 3-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded n-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components correspond to equilateral knot types. When n ≤ 5, the space of equilateral knots is connected. Therefore, we examine the space of equilateral hexagons. Using techniques from symplectic geometry, we can parametrize the space of equilateral hexagons with a set of measure preserving action-angle coordinates. With this coordinate system, we provide new bounds on the knotting probability of equilateral hexagons.

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