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For every natural number n, we exhibit a graph with the property that every embedding of it in M 3 contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in M 3 that contains no non-split n-component link. 1.1991 Mathematics Subject Classification. 57M25, 57M15.

Abstract. The topological symmetry group of a graph embedded in the 3-sphere is the group consisting of those automorphisms of the graph which are induced by some homeomorphism of the ambient space. We prove strong restrictions on the groups that can occur as the topological symmetry group of some embedded graph. In addition, we characterize the orientation preserving topological symmetry groups of embedded 3-connected graphs in the 3-sphere. Mathematics Subject Classification (2000). 05C10, 57M15; 05C25, 57M25, 57N10.

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In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product, we assign a rational number μ ( σ ) \mu (\sigma ) to each rational polyhedral cone σ \sigma in the lattice, such that for any toric variety X X with fan Σ \Sigma in the lattice, we have \[ Td ⁡ ( X ) = ∑ σ ∈ Σ μ ( σ ) [ V ( σ ) ] . \operatorname {Td}(X)=\sum _{\sigma \in \Sigma } \mu (\sigma ) [V(\sigma )]. \] This constitutes an improved answer to an old question of Danilov. In a similar way, beginning from the choice of a complete flag in the lattice, we obtain the cycle Todd classes constructed by Morelli. Our construction is based on an intersection product on cycles of a simplicial toric variety developed by the second author. Important properties of the construction are established by showing a connection to the canonical representation of the Todd class of a simplicial toric variety as a product of torus-invariant divisors developed by the first author.