We list more than 200 new examples of minor minimal intrinsically knotted graphs and describe many more that are intrinsically knotted and likely minor minimal.Comment: 19 pages, 16 figures, Appendi

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.

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Abstract. We show that, given any n and α, any embedding of any sufficiently large complete graph in R 3 contains an oriented link with components Q1, . . . , Qn such that for every i = j, |lk(Qi, Qj)| ≥ α and |a2(Qi)| ≥ α, where a2(Qi) denotes the second coefficient of the Conway polynomial of Qi.1. Introduction. The study of embeddings of graphs in R 3 is a natural extension of knot theory. However, in contrast with knots, whose properties depend only on their extrinsic topology, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in R 3 . Conway and Gordon [1] obtained groundbreaking results of this nature by showing that every embedding of the complete graph K 6 in R 3 contains a non-trivial link and every embedding of K 7 in R 3 contains a non-trivial knot. Because this type of linking and knotting is intrinsic to the graph itself rather than depending on the particular embedding of the graph in R 3 , K 6 is said to be intrinsically linked and K 7 is said to be intrinsically knotted . On the other hand, Conway and Gordon [1] illustrated an embedding of K 6 such that the only non-trivial link L 1 ∪ L 2 contained in K 6 is the Hopf link (which has |lk(L 1 , L 2 )| = 1); and they illustrated an embedding of K 7 such that the only non-trivial knot Q contained in K 7 is the trefoil knot (which has |a 2 (Q)| = 1, where a 2 (Q) denotes the second coefficient of the Conway polynomial of Q). In this sense, we see that K 6 exhibits the simplest type of intrinsic linking and K 7 exhibits the simplest type of intrinsic knotting.More recently, it has been shown that for sufficiently large values of r, the complete graph K r exhibits more complex types of intrinsic linking and knotting. In particular, Flapan [2] showed that for every λ ∈ N, there is a

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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We determine for which m the complete graph K m has an embedding in S 3 whose topological symmetry group is isomorphic to one of the polyhedral groups A 4 , A 5 or S 4 .57M15, 57M25; 05C10

Abstract. It is shown that for any locally knotted edge of a 3-connected graph in S 3 , there is a ball that contains all of the local knots of that edge which is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S 3 .Schubert's 1949 result [9] that every non-trivial knot can be uniquely factored into prime knots is a fundamental result in knot theory. Hashizume [6], extended Schubert's result to links in 1958. Then in 1987, Suzuki [12] generalized Schubert's result to spatial graphs by proving that every connected graph embedded in S 3 can be split along spheres meeting the graph in 1 or 2 points to obtain a unique collection of prime embedded graphs together with some trivial graphs.Although the set of prime factors of a knot or embedded graph is unique up to equivalence, the set of splitting spheres is generally not unique up to an isotopy setwise fixing the knot or graph. For example, consider the embedding of the complete graph K 6 which is illustrated on both the left and right sides of Figure 1. The edge e = 14 contains two trefoil knots. The spheres T 1 and T 2 (illustrated on the left) are splitting spheres for these two knots. However, one of the balls bounded by F (illustrated on the right) meets e in an arc whose union with an arc in F is a single trefoil knot. Thus F is also a splitting sphere for one of the two local knots in e. However, F is not isotopic (fixing the embedded graph setwise) to either of the spheres T 1 or T 2 .By contrast, in this paper we show that for any locally knotted edge of an embedded 3-connected graph, there is a ball meeting the graph in an arc containing all of the local knots of that edge which is unique up to an isotopy fixing the graph. We call such a ball an unknotting ball for that edge. Our main theorem is the following.