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Mathematics (Pomona)

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knots, non-hyperbolic knot


In contrast with knots, whose properties depend only on their extrinsic topology in , there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in . For example, it was shown by Conway and Gordon that every embedding of the complete graph K_7 in contains a non-trivial knot. Later it was shown that for every m ∈ N there is a complete graph K_n such that every embedding of K_n in contains a knot Q whose minimal crossing number is at least m. Thus there are arbitrarily complicated knots in every embedding of a sufficiently large complete graph in . We prove the contrasting result that every graph has an embedding in such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in which contains no composite or satellite knots.

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