2009/06/11 by Erica Flapan, Flapan, Erica, Hugh Howards +1
Computer Science · Mathematics · #57M99 #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT #msc:57M99
paper · pdf · doi:10.48550/arxiv.0906.2229
12 pages 4 figures
arxiv created 2009/06/11 · openalex publication_date 2009/06/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In contrast with knots, whose properties depend only on their extrinsic topology in S3, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in S3 . For example, it was shown in [2] that every embedding of the complete graph K7 in S3 contains a non-trivial knot. Later in it was shown that for every m ∈ N, there is a complete graph Kn such that every embedding of Kn in S3 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 S3. We prove here the contrasting result that every graph has an embedding in S3 such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in S3 which contains no composite or satellite knots.