Spatial graphs with local knots

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 and it 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. Response to Reviewers: See attachment. SPATIAL GRAPHS WITH LOCAL KNOTS ERICA FLAPAN, BLAKE MELLOR, AND RAMIN NAIMI Abstract. It is shown that for any locally knotted edge of a 3-connected graph in S, there is a ball that contains all of the local knots of that It is shown that for any locally knotted edge of a 3-connected graph in S, 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. 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 S3 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.