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Full-Text Articles in Mathematics
Counting Links And Knots In Complete Graphs, Loren Abrams, Blake Mellor, Lowell Trott
Counting Links And Knots In Complete Graphs, Loren Abrams, Blake Mellor, Lowell Trott
Mathematics Faculty Works
We investigate the minimal number of links and knots in embeddings of complete partite graphs in S3. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of K4,4,1 is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.
Intrinsic Linking And Knotting Are Arbitrarily Complex, Erica Flapan, Blake Mellor, Ramin Naimi
Intrinsic Linking And Knotting Are Arbitrarily Complex, Erica Flapan, Blake Mellor, Ramin Naimi
Mathematics Faculty Works
We show that, given any n and α, every embedding of any sufficiently large complete graph in R3 contains an oriented link with components Q1, ..., Qn such that for every i≠j, $|\lk(Q_i,Q_j)|\geq\alpha$ and |a2(Qi)|≥α, where a2(Qi) denotes the second coefficient of the Conway polynomial of Qi.
Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor
Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor
Mathematics Faculty Works
We prove that a graph is intrinsically linked in an arbitrary 3–manifold MM if and only if it is intrinsically linked in S3. Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S3.