Intrinsic Knotting and Linking of Complete Graphs

Intrinsic Linking and Knotting of Graphs

An analog to intrinsic linking, intrinsic even linking, is explored in the first half of this paper. Four graphs are established to be minor minimal intrinsically even linked, and it is conjectured that they form a complete minor minimal set. Some characterizations are given, using the simplest of the four graphs as an integral part of the arguments, that may be useful in proving the conjecture...

Intrinsic Linking and Knotting in Virtual Spatial Graphs

We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, ...

Intrinsic Linking and Knotting of Graphs in Arbitrary 3â•fimanifolds

Intrinsic Linking and Knotting of Graphs in Arbitrary 3–manifolds

We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S. Also, assuming the Poincaré Conjecture, a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S.

Intrinsic Linking and Knotting Are Arbitrarily Complex

We show that, given any n and α, every embedding of any sufficiently large complete graph in R contains an oriented link with components Q1, . . . , Qn such that for every i 6= j, |lk(Qi, Qj)| ≥ α and |a2(Qi)| ≥ α, where a2(Qi) denotes the second coefficient of the Conway polynomial of Qi.