Joel Foisy

INTRINSICALLY n-LINKED GRAPHS

For every natural number n, we exhibit a graph with the property that every embedding of it in ℝ3 contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in ℝ3 that contains no non-split n-component link.

SOME RESULTS ON INTRINSICALLY KNOTTED GRAPHS

We show that graphs of the form G * K2 are intrinsically knotted if and only if G is nonplanar. This can be extended to show that G * K5m+1 is intrinsically (m + 2)-linked when G is nonplanar. We also apply this result to classify all complete n-partite graphs with respect to intrinsic knotting and show that this family does not produce any new minor-minimal examples. Finally, we categorize all minor-minimal intrinsically knotted graphs on 8 or fewer vertices.

SOME NEW INTRINSICALLY 3-LINKED GRAPHS

In [2], it was shown that every spatial embedding of K10, the complete graph on ten vertices, contains a non-split 3-component link (K10 is intrinsically 3-linked). We improve this result by exhibiting two different subgraphs of K10 that also have this property. In addition, we also exhibit several families of graphs that are intrinsically 3-linked.

GRAPHS WITH DISJOINT LINKS IN EVERY SPATIAL EMBEDDING

We exhibit a graph, G12, that in every spatial embedding has a pair of non-splittable 2 component links sharing no vertices or edges. Surprisingly, G12 does not contain two disjoint copies of graphs known to have non-splittable links in every embedding. We exhibit other graphs with this property that cannot be obtained from G12 by a finite sequence of Δ-Y and/or Y-Δ exchanges. We prove that G12 is minor minimal in the sense that every minor of it has a spatial embedding that does not contain a pair of non-splittable 2 component links sharing no vertices or edges.

Intrinsically linked graphs in projective space

We examine graphs that contain a non-trivial link in every embedding into real projective space, using a weaker notion of unlink than was used in [5]. We call such graphs intrinsically linked in RP 3 . We fully characterize such graphs with connectivity 0,1 and 2. We also show that only one Petersen-family graph is intrinsically linked in RP 3 and prove that K7 minus any two edges is also minor-minimal intrinsically linked. In all, 594 graphs are shown to be minor-minimal intrinsically linked in RP 3 .

GRAPHS WITH A KNOT OR 3-COMPONENT LINK IN EVERY SPATIAL EMBEDDING

We describe a family of graphs that contain a knot or a non-split 3-component link in every spatial embedding. We exhibit a graph in this family that has a knotless embedding, and a 3-component linkless embedding.

Intrinsically triple-linked graphs in ℝP3

Flapan, et al [8] showed that every spatial embedding of K10, the complete graph on ten vertices, contains a non-split three-component link (K10 is intrinsically triple-linked). The papers [2] and [7] extended the list of known intrinsically triple-linked graphs in R 3 to include several other families of graphs. In this paper, we will show that while some of these graphs can be embedded 3-linklessly in RP 3 , K10 is intrinsically triple-linked in RP 3 .

Intrinsically linked signed graphs in projective space

Abstract We define a signed embedding of a signed graph into real projective space to be an embedding such that an embedded cycle is 0-homologous if and only if it is balanced. We characterize signed graphs that have a linkless signed embedding. In particular, we exhibit 46 graphs that form the complete minor-minimal set of signed graphs that contain a non-split link for every signed embedding. With one trivial exception, these graphs are derived from different signings of the seven Petersen family graphs.