91 research outputs found
Intrinsically triple-linked graphs in RP^3
Flapan--Naimi--Pommersheim showed that every spatial embedding of ,
the complete graph on ten vertices, contains a non-split three-component link;
that is, is intrinsically triple-linked in . The work of
Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known
intrinsically triple-linked graphs in 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 , is intrinsically
triple-linked in .Comment: 23 pages, 6 figures; v2: revised introduction, minor corrections, new
outlines to longer proof
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^3. Also, assuming the Poincare
Conjecture, we prove that a graph is intrinsically knotted in M if and only if
it is intrinsically knotted in S^3.Comment: This is the version published by Algebraic & Geometric Topology on 9
August 200
Combinatorics of embeddings
We offer the following explanation of the statement of the Kuratowski graph
planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas
intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to
every cell there corresponds a unique cell with the complementary set of
vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere;
there exist precisely two 3-dimensional dichotomial cell complexes, and their
1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their
1-skeleta all but one graphs of the Petersen family.
In higher dimensions n>2, we observe that in order to characterize those
compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited
minors", it suffices to establish finiteness of the list of all (n-1)-connected
n-dimensional finite cell complexes that do not embed in S^{2n} yet all their
proper subcomplexes and proper cell-like combinatorial quotients embed there.
Our main result is that this list contains the n-skeleta of (2n+1)-dimensional
dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell
complexes include (apart from the three joins of the i-skeleta of
(2i+2)-simplices) at least ten non-simplicial complexes.Comment: 49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on
transforms of dichotomial spheres reworked to include more details;
subsection 2.D "Algorithmic issues" added, etc
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