48 research outputs found
Topology Explains Why Automobile Sunshades Fold Oddly
We use braids and linking number to explain why automobile shades fold into
an odd number of loops.Comment: 8 pages, 9 figure
An Algorithm for Detecting Intrinsically Knotted Graphs
We describe an algorithm that recognizes some (perhaps all) intrinsically
knotted (IK) graphs, and can help find knotless embeddings for graphs that are
not IK. The algorithm, implemented as a Mathematica program, has already been
used by Goldberg, Mattman, and Naimi [6] to greatly expand the list of known
minor minimal IK graphs, and to find knotless embeddings for some graphs that
had previously resisted attempts to classify them as IK or non-IK.Comment: 9 pages, 4 figure
Linear embeddings of are triple linked
We use the theory of oriented matroids to show that any linear embedding of
, the complete graph on nine vertices, contains a non-split link with
three components.Comment: 7 pages, 1 figure. An updated Mathematica program and five files
containing evidence that the program works correctly are available as
ancillary files associated with this articl
On the number of links in a linearly embedded
We show there exists a linear embedding of with n nontrivial
2-component links if and only if n = 1, 2, 3, 4, or 5.Comment: 20 pages, 6 figure
List Coloring and -monophilic graphs
In 1990, Kostochka and Sidorenko proposed studying the smallest number of
list-colorings of a graph among all assignments of lists of a given size
to its vertices. We say a graph is -monophilic if this number is
minimized when identical -color lists are assigned to all vertices of .
Kostochka and Sidorenko observed that all chordal graphs are -monophilic for
all . Donner (1992) showed that every graph is -monophilic for all
sufficiently large . We prove that all cycles are -monophilic for all
; we give a complete characterization of 2-monophilic graphs (which turns
out to be similar to the characterization of 2-choosable graphs given by Erdos,
Rubin, and Taylor in 1980); and for every we construct a graph that is
-choosable but not -monophilic
On intrinsically knotted and linked graphs
We give a brief survey of some known results on intrinsically linked or
knotted graphs.Comment: 8 pages, 3 figure
Escher squares and lattice links
We give a shorter and simpler proof of the result of [2], which gives a
necessary and sufficient condition for when a lattice diagram is the projection
of a lattice link.Comment: 4 pages, 3 figure
Deleting an edge of a 3-cycle in an intrinsically knotted graph gives an intrinsically linked graph
We show that deleting an edge of a 3-cycle in an intrinsically knotted graph
gives an intrinsically linked graph.Comment: 7 pages, 2 figure
Intrinsic linking and knotting are arbitrarily complex in directed graphs
Fleming and Foisy recently proved the existence of a digraph whose every
embedding contains a -component link, and left open the possibility that a
directed graph with an intrinsic -component link might exist. We show that,
indeed, this is the case. In fact, much as Flapan, Mellor, and Naimi show for
graphs, knotting and linking are arbitrarily complex in directed graphs.
Specifically, we prove the analog for digraphs of the main theorem of their
paper: for any and , every embedding of a sufficiently large
complete digraph in contains an oriented link with components
such that, for every , and , where denotes the second
coefficient of the Conway polynomial of .Comment: 9 pages, 2 figure
Classification of topological symmetry groups of
In this paper we complete the classification of topological symmetry groups
for complete graphs by characterizing which can have a cyclic
group, a dihedral group, or a subgroup of where is odd, as
its topological symmetry group.Comment: 19 pages; v2 lists authors correctly on arXiv; v3 substantially
revises the introductio