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 K9K_9 are triple linked

We use the theory of oriented matroids to show that any linear embedding of $K_9$, 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 K3,3,1K_{3,3,1}

We show there exists a linear embedding of $K_{3,3,1}$ 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 nn-monophilic graphs

In 1990, Kostochka and Sidorenko proposed studying the smallest number of list-colorings of a graph $G$ among all assignments of lists of a given size $n$ to its vertices. We say a graph $G$ is $n$-monophilic if this number is minimized when identical $n$-color lists are assigned to all vertices of $G$. Kostochka and Sidorenko observed that all chordal graphs are $n$-monophilic for all $n$. Donner (1992) showed that every graph is $n$-monophilic for all sufficiently large $n$. We prove that all cycles are $n$-monophilic for all $n$; 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 $n$ we construct a graph that is $n$-choosable but not $n$-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 $4$-component link, and left open the possibility that a directed graph with an intrinsic $n$-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 $n$ and $\alpha$, every embedding of a sufficiently large complete digraph in $\mathbb{R}^3$ contains an oriented link with components $Q_1, \ldots, Q_n$ such that, for every $i \neq j$, $|\mathrm{lk}(Q_i,Q_j)| \geq \alpha$ and $|a_2(Q_i)| \geq \alpha$, where $a_2(Q_i)$ denotes the second coefficient of the Conway polynomial of $Q_i$.Comment: 9 pages, 2 figure

Classification of topological symmetry groups of KnK_n

In this paper we complete the classification of topological symmetry groups for complete graphs $K_n$ by characterizing which $K_n$ can have a cyclic group, a dihedral group, or a subgroup of $D_m \times D_m$ where $m$ is odd, as its topological symmetry group.Comment: 19 pages; v2 lists authors correctly on arXiv; v3 substantially revises the introductio