arXiv subjects

Kenji Kozai

Publications and source records attributed to Kenji Kozai.

6 recordsLinked to original sources

arXiv2023

Linking number of monotonic cycles in random book embeddings of complete graphs

A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining "interior" edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of graphs. In particular, we show that the distribution of linking numbers can be described in terms of Eulerian numbers. Consequently, the mean of the squared linking number over all random embeddings is $\frac{i}{6}$, where $i$ is the number of interior edges in the cycles. We also show that the mean of the squared linking number over all pairs of $n$-cycles in $K_{2n}$ grows linearly in $n$.

math.GT↗

arXiv2021

Deformations of reducible SL(n,C) representations of fibered 3-manifold groups

Let $M_ϕ$ be a surface bundle over a circle with monodromy $ϕ:S \rightarrow S$. We study deformations of certain reducible representations of $π_1(M_ϕ)$ into $\text{SL}(n,\mathbb{C})$, obtained by composing a reducible representation into $\text{SL}(2,\mathbb{C})$ with the irreducible representation $\text{SL}(2,\mathbb{C}) \rightarrow \text{SL}(n,\mathbb{C})$. In particular, we show that under certain conditions on the eigenvalues of $ϕ^*$, the reducible representation is contained in a $(n+1+k)(n-1)$ dimensional component of the representation variety, where $k$ is the number of components of $\partial M_ϕ$. This result applies to mapping tori of pseudo-Anosov maps with orientable invariant foliations whenever 1 is not an eigenvalue of the induced map on homology, where the reducible representation is also a limit of irreducible representations.

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arXiv2021

Stick number of non-paneled knotless spatial graphs

We show that the minimum number of sticks required to construct a non-paneled knotless embedding of $K_4$ is 9 and of $K_5$ is 12 or 13. We use our results about $K_4$ to show that the probability that a random linear embedding of $K_{3,3}$ in a cube is in the form of a Möbius ladder is $0.97380\pm 0.00003$, and offer this as a possible explanation for why $K_{3,3}$ subgraphs of metalloproteins occur primarily in this form.

math.GT↗

arXiv2016

Linking number and writhe in random linear embeddings of graphs

In order to model entanglements of polymers in a confined region, we consider the linking numbers and writhes of cycles in random linear embeddings of complete graphs in a cube. Our main results are that for a random linear embedding of $K_n$ in a cube, the mean sum of squared linking numbers and the mean sum of squared writhes are of the order of $θ(n(n!))$. We obtain a similar result for the mean sum of squared linking numbers in linear embeddings of graphs on $n$ vertices, such that for any pair of vertices, the probability that they are connected by an edge is $p$. We also obtain experimental results about the distribution of linking numbers for random linear embeddings of these graphs. Finally, we estimate the probability of specific linking configurations occurring in random linear embeddings of the graphs $K_6$ and $K_{3,3,1}$.

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arXiv2015

Hyperbolic structures from Sol on pseudo-Anosov mapping tori

The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov $ϕ:S\rightarrow S$ has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on the closed surface, then the Sol structure can be deformed to nearby cone hyperbolic structures, in the sense of projective structures. The cone angles can be chosen to be decreasing from multiples of $2π$.

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arXiv2008

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 by Flapan, et al. We call such graphs intrinsically linked in projective space. We fully characterize such graphs with connectivity 0,1 and 2. We also show that only one Petersen-family graph is intrinsically linked in projective space 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 projective space.

math.GT↗