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Register a new userCapturing links in spatial complete graphs
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Ryo Nikkuni
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We say that a set of pairs of disjoint cycles $Lambda(G)$ of a graph $G$ is linked if for any spatial embedding $f$ of $G$ there exists an element $lambda$ of $Lambda(G)$ such that the $2$-component link $f(lambda)$ is nonsplittable, and also say minimally linked if none of its proper subsets are linked. In this paper, we show that: (1) the set of all pairs of disjoint cycles of $G$ is minimally linked if and only if $G$ is essentially same as a graph in the Petersen family, and (2) for any two integers $p,qge 3$, we exhibit a minimally linked set of Hamiltonian $(p,q)$-pairs of cycles of the complete graph $K_{p+q}$ with at most eighteen elements.
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There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzukis theta-curves and show that the invariants are nontrivial for Suzukis theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on abelianizations and calculate the invariant of the handlebody-knots up to six crossings. For surface-links, we correct Yoshikawas table and calculate the invariants of the surface-links in the table.
We introduce a new equivalence relation on decorated ribbon graphs, and show that its equivalence classes directly correspond to virtual links. We demonstrate how this correspondence can be used to convert any invariant of virtual links into an invariant of ribbon graphs, and vice versa.
This is a short review article on invariants of spatial graphs, written for A Concise Encyclopedia of Knot Theory (ed. Adams et. al.). The emphasis is on combinatorial and polynomial invariants of spatial graphs, including the Alexander polynomial, the fundamental quandle of a graph, and the Yamada polynomial.
The symmetries of complex molecular structures can be modeled by the {em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to $A_4$, $S_4$ or $A_5$; in this paper we determine which complete bipartite graphs have an embedding in $S^3$ whose topological symmetry group is isomorphic to $mathbb{Z}_m$, $D_m$, $mathbb{Z}_r times mathbb{Z}_s$ or $(mathbb{Z}_r times mathbb{Z}_s) ltimes mathbb{Z}_2$.
A {em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and $p$-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which $p$ the graph is $p$-colorable, and that a $p$-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group $Gamma(p,m,k)$. We finish by proving some properties of the Alexander polynomial.
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