新國 亮

We say that a set of pairs of disjoint cycles [Formula: see text] of a graph [Formula: see text] is linked if for any spatial embedding [Formula: see text] of [Formula: see text] there exists an element [Formula: see text] of [Formula: see text] such that the [Formula: see text]-component link [Formula: see text] is nonsplittable, and also say minimally linked if none of its proper subsets are linked. In this paper, (1) we show that the set of all pairs of disjoint cycles of [Formula: see text] is minimally linked if and only if [Formula: see text] is essentially same as a graph in the Petersen family, and (2) for any two integers [Formula: see text], we exhibit a minimally linked set of Hamiltonian [Formula: see text]-pairs of cycles of the complete graph [Formula: see text] with at most 18 elements.

The Jones polynomial V-L(t) for an oriented link L is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer n >= 3, we show that: (1) the difference of Jones polynomials for two oriented links which are C-n-equivalent is divisible by (t - 1)(n) (t(2) + t + 1)(t(2) + 1), and (2) there exists a pair of two oriented knots which are C-n-equivalent such that the difference of the Jones polynomials for them equals (t - 1)(n) (t(2) + t + 1)(t(2) + 1).

This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider spatial graphs in S3 as well as in other 3-manifolds.

A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to the first homology group of each of the connected components. This also leads a kind of homotopy classification of 2-component handlebody-links. (C) 2015 Elsevier B.V. All rights reserved.

We introduce invariants of graphs embedded in S-3 which are related to the Wu invariant and the Simon invariant. Then we use our invariants to prove that certain graphs are intrinsically chiral, and to obtain lower bounds for the minimal crossing number of particular embeddings of graphs in S-3.

We give a Conway-Gordon type formula for invariants of knots and links in a spatial complete four-partite graph K-3,K-3,K-1,K-1 in terms of the square of the linking number and the second coefficient of the Conway polynomial. As an application, we show that every rectilinear spatial K-3,K-3,K-1,K-1 contains a nontrivial Hamiltonian knot.

For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of the Conway polynomial.

Conway-Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this paper, we give a Conway-Gordon type theorem for any graph which is obtained from the complete graph on six or seven vertices by a finite sequence of Delta Y-exchanges.

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link each of whose 2-component sublinks is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of ΔY-exchanges and YΔ-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph. © 2011 by Pacific Journal of Mathematics.

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper, we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot.

We give an explicit calculation of the Wu invariants for immersions of a finite graph into the plane and classify all generic immersions of a graph into the plane up to regular homotopy by the Wu invariant. This result is a generalization of the fact that two plane curves are regularly homotopic if and only if they have the same rotation number.

Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split links. In this paper, we construct some new edge (resp. vertex)-homotopy invariants of spatial graphs without any restriction of linking numbers of the constituent 2-component links by applying the generalized Sato-Levine invariant.

In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this article, we give integral lifts of the Conway-Gordon theorems above in terms of the square of the linking number and the second coefficient of the Conway polynomial. As applications, we give alternative topological proofs of theorems of Brown-Ramirez Alfonsin and Huh-Jeon for rectilinear spatial complete graphs which were proved by computational and combinatorial methods. (C) 2009 Elsevier B.V. All rights reserved.

Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples of non-splittable spatial graphs up to edge (resp. vertex)-homotopy, all of whose constituent links are link-homotopically trivial.

We say that a graph is intrinsically nontrivial if every spatial embedding of the graph contains a nontrivial spatial subgraph. We prove that an intrinsically nontrivial graph is intrinsically linked, namely every spatial embedding of the graph contains a nonsplittable 2-component link. We also show that there exists a graph such that every spatial embedding of the graph contains either a nonsplittable 3-component link or an irreducible spatial handcuff graph whose constituent 2-component link is split.

An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3 + 3 vertices in detail, and determine necessary conditions on linking numbers and Simon invariants for such links and spatial graphs to be symmetric.

Two spatial embeddings of a graph are said to be delta (resp. sharp) edge-homotopic if they are transformed into each other by self delta (resp. sharp) moves and ambient isotopies. We show that any two spatial embeddings of a graph are delta (resp. sharp) edge-homotopic if and only if the graph does not contain a subgraph which is homeomorphic to the theta graph or the disjoint union of two 1-spheres, or equivalently G is homeomorphic to a bouquet. (c) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

A generic immersion of a finite graph into the 2-space with p double points is said to be completely distinguishable if any two of the 2(P) embeddings of the graph into the 3-space obtained from the immersion by giving over/under information to each double point are not ambient isotopic in the 3-space. We show that only non-trivializable graphs and non-planar graphs have a non-trivial completely distinguishable immersion. We give examples of non-trivial completely distinguishable immersions of several non-trivializable graphs, the complete graph on n vertices and the complete bipartite graph on m + n vertices.

A spatial embedding of a graph is called a partial derivative-spatial embedding if all knots in the embedding bound Seifert surfaces simultaneously such that the interiors of the surfaces are mutually disjoint and disjoint from the image of the embedding. This is a generalization of the boundary link. In this paper, we give a complete characterization of a graph which has a partial derivative-spatial embedding and we classify partial derivative-spatial embeddings completely up to self pass-moves and ambient isotopies. In particular, any partial derivative-spatial embedding of a graph is trivial up to edge-homotopy.

A planar graph is said to be trivializable if every regular projection of the graph produces a trivial spatial embedding by giving some over/under informations to the double points. Every minor of a trivializable graph is also trivializable, thus the set of forbidden graphs is finite. Seven forbidden graphs for the trivializability were previously known. In this paper, we exhibit nine more forbidden graphs.

Two spatial embeddings of a graph are said to be delta edge-homotopic if they axe transformed into each other by self delta moves and ambient isotopies. In this paper we classify theta curves up to delta edge-homotopy in terms of the third coefficient of the Conway polynomial of an associated 2-component link. In particular, we show that every boundary theta curve is delta edge-homotopically trivial, and two cobordant theta curves are delta edge-homotopic.

We investigate how a self-delta move, which is a delta move on the same component, influences the HOMFLY polynomial of a link. Then we reveal some relationships among finite type invariants, which are coming from the derivatives of the Jones polynomials and the first HOMFLY coefficient polynomials, of the four links involving in a self-delta move. (C) 2004 Published by Elsevier B.V.

Two spatial embeddings of a graph are said to be edge-homotopic if they are transformed into each other by self-crossing changes and ambient isotopies. We show that two spatial embeddings of the complete graph on four vertices are edge-homotopic if and only if they have the same alpha-invariant.

We characterize the knot types in a spatial embedding of a graph which is obtained from the 5-cycle by duplicating each edge. It is characterized in terms of the second coefficients of the Conway polynomials and the third derivatives at 1 of the Jones polynomials of knots.

In this paper our main object is the graph embedded into the 3-space, called the spatial graph. We study a clasp-pass equivalence on spatial graphs, which is an equivalence relation generated by clasp-pass moves and ambient isotopies. Our approach is an analogy of the delta equivalence classification on spatial graphs, where a delta equivalence is an equivalence relation generated by delta moves and ambient isotopies and implied by a clasp-pass equivalence. Consequently, clasp-pass classifications on spatial embeddings of several non-planar graphs and a specified planar graph are given. This is a preliminary report on our recent work and details will appear elsewhere.

Let L(G) be the second skew-symmetric cohomology group of the residual space of a graph G. We determine L(G) in the case G is a 3-connected simple graph, and give the structure of L(G) in the case G is a complete graph, and a complete bipartite graph. By using these results, we determine the Wu. invariants in L(G) of the spatial embeddings of the complete graph and those of the complete bipartite graph, respectively. Since the Wu invariant of a spatial embedding is a complete invariant up to homology which is an equivalence relation on spatial embeddings introduced in [12], we give a homology classification of the spatial embeddings of such graphs.