An n-string tangle is a pair (B, A) such that A is a disjoint union of properly embedded n arcs in a topological 3-ball B. And an n-string tangle is said to be trivial (or rational) a, if it is homeomorphic to (D x I, {x(1),..., x(n)} x I) as a pair, where D is a 2-disk, I is the unit interval and each x(i) is a point in the interior of D. A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an n-string stick tangle its stick-order is defined to be a nonincreasing sequence (s(1), s(2),..., s(n)) of natural numbers such that, under an ordering of the arcs of the tangle, each si denotes the number of sticks constituting the ith arc of the tangle. And a stick-order S is said to be trivial, if every stick tangle of the order S is trivial.
In this paper, restricting the 3-ball B to be the standard 3-ball, we give the complete list of trivial stick-orders.

Strong and weak (1, 3) homotopies are equivalence relations on knot projections, defined by the first flat Reidemeister move and each of two different types of the third flat Reidemeister moves. In this paper, we introduce the cross chord number that is the minimal number of double points of chords of a chord diagram. Cross chord numbers induce a strong (1, 3) invariant. We show that Hanaki's trivializing number is a weak (1, 3) invariant. We give a complete classification of knot projections having trivializing number two up to the first flat Reidemeister moves using cross chord numbers and the positive resolutions of double points. Two knot projections with trivializing number two are both weak (1, 3) homotopy equivalent and strong (1, 3) homotopy equivalent if and only if they can be related by only the first flat Reidemeister moves. Finally, we determine the strong (1, 3) homotopy equivalence class containing the trivial knot projection and other classes of knot projections.

By a fixed continuous map from a 3-space to itself, a knot in the 3-space may be mapped to another knot in the 3-space. We analyze possible knot types of them. Then we map a knot repeatedly by a fixed continuous map and analyze possible infinite sequences of knot types.

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.

We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another. Based on this multiplicity, we define a pseudo distance on the class of objects. We define and study several multiplicities in the category of topological spaces and continuous maps, the category of groups and homomorphisms, the category of finitely generated R-modules and R-linear maps over a principal ideal domain R, and the neighbourhood category of oriented knots in the 3-sphere.

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 Delta Y-exchanges and Y Delta-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.

We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link, it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy does not hold for spatial graphs.

We say that a link L(1) is an s-major of a link L(2) if any diagram of L(1) can be transformed into a diagram of L(2) by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knots and links with the crossing number less than or equal to six. The proofs are given by graph-theoretic methods. (C) 2010 Elsevier B.V. All rights reserved.

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

We give a complete characterization of a circle immersion that call be divided into two arc embeddings in terms of its chord diagram.

We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

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.

We consider a tiling of a square by finitely many tiles each of which is a rectangle. We do not assume that the tiles are mutually congruent. Such a tiling is called irreducible if for any two tiles the union of them is not a rectangle. A tiling is called generic if no four tiles meet in a point. A tilling is trivial if it has only one tile. A tile r in a generic tiling of a square is called a spiral if it is contained in the interior of the square and for each edge e of r there is a tile s adjacent to r such that the straight line containing e intersects the interior of s. We show that a nontrivial generic irreducible tiling of a square has a spiral. © 2008 Birkhäuser Verlag Basel/Switzerland.

An embedded graph G in the 3-sphere S-3 is called 2-irreducible if there are no separating spheres, cutting spheres, singular separating spheres, singular cutting spheres or 2-cutting spheres of G. Let D be a 2-disk in S-3 that is very good for G. Let G' be an embedded graph in S-3 obtained from G by contracting D to a point. We show that if G' is 2-irreducible then G is 2-irreducible. By this criterion certain graphs are easily shown to be 2-irreducible. As an application we show a pair of embedded graphs in the 3-sphere which is distinguished by 2-irreducibility.

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.

A generic map from a finite graph to the 2-space is called identifiable if any two embeddings of the graph into the 3-space obtained by lifting the map with respect to the natural projection from the 3-space to the 2-space are ambient isotopic in the 3-space. We show that only planar graphs have identifiable maps. We characterize the identifiable maps for some planar graphs.

Let k be a non-negative integer. Then any embedding of the complete graph on 6 . 2(k) vertices into a three-space contains a two-component link with the absolute value of its linking number greater than or equal to 2(k). Let j be a non-negative integer. Then any embedding of the complete graph on 48 . 2(j) vertices into a three-space contains a knot with the absolute value of the second coefficient of its Conway polynomial greater than or equal to 2(2j).

We show that two embeddings f and g of a finite graph G into the 3-space are spatial-graph-homologous if and only if for each subgraph H of G that is homeomorphic to a disjoint union of two circles, the restriction maps f\(H) and g\(H) have the same linking number, and for each subgraph H of G that is homeomorphic to a complete graph K-5 or a complete bipartite graph K-3,K-3, the restriction maps f\(H) and g\(H) have the same Simon invariant. (C) 2003 Elsevier B.V. All rights reserved.

We de. ne A(k)-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let A(k)-equivalence denote an equivalence relation generated by A(k)-moves and ambient isotopy. A(k)-equivalence implies A(k)-1-equivalence. Let F be an A(k)-1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let G be the quotient set of F under A(k)-equivalence. We show that the set G forms an abelian group under a certain geometric operation. We define finite type invariants on F of order (n; k). And we show that if any finite type invariant of order (1; k) takes the same value on two elements of F, then they are A(k)-equivalent. A(k)-move is a generalization of C-k-move defined by K. Habiro. Habiro showed that two oriented knots are the same up to C-k-move and ambient isotopy if and only if any Vassiliev invariant of order less than or equal to k - 1 takes the same value on them. The 'if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be C-k-equivalent.

A clasp-pass move is a local move on oriented links introduced by Habiro in 1993. He showed that two knots are transformed into each other by clasp-pass moves if and only if they have the same second coefficient of the Conway polynomial. We extend his classification to two-component links, three-component links, algebraically split links, and spatial embeddings of a planar graph that does not contain disjoint cycles. These are classified in terms of linking numbers, the second coefficient of the Conway polynomial, the Arf invariant, and the Milnor mu-invariant. (C) 2002 Elsevier Science B.V. All rights reserved.

Let X be a topological space represented by a locally finite one-dimensional simplicial complex consisting of at most countably many simplices. Let n be a natural number. Then there are mutually disjoint and mutually homeomorphic subspaces A(1), . . . , A(n) of X such that X = A(1) boolean OR (. . .) boolean OR A(n). (C) 2002 Elsevier Science B.V. All rights reserved.

A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We show that irreducibility is preserved under certain deformations of embedded graphs. We show that certain embedded graphs are irreducible.

We show that for any natural number n and any knot K, there are infinitely many unknotting number one knots, all of whose Vassiliev invariants of order less than or equal to n coincide with those of K. © 2002 by the University of Notre Dame. All rights reserved.

We show that the Vassiliev invariants of the knots contained in an embedding of a graph G into R-3 satisify certain equations that are independent of the choice of the embedding of G. By a similar observation we de ne certain edge-homotopy invariants and vertex-homotopy invariants of spatial graphs based on the Vassiliev invariants of the knots contained in a spatial graph. A graph G is called adaptable if, given a knot type for each cycle of G, there is an embedding of G into R-3 that realizes all of these knot types. As an application we show that a certain planar graph is not adaptable.

For a graph G, let Gamma be either the set Gamma (1) of cycles of G or the set Gamma (2) of pairs of disjoint cycles of G. Suppose that for each gamma is an element of Gamma, an embedding phi (gamma) : gamma --> S-3 is given, A set {phi (gamma) \ gamma is an element of Gamma) is realizable if there is an embedding f:G --> S-3 such that the restriction map f\gamma is ambient isotopic to phi (gamma) for any gamma is an element of Gamma. A graph is adaptable if any set {phi (gamma) \ gamma is an element of Gamma (1)} is realizable. In this paper, we have the following three results:
(1) For the complete graph K-5 on 5 vertices and the complete bipartite graph K-3,K-3 on 3 + 3 vertices, we give a necessary and sufficient condition for {phi (gamma) \ gamma is an element of Gamma (1)} to be realizable in terms of the second coefficient of the Conway polynomial.
(2) For a graph in the Petersen family, we give a necessary and sufficient condition for {phi (gamma) \ gamma is an element of Gamma (2)} to be realizable in terms of the linking number.
(3) The set of non-adaptable graphs all of whose proper miners are adaptable contains eight specified planar graphs. (C) 2001 Elsevier Science B.V. All rights reserved.

We show that a Brunnian link of n components and the n component trivial link share the same first n - 1 coefficients of the Jones-Conway (Homflypt) polynomial (answering the question of Kanenobu and Miyazawa). We prove also the similar result for the Kauffman polynomial of Brunnian links. We place our solution in the context of Vassiliev-Gusarov skein modules based on mixed singular crossings.

We show that any embedding of the n-skeleton of a (2n + 3)-dimensional simplex into the (2n + 1)-dimensional sphere contains a nonsplittable link of two n-dimensional spheres.

We construct an infinite set of knotted handcuff graphs such that the set of the regular projections of the handcuff graphs in the set equals the set of the regular projections of all knotted handcuff graphs. We also show that no finite set of knotted handcuff graphs have this property.

In this paper we define the total curvature of a polygonal map from a finite graph G to a Euclidean space E-n. We characterize for certain G the polygonal maps with minimal total curvature. When G is homeomorphic to a circle the result is the piecewise linear version of the generalized Fenchel theorem on the total curvature of a smooth closed curve in a Euclidean space.

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2,n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.

In this paper we describe a regular projection of a planar graph that is achieved only by knotted embeddings in 3-space. We also give conditions under which any regular projection of such graphs can be achieved by unknotted embeddings.

In this paper we give a homology classification of spatial embeddings of a graph by an invariant defined by Wu.

In this paper we show that the cobordism classes of theta curves in S3 form a group under vertex connected sum. We investigate this group by means of knot cobordism and link cobordism.

In this paper we introduce a new partial order ≦ on the set of all classical knots. We show for example that every nontrivial knot ≧ the trefoil knot. © 1989, International Academic Printing Co. Ltd., All rights reserved.

In the previous paper [5] we introduced a new partial order of knots and links, and studied about knots. In this paper we study the partial order of two-component links. © 1989, International Academic Printing Co. Ltd., All rights reserved.