Kouki Taniyama

Realization of knots and links in a spatial graph

Abstract For a graph G , let Γ be either the set Γ 1 of cycles of G or the set Γ 2 of pairs of disjoint cycles of G . Suppose that for each γ∈Γ , an embedding φ γ :γ→S 3 is given. A set {φ γ ∣γ∈Γ} is realizable if there is an embedding f :G→S 3 such that the restriction map f|γ is ambient isotopic to φ γ for any γ∈Γ . A graph is adaptable if any set {φ γ ∣γ∈Γ 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 3,3 on 3+3 vertices, we give a necessary and sufficient condition for {φ γ ∣γ∈Γ 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 {φ γ ∣γ∈Γ 2 } to be realizable in terms of the linking number. (3) The set of non-adaptable graphs all of whose proper minors are adaptable contains eight specified planar graphs.

Homology classification of spatial embeddings of a graph

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

Clasp-pass moves on knots, links and spatial graphs

Abstract 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 μ -invariant.

Knotted projections of planar graphs

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.

On intrinsically knotted or completely 3-linked graphs

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 4Y-exchanges and Y4-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.

ALMOST POSITIVE LINKS HAVE NEGATIVE SIGNATURE

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.

Band description of knots and Vassiliev invariants

In the 1990s, Habiro defined C k -move of oriented links for each natural number k [ 5 ]. A C k -move is a kind of local move of oriented links, and two oriented knots have the same Vassiliev invariants of order [les ] k −1 if and only if they are transformed into each other by C k -moves. Thus he has succeeded in deducing a geometric conclusion from an algebraic condition. However, this theorem appears only in his recent paper [ 6 ], in which he develops his original clasper theory and obtains the theorem as a consequence of clasper theory. We note that the ‘if’ part of the theorem is also shown in [ 4 ], [ 9 ], [ 10 ] and [ 16 ], and in [ 13 ] Stanford gives another characterization of knots with the same Vassiliev invariants of order [les ] k −1.

Symmetries of spatial graphs and Simon invariants

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 the necessary conditions on linking numbers and Simon invariants for such links and spatial graphs to be symmetric.

Regular projections of knotted handcuff graphs

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.

Total curvature of graphs in Euclidean spaces

Abstract 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.