Akira Yasuhara

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.

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.

Four-genus and four-dimensional clasp number of a knot

For a knot K in the 3-sphere, by using the linking form on the first homology group of the double branched cover of the 3-sphere, we investigate some numerical invariants, 4-genus g* (K), nonorientable 4-genus -y* (K) and 4dimensional clasp number c* (K), defined from the four-dimensional viewpoint. T. Shibuya gave an inequality g* (K) < c* (K), and asked whether the equality holds or not. From our result in this paper, we find that the equality does not hold in general.

Self delta-equivalence for links whose Milnor’s isotopy invariants vanish

For an n-component link, Milnors isotopy invariants are defined for each multi-index I = i 1 i 2 ...i m (i j ∈ {1,...,n}). Here m is called the length. Let r(I) denote the maximum number of times that any index appears in I. It is known that Milnor invariants with r = 1, i.e., Milnor invariants for all multi-indices I with r(I) = 1, are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with r = 1 coincide. This gives us that a link in S 3 is link-homotopic to a trivial link if and only if all Milnor invariants of the link with r = 1 vanish. Although Milnor invariants with r = 2 are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with r≤ 2 are self Δ-equivalence invariants. In this paper, we give a self Δ-equivalence classification of the set of n-component links in S 3 whose Milnor invariants with length < 2n, - 1 and r < 2 vanish. As a corollary, we have that a link is self Δ-equivalent to a trivial link if and only if all Milnor invariants of the link with r < 2 vanish. This is a geometric characterization for links whose Milnor invariants with r < 2 vanish. The chief ingredient in our proof is Habiros clasper theory. We also give an alternate proof of a link-homotopy classification of string links by using clasper theory.

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.

Milnor's invariants and self

It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor invariants with repeated indices are invariants not only of isotopy, but also of self Ck-equivalence. Here self C k -equivalence is a natural generalization of link homotopy based on certain degree k clasper surgeries, which provides a filtration of link homotopy classes.

C_{k}

Self ?-equivalence is an equivalence relation for links, which is stronger than link-homotopy defined by J. W. Milnor. It was shown that any boundary link is link-homotopic to a trivial link by L. Cervantes and R. A. Fenn and by D. Dimovski independently. In this paper we will show that any boundary link is self ?-equivalent to a trivial link

-equivalence

Self A-equivalence is an equivalence relation for links, which is stronger than the link-homotopy defined by J. Milnor. It is known that cobordant links are link-homotopic and that they are not necessarily self A-equivalent. In this paper, we will give a sufficient condition for cobordant links to be self A-equivalent.

Boundary links are self delta-equivalent to trivial links

The method of distinguishing knots and links using the colorability of their diagrams was invented by Ralph Fox [2]. As generalization of this method, we introduce certain method of distinguishing spatial graphs.

Self delta-equivalence of cobordant links

In this paper, we give a complete set of finite type string link in- variants of degree < 5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain clo- sure of (cabled) string links. We show that finite type invariants classify string links up to Ck-moves for k � 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar characterization of finite type concordance invariants of degree < 6.