A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants of covering links is a cobordism invariant of a link, and this invariant can detect some links undetected by the ordinary Milnor invariants. Moreover, for a Brunnian link L, the first non-vanishing Milnor invariant of L is modulo-2 congruent to a sum of Milnor invariants of covering links. As a consequence, a sum of linking numbers of 'iterated' covering links gives the first non-vanishing Milnor invariant of L modulo 2. (C) 2017 Elsevier B.V. All rights reserved.

Levine introduced clover links to investigate the indeterminacy of Milnor invariants of links. He proved that for a clover link, Milnor numbers of length up to 2k + 1 are well-defined if those of length <= k vanish, and that Milnor numbers of length at least 2k + 2 are not well-defined if those of length k + 1 survive. For a clover link c with vanishing Milnor numbers of length <= k, we show that the Milnor number mu(c)(I) for a sequence I is well-defined by taking modulo the greatest common divisor of the mu(c)(J)' s, where J is any proper subsequence of I obtained by removing at least k + 1 indices. Moreover, if I is a non-repeated sequence of length 2k + 2, the possible range of mu(c)(I) is given explicitly. As an application, we give an edge-homotopy classification of 4-clover links.

The set SL(n) of n-string links has a monoid structure, given by the stacking product. When considered up to concordance, SL(n) becomes a group, which is known to be abelian only if n = 1. In this paper, we consider two families of equivalence relations which endow SL(n) with a group structure, namely the C-k-equivalence introduced by Habiro in connection with finite-type invariants theory, and the C-k-concordance, which is generated by C-k-equivalence and concordance. We investigate under which condition these groups are abelian, and give applications to finite-type invariants.

Gusarov and Habiro introduced a C-m move, that is strongly related to Vassiliev invariants. In this note, we study a special kind of C-m move, called a non-self C-m move. We show that two links can be transformed into each other by a finite sequence of non-self C-m moves if and only if (1) the two links can be transformed into each other by a finite sequence of C-m moves, and (2) the knot types of corresponding components coincide.

We give formulas expressing Milnor invariants of an n-component link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant (mu) over bar (L)(J) vanishes for any sequence J with length at most k, then any Milnor (mu) over bar -invariant (mu) over bar (L)(I) with length between 3 and 2 k C 1 can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations. In particular, the "first nonvanishing" Milnor invariants can be always represented as such a linear combination.

A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in S3 to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given. © 2011 World Scientific Publishing Company.

We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length <= k are all zero into a link with vanishing Milnor invariants of length <= 2k + 1. and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Delta-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if the link L with a single component Whitehead doubled is self Delta-equivalent to the unlink.

We give a complete set of finite type string link invariants of degree < 5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closures of (cabled) string links. We show that finite type invariants classify string links up to C(k)-moves for k <= 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar classification of string links up to C(k)-moves and concordance for k <= 6.

Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain component of a link with fixing the other components, in other words, homotopy and Delta-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be Delta-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to Delta-equivalence and concordance. (C) 2010 Elsevier B.V. All rights reserved.

For all n-component link, Milnor'z isotopy invariants are defined for each multi-index I = i(1) i(2)...i(m) (i(j) is an element of {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 Delta-equivalence invariants. In this paper, we give a self Delta-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 Delta-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 Habiro's clasper theory. We also give all alternate proof of a link-homotopy classification of string links by using clasper theory.

It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor's invariants with repeated indices are invariants not only of isotopy, but also of self C-k-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.

For an n-component string link, the Milnor's concordance invariant is defined for each sequence I = i1i2...im(ij,in {1,...,n}) Let r(I) denote the maximum number of times that any index appears. We show that two string links are equivalent up to self Δ-moves and concordance if and only if their Milnor invariants coincide for all sequences I with r(I) ≤ 2. © 2009 Mathematical Sciences Publishers.

A C-n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a 'higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to C-n-moves, under certain restrictions. Namely, we give a classification up to C-4-moves of 2-component links, 3-component Brunnian links and n-component C-3-trivial links. We also classify n-component link-homotopically trivial Brunnian links up to Cn+1-moves.

A surface in a smooth 4-manifold is called flexible if, for any diffeomorphism phi on the surface, there is a diffeomorphism on the 4-manifold whose restriction on the surface is phi and which is isotopic to the identity. We investigate a sufficient condition for a smooth 4-manifold M to include flexible knotted surfaces, and introduce a local operation in simply connected 4-manifolds for obtaining a flexible knotted surface from any knotted surface. (C) 2007 Elsevier Ltd. All rights reserved.

Self Delta-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 Delta-equivalent to a trivial link.

Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the mu-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of Z[t, t(-1)]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by Cochran. In fact, the coefficients of the powers of t-1 will be the linking numbers of certain derived links in S3. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S-3. This generalizes a result of Hoste.

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, C-n-moves, and showed that two knots have the same Vassiliev invariants of order <= n - 1 if and only if they are transformed into each other by C-n-moves. We define a local move, beta(n)-move, which is obtained from a C-n-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a beta(n)-move and that a beta(n)-move is realized by twice C-n-moves. In this note we study the relation between C-n-move and On-move, and in particular, give answers to the following questions: (1) Is a beta(n)-move realized by a finite sequence of Cn+-moves? (2) Is C-n-move realized by a finite sequence of on-moves?

We give a classification of n-component links up to C-n-move. In order to prove this classification, we characterize Brunnian links, and have that a Brunnian link is ambient isotopic to a band sum of a trivial link and Milnor's links. (C) 2005 Elsevier B.V. All rights reserved.

Self Delta-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 Delta-equivalent. In this paper, we will give a sufficient condition for cobordant links to be self Delta-equivalent.

K. Habiro gave a neccesary and sufficient condition for knots to have the same Vassiliev invariants in terms of C-k-moves. In this paper we give another geometric condition in terms of Brunnian local moves. The proof is simple and self-contained.

Nakanishi and Shibuya gave a relation between link homotopy and quasi self delta equivalence. And they also gave a necessary condition for two links to be self delta equivalent by using the multivariable Alexander polynomial. Link homotopy and quasi self delta-equivalence are also called self Cl-equivalence and quasi self C-2-equivalence respectively. In this paper, we generalize their results. In Sec. 1, we give a relation between self C-k-equivalence and quasi self Ck+1-equivalence. In Secs. 2 and 3, we give necessary conditions for two links to be self C-k-equivalent by using the multivariable Conway potential function and the Conway polynomial respectively.

We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in Q and in Q(Z[t, t(-1)]), respectively, where Q(Z[t, t(-1)]) denotes the quotient field of Z[ t; t 1]. It is known that the modulo-Z linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-Z[t, t(-1)] linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate 'modulo Z' and 'modulo Z[t, t(-1)](,). When the finite cyclic cover of the 3-sphere branched over a knot is a rational homology 3-sphere, the linking number of a pair in the preimage of a link in the 3-sphere is determined by the Goeritz/Seifert matrix of the knot.

Rotors were introduced as a generalization of mutation by Anstee, Przytycki and Rolfsen in 1987. In this paper we show that the Tristram-Levine signature is preserved by orient at ion-preserving rotations. Moreover, we show that any link invariant obtained from the characteristic polynomial of the Goeritz matrix, including the Murasugi-Trotter signature, is not changed by rotations. In 2001, P. Traczyk showed that the Conway polynomials of any pair of orientation-preserving rotants coincide. We show that there is a pair of orientation-reversing rotants with different Conway polynomials.

A pass-move and a #-move are local moves on oriented links defined by L. H. Kauffman and H. Murakami respectively. Two links are self pass-equivalent (resp. self #-equivalent) if one can be deformed into the other by pass-moves (resp. #-moves), where none of them can occur between distinct components of the link. These relations are equivalence relations on ordered oriented links and stronger than link-homotopy defined by J. Milnor. We give two complete classifications of links with arbitrarily many components up to self pass-equivalence and up to self #-equivalence respectively. So our classifications give subdivisions of link-homotopy classes.

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.

We give an algorithm for a surgery description of a p-fold cyclic branched cover of B-3 branched along a tangle. We generalize constructions of Montesinos and Akbulut-Kirby.

We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of 'symmetric' links in S-3.

The C-k-equivalence is an equivalence relation generated by C-k-moves defined by Habiro. Habiro showed that the set of C-k-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order less than or equal to k - 1. We see that the set of C-k-equivalence classes of the spatial theta-curves forms a group under the vertex connected sum and that if the group is abelian, then it can be classified by the additive Vassiliev invariant of order less than or equal to k - 1. However the group is not necessarily abelian. In fact, we show that it is nonabelian for k greater than or equal to 12. As an easy consequence, we have the set of C-k-equivalence classes of m-string links, which forms a group under the composition, is nonabelian for k greater than or equal to 12 and m greater than or equal to 2. (C) 2002 Elsevier Science B.V. All rights reserved.

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.

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.

In this paper we study certain kinds of links; proper links, algebraically split links and Z(2)-algebraically split links. These links have 'algebraic' definitions. In fact these are defined in terms of the linking number. We shall give these links certain 'geometric' definitions. By using the geometric definitions, we study the Arf invariants of these links.

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 gamma*(K) and 4-dimensional clasp number c*(K), defined from the four-dimensional viewpoint. T. Shibuya gave an inequality g*(K) less than or equal to 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.

Let M be a 4-manifold with partial derivative M congruent to S-3 and L subset of partial derivative M a link. The link L is null-homologous in M if L bounds a disjoint union of once-punctured, orientable surfaces in M. In a previous paper [1] the author defined null-homologous link in 4-manifolds and gave a necessary and sufficient condition for links to be null-homologous in I-manifolds. By using this condition, we investigate the sets of null-homologous links in punctured CP2, <(CP)over bar>(2), CP2 #<(CP)over bar>(2) and S-2 x S-2.

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.

We define link homology in 4-manifolds, and show that it has a close connection to linking numbers and intersection matrices of 4-manifolds. We also define null-homologous links in 4-manifolds. We give a necessary and sufficient condition for links to be null-homologous in 4-manifolds. This condition implies that for any 4-manifold with second Betti number n, there are (n + 2)-component links which are not null-homologous in the 4-manifold.

Using disk/band surfaces, we investigate spatial-graph homology. A necessary and sufficient condition for spatial embeddings of a graph to be homologous is expressed in terms of disk/band surfaces. By this condition we can decide practically whether two given spatial embeddings are homologous or not. We equip the set of spatial-graph homology classes with an abelian group structure and give a method to calculate this group by using the condition above.

We consider surfaces in simply connected 4-manifolds. We estimate the normal Euler numbers of bounded non-orientable surfaces and consider the problem of representing characteristic homology classes by orientable surfaces. To do so, we develop techniques connecting the above problems for given surfaces with the problems for surfaces with fewer first Betti numbers. © 1996 by the University of Notre Dame. All rights reserved.

B. E. Clark defined the crosscap number of a knot to be the minimum number of the first Betti numbers of non-orientable surfaces bounding it. In this paper, we investigate the crosscap numbers of knots. We show that the crosscap number of 7(4) is equal to 3. This gives an affirmative answer to a question given by Clark. In general, the crosscap number is not additive under the connected sum. We give a necessary and sufficient condition for the crosscap number to be additive under the connected sum.