Link Cobordism Research Articles

Cobordism of theta curves in S3

AbstractIn 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.

Real algebraic curves and link cobordism

If A is a nonsingular real algebraic curve of degree m, we show that there is a link cobordism of specified topological type between a link which depends only on the isotopy type of A and a link which depends only on m. We prove a generalization of the ΊϊistramMurasugi inequalities for link cobordisms and apply it to this situation. 0. Introduction. In this paper, we will show that if a certain collection of simple closed curves, C, can be realized as a real algebraic curve of degree m then there is a cobordism of specified topological type between two links in a 3-manifold Q. Q may be described as S3 modulo the quaternion eight group. One link, L(C), depends only on C, and the other link depends only on m. This is our main result. It was announced at the Durham 4-manifolds conference in 1982. Since then it has been reformulated and improved with two addenda. Many of the known restrictions on link cobordisms in S3 may be generalized to obtain restrictions on cobordisms in Q-homology 3-spheres. In [G3], we will develop the theory of 2-signatures and 2-nullities in a rational homology 3-sphere, and derive the generalized TristramMurasugi inequalities in this context. We will consider non-orientable as well as orientable cobordisms. Applying these results to the cobordism in / x Q, we have obtained proofs of the strengthened Pretrovski inequalities, and the strengthened Arnold inequalities ((3.4), (3.5) (3.6) (3.7) (3.8) (3.9) of Viro's survey article [V]). Applying this same result to the lift of this cobordism to a covering space of Q, we have obtained Zvonilov's inequality ((3.23) of [V]). These derivations will appear in a sequel to this paper [G4]. Another result for links in S3 states that the Arf invariants of two proper links in S3, which are related by a planar cobordism, must be the same. The correct generalization of this result to links in a rational homology sphere will appear in [G3]. Applying these results to our cobordism in Q and lifts of it to certain covers, we have rederived Rokhlin's congruence for M-curves, the related congruence for M - 1 curves. (3.2) (3.3) [V] and Fiedler's congruence [Fl]. These results will be included in

A note on cobordism of surface links in $S\sp 4$

Sato’s idea of the asymmetric linking number is used in cyclic branched coverings to give an invariant of the cobordism of embedded surfaces in the $4$-sphere.

A Seifert-matrix interpretation of Cappell and Shaneson's approach to link cobordisms

We classified the set of Fm-cobordism classes of Fm-links by their Seifert matrices in [5]. On the other hand Cappell and Shaneson identified them with essentially a quotient group of their homology surgery obstruction group [2]. In this paper, we will find a description of their surgery obstruction in terms of a Seifert matrix. In relation to Ledimet's recent results [7], we hope this might provide some clue to whether Fm-cobordism or boundary cobordism is stronger than ordinary link cobordism. It also seems to be an interesting algebraic question to find an algorithm for obtaining a Seifert matrix from their surgery obstruction.

A Generalization of the Levine-Tristram Link Invariant

Invariants to $m$-component links are defined and are shown to be link cobordism invariants under certain conditions. Examples are given.

A generalization of the Levine-Tristram link invariant

Invariants to m m -component links are defined and are shown to be link cobordism invariants under certain conditions. Examples are given.

Seifert Matrices and Boundary Link Cobordisms

To an $m$-component boundary link of odd dimension, a matrix is associated by taking the Seifert pairing on a Seifert surface of the link. An algebraic description of the set of boundary link cobordism classes of boundary links is obtained by using this matrix invariant.

Seifert matrices and boundary link cobordisms

To an m m -component boundary link of odd dimension, a matrix is associated by taking the Seifert pairing on a Seifert surface of the link. An algebraic description of the set of boundary link cobordism classes of boundary links is obtained by using this matrix invariant.

On an invariant of link cobordism in dimension four

We conduct a detailed investigation of β( L), the Sato-Levine concordance invariant of a 2-component link of 2-spheres in S 4. It is shown that β( L) vanishes for broad categories of links, the most striking result being that it vanishes if one component of L is unknotted.

Link cobordism

Link cobordism

Link Cobordism and Milnor's Invariant

Link Cobordism and Milnor's Invariant

A Concept of Cobordism between Links

and in ? 2, a slight modification of the suggested definition is introduced. In ?? 3 and 4, we prove that this is suitable in the sense described above. In ? 5, it is shown that the total linking number of a link type is an invariant under this link cobordism, and that the link cobordism group splits into the direct sum of the knot cobordism group and an infinite cyclic group. In ? 6, we investigate the relation between slice link in the weak sense, slice link, slice link in the strong sense, and link which is cobordic to zero. Throughout this paper we suppose that all knots and links are oriented and tame. Oriented knot and link types will be denoted by Greek letters; representative oriented knots and links, by the corresponding roman letters. The author is indebted very much to R. H. Fox for his kind and helpful suggestions. 1. It is well known that the oriented knot types form a commutative semi-group under the operation of composition #. The zero of this semi-group