Narrow Results

We show that for each $k\in \mathbb{N}$, a link $L\subset S^3$ bounds a degree $k$ Whitney tower in the 4-ball if and only if it is $C_k$-concordant to the unlink. This means that $L$ is obtained from the unlink by a finite sequence of concordances and degree $k$ clasper surgeries. In our construction, the trees associated to the Whitney towers coincide with the trees associated to the claspers. As a corollary to our previous obstruction theory for Whitney towers in the 4-ball, it follows that the $C_k$-concordance filtration of links is classified in terms of Milnor invariants, higher-order Sato–Levine and Arf invariants. Using a new notion of $k$-repeating twisted Whitney towers, we also classify a natural generalization of the notion of link homotopy, called twisted self$C_k$-concordance, in terms of $k$-repeating Milnor invariants and $k$-repeating Arf invariants.

For Σ a compact connected oriented surface, we consider homology cylinders over Σ: these are homology cobordisms with an extra homological triviality condition. When considered up to Y₂-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group.

In this paper, when Σ has one or zero boundary component, we define a surgery map from a certain space of graphs to this group. This map is shown to be an isomorphism, with inverse given by some extensions of the first Johnson homomorphism and Birman-Craggs homomorphisms.

We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant V₂, are necessary and sufficient to express any string link Vassiliev invariant of order two. Explicit combinatorial formulas are given for these invariants. This result is applied to the theory of claspers for string links.

Habiro found in his thesis a topological interpretation of finite type invariants of knots in terms of local moves called Habiro's C_k-moves. C_k-moves are defined by using his claspers. In this paper we define "oriented" claspers and RC_k-moves among ribbon 2-knots as modifications of Habiro's notions to give a similar interpretation of Habiro–Kanenobu–Shima's finite type invariants of ribbon 2-knots. It works also for ribbon 1-knots. Furthermore, by using oriented claspers for ribbon 1-knots, we can prove Habiro–Shima's conjecture in the case of ℚ-valued invariants, saying that ℚ-valued Habiro–Kanenobu–Shima finite type invariant and ℚ-valued Vassiliev–Goussarov finite type invariant are the same thing.

In 2003, Naik and Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called strong S-equivalence, and prove that two oriented, ordered links L and L′ are related by a sequence of doubled-delta moves if and only if they are strongly S-equivalent. We also show that this is equivalent to the fact that L′ can be obtained from L through a sequence of Y-clasper surgeries, where each clasper leaf has total linking number zero with L.

Polyak showed that any Milnor’s $\overline{\mu }$-invariant of length 3 can be represented as a combination of the Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called $\mu$-invariants. We show that any Milnor’s $\mu$-invariant of length $\le k+2$ can be represented as a combination of the HOMFLYPT polynomials of knots obtained from the string link by some operation, if all $\mu$-invariants of length $\le k$ vanish. Moreover, $\mu$-invariants of length $3$ are given by a combination of the Conway polynomials and linking numbers without any vanishing assumption.

Two links are link-homotopic if they are transformed into each other by a sequence of self-crossing changes and ambient isotopies. The link-homotopy classes of links with up to three components were classified by the Milnor homotopy invariants. Levine investigated the indeterminacy of Milnor homotopy invariants and gave a classification of the link-homotopy classes of 4-component links. In this paper, we give another classification of the link-homotopy classes of 4-component links by using the clasper theory. This classification is obtained through a tetrahedron standard form of the link-homotopy classes of 4-component links. As an application, we give several new subsets of the link-homotopy classes of 4-component links which are classified by computable invariants. Moreover, the classification allows us to run the Habegger–Lin algorithm which determines whether two given links are link-homotopic or not.