The intersection of spheres in a sphere and a new geometric meaning of the Arf invariant

Let S^3_i be a 3-sphere embedded in the 5-sphere S^5 (i=1,2). Let S^3_1 and S^3_2 intersect transversely. Then the intersection C of S^3_1 and S^3_2 is a disjoint collection of circles. Thus we obtain a pair of 1-links, C in S^3_i (i=1,2), and a pair of 3-knots, S^3_i in S^5 (i=1,2). Conversely let (L_1,L_2) be a pair of 1-links and (X_1,X_2) be a pair of 3-knots. It is natural to ask whether the pair of 1-links (L_1,L_2) is obtained as the intersection of the 3-knots X_1 and X_2 as above. We give a complete answer to this question. Our answer gives a new geometric meaning of the Arf invariant of 1-links. Let f be a smooth transverse immersion S^3 into S^5. Then the self-intersection C consists of double points. Suppose that C is a single circle in S^5. Then f^{-1}(C) in S^3 is a 1-knot or a 2-component 1-link. There is a similar realization problem. We give a complete answer to this question.Comment: 15 pages 13 figure

Ribbon-moves of 2-knots: the torsion linking pairing and the η~\widetilde\eta-invariants of 2-knots

We discuss the ribbon-move for 2-knots, which is a local move. Let $K$ and $K'$ be 2-knots. Then we have: Suppose that $K$ and $K'$ are ribbon-move equivalent. (1) Let ${\mathrm {Tor}} H_1(\widetilde X_K; {\Z})$ (resp. ${\mathrm {Tor}} H_1(\widetilde X_{K'}; {\Z})$) be the $\Z$-torsion submodule of the Alexander module $H_1(\widetilde X_K; {\Z})$ (resp. $H_1(\widetilde X_{K'}; {\Z})$). Then ${\mathrm {Tor}} H_1(\widetilde X_K; {\Z})$ is isomorphic to ${\mathrm {Tor}} H_1(\widetilde X_{K'}; {\Z})$ not only as $\Z$-modules but also as ${\Z}[t,t^{-1}]$-modules. (2) The Farber-Levine pairing for $K$ is equivalent to that for $K'$. (3) The set of the values of the \Q/\Z-valued $\tilde\eta$ invariants for $K$ is equivalent to that for $K'$.Comment: 8 pages 2 figure

A new obstruction for ribbon-moves of 2-knots: 2-knots fibred by the punctured 3-tori and 2-knots bounded by homology spheres

This paper gives a new obstruction for ribbon-move equivalence of 2-knots. Let $K$ and $K'$ be 2-knots. Let $K$ and $K'$ are ribbon-move equivalent. One corollary to our main theorem is as follows. A 2-dimensional fibered knot whose fiber is the punctured 3-dimensional torus is not ribbon-move equivalent to any 2-dimensional knot whose Seifert hypersurface is a punctured homology sphere.Comment: 27pages 5 figure

Intersectional pairs of nn-knots, local moves of nn-knots, and their associated invariants of nn-knots

Let $n$ be an integer$\geqq0$. Let $S^{n+2}_1$ (respectively, $S^{n+2}_2$) be the $(n+2)$-sphere embedded in the $(n+4)$-sphere $S^{n+4}$. Let $S^{n+2}_1$ and $S^{n+2}_2$ intersect transversely. Suppose that the smooth submanifold, $S^{n+2}_1 \cap S^{n+2}_2$ in $S^{n+2}_i$ is PL homeomophic to the $n$-sphere. Then $S^{n+2}_1$ and $S^{n+2}_2$ in $S^{n+2}_i$ is an $n$-knot $K_i$. We say that the pair $(K_1,K_2)$ of n-knots is realizable. We consider the following problem in this paper. Let $A_1$ and $A_2$ be n-knots. Is the pair $(A_1,A_2)$ of $n$-knots realizable? We give a complete characterization.Comment: 22 pages, 1 figure,Chapter I: Mathematical Research Letters, 1998, 5, 577-582. Chapter II: University of Tokyo preprint series UTMS 95-50. This paper is beased on the author's master thesis 1994, and his PhD thesis 199

n-dimensional links, their components, and their band-sums

We prove the following results (1) (2) (3) on relations between $n$-links and their components. (1) Let L=(L_1, L_2) be a (4k+1)-link (4k+1\geq 5). Then we have Arf L=Arf L_1+Arf L_2. (2) Let L=(L_1, L_2) be a (4k+3)-link (4k+3\geq3). Then we have \sigma L=\sigma L_1+\sigma L_2. (3) Let n\geq1. Then there is a nonribbon n-link L=(L_1, L_2) such that L_i is a trivial knot. We prove the following results (4) (5) (6) (7) on band-sums of n-links. (4) Let L=(L_1, L_2) be a (4k+1)-link (4k+1\geq 5). Let K be a band-sum of L. Then we have Arf K=Arf L_1+Arf L_2. (5) Let L=(L_1, L_2) be a (4k+3)-link (4k+3\geq3). Let K be a band-sum of L. Then we have \sigma K=\sigma L_1+ \sigma L_2. The above (4)(5) imply the following (6). (6) Let 2m+1\geq3. There is a set of three (2m+1)-knots K_0, K_1, K_2 with the following property: K_0 is not any band-sum of any n-link L=(L_1, L_2) such that L_i is equivalent to K_i (i=1,2). (7) Let n\geq1. Then there is an n-link L=(L_1, L_2) such that L_i is a trivial knot (i=1,2) and that a band-sum of $L$ is a nonribbon knot. We prove a 1-dimensional version of (1). (8) Let L=(L_1, L_2) be a proper 1-link. Then Arf L =Arf L_1+ Arf L_2+{1/2}\{\beta^*(L)$+mod4$\{{1/2}lk (L)\}\} =Arf L_1+Arf L_2+mod2 \{\lambda (L)\}, where \beta^*(L) is the Saito-Sato-Levine invariant and \lambda(L) is the Kirk-Livingston invariant.Comment: 16 pages, no figur

A new invariant associated with decompositions of manifolds

We introduce a new topological invariant, which is a nonnegative integer, of compact manifolds with boundaries associated with a kind of decomposition of them. Let M and N be m-dimensional compact connected manifolds with boundaries. The new invariant of any boundary union of M and N is less than or equal to that of M (resp. that of N).Comment: 13 page

Remarks on nn-dimensional Feynman diagrams, for example, which will appear in M-theory and in F-theory

We state some remarks on `$n$-dimensional Feynman diagrams' ($n\in\N$). There are different features between in the case of $n$-dimensional Feynman diagrams($n\geqq3$) and in the 1-, 2-dimensional case.Comment: 5 pages, no figur

A new pair of non-cobordant surface-links which the Orr invariant, the Cochran sequence, the Sato-Levine invariant, and the alinking number cannot find

We submit a new way to detect pairs of non-cobordant surface-links. We find a new example of a pair of non-cobordant surface-links with the following properties: Orr invariant, Cochran sequence, Sato-Levine invariant, the alinking number and one of Stallings's theorems cannot distinguish them. However our new way can distinguish them.Comment: 28pages, many figure

Local-move-identities for the Z[t,t^{-1}]-Alexander polynomials of 2-links, the alinking number, and high dimensional analogues

A well-known identity (Alex+) - (Alex-)=(t^{1/2}-t^{-1/2}) (Alex0) holds for three 1-links L+, L-, and L0 which satisfy a famous local-move-relation. We prove a new local-move-identity for the Z[t,t^{-1}]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander-Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the Z[t,t^{-1}]-Alexander polynomials of 2-links and the alinking number of 2-links. We show high dimensional analogues of these results. Furthermore we prove that in the 2-dimensional case we cannot normalize the Z[t,t^{-1}]-Alexander polynomials to be compatible with our identity but that in a high-dimensional case we can do that to be compatible with our new identity.Comment: 48pages, many figure

Ribbon-moves of 2-links preserve the \mu-invariant of 2-links

We introduce ribbon-moves of 2-knots, which are operations to make 2-knots into new 2-knots by local operations in B^4. (We do not assume the new knots is not equivalent to the old ones.) Let L_1 and L_2 be 2-links. Then the following hold. (1) If L_1 is ribbon-move equivalent to L_2, then we have \mu(L_1)=\mu(L_2). (2) Suppose that L_1 is ribbon-move equivalent to L_2. Let W_i be arbitrary Seifert hypersurfaces for L_i. Then the torsion part of H_1(W_1)+H_1(W_2) is congruent to G+G for a finite abelian group G. (3) Not all 2-knots are ribbon-move equivalent to the trivial 2-knot. (4) The inverse of (1) is not true. (5) The inverse of (2) is not true. Let L=(L_1,L_2) be a sublink of homology boundary link. Then we have: (i) L is ribbon-move equivalent to a boundary link. (ii) \mu(L)= \mu(L_1) + \mu(L_2). We would point out the following facts by analogy of the discussions of finite type invariants of 1-knots although they are very easy observations. By the above result (1), we have: the \mu-invariant of 2-links is an order zero finite type invariant associated with ribbon-moves and there is a 2-knot whose \mu-invariant is not zero. The mod 2 alinking number of (S^2, T^2)-links is an order one finite type invariant associated with the ribbon-moves and there is an (S^2, T^2)-link whose mod 2 alinking number is not zero.Comment: 13 pages 21 figure