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

Singularities of the projections of nn-dimensional knots

Let n be aninteger>4. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball in (n+2)-space.) Furthermore, the projection is not the projection of any unknotted sphere in the (n+2)-space. There are two inequivalent embeddings of an n-manifold in the (n+2)-space such that the projection of one of these in (n+1)-space has no double points and the projection of the other has a connected embedded double point set.Comment: 9 papges, 4 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

'Topological parallel world' constructed by modification of space-time along observables

We introduce a new concept, `(topological) (vacuum) parallel world, ' which is a new tool to research submanifolds. Roughly speaking, `Observables in (T)QFT' is equal to `a (topological) modification of space-time.' In other words, we give a new interpretation of observables. We give some examples associated with the Alexander polynomial, the Jones polynomial.Comment: 6page

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

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

The projections of n-knots which are not the projection of any unknotted knot

Let n be any integer greater than two. We prove that there exists a projection P having the following properties. (1) P is not the projection of any unknotted knot. (2) The singular point set of P consists of double points. (3) P is the projection of an n-knot which is diffeomorphic to the standard sphere. We prove there exists an immersed n-sphere (in R^{n+1}\times{0}) which is not the projection of any n-knot (n>2). Note that the second theorem is different from the first one.Comment: 12 pages, no figur

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

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

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