arXiv subjects
Publications and source records attributed to Eiji Ogasa.
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.
The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the following facts. For the ribbon-move on 2-knots, which is a kind of local-move on knots, we have the following: There is a ribbon-move-unknotting-number-two 2-knot. The ribbon-move-unknotting-number of 2-knots is unbounded. For the pass-move on 1-knots, which is a kind of local-move on knots, we have the following: There is a pass-move-unknotting-number-two 1-knot whose (ordinary) unknotting-number is 4. For any natural number n, there is a 1-knot whose pass-move-unknotting-number is>n and whose (ordinary) unknotting-number is 4n. For the high-dimensional-pass-move on high-dimensional knots, which is a kind of local-move on knots, we have the following: There is a (2k+1, 2k+2)-pass-move-unknotting-number-two (4k+2)-knot. The (2k+1, 2k+2)-pass-move-unknotting-number of (4k+2)-knot is unbounded. There is a (2k+1, 2k+1)-pass-move-unknotting-number-two (4k+1)-knot. The (2k+1, 2k+1)-pass-move-unknotting-number of (4k+1)-knot is unbounded. There is a (4k+1)-knot whose twist-move-unknotting-number is n for any natural number n.
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.
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).
We show several relations between local moves on 1-dimensional knots and those on high dimensional knots related by products of knots.
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.
It is well-known:Suppose there are three 1-dimensional links $K_+$, $K_-$, $K_0$ such that $K_+$, $K_-$, and $K_0$ coincide out of a 3-ball $B$ trivially embedded in $S^3$ and that $K_+\cap B$, $K_-\cap B$, and $K_0\cap B$ are drawn as follows. Then $Δ_{K_+}-Δ_{K_+}=(t-1)\cdotΔ_{K_0}$, where $Δ_{K}$ is the Alexander polynomial of $K$. We know similar formulae of other invariants of 1-dimensional knots and links. (The Jones polynomial etc.) It is natural to ask: Suppose there are two $n$-dimensional knots $K_+$, $K_-$ and a submanifold $K_0$ such that $K_+$, $K_-$, and $K_0$ coincide out of a $n$-ball $B$ trivially embedded in $S^{n+2}$. Then is there a relation in $K_+\cap B$, $K_-\cap B$, and $K_0\cap B$ with the following property(*)? (*)If $K_+$, $K_-$, and $K_0$ satisfy this relation, an invariant of $K_+$, that of $K_-$, and that of $K_0$ satisfy a fixed relation. In this paper we pove there are such a relation where $K_+$, $K_-$, and $K_0$ satisfy the formula $Δ_{K_+}-Δ_{K_+}=(t-1)\cdotΔ_{K_0}$, where $Δ_{K}$ is a polynomial to represent the Alexander polynomial of $K$. We show another relation where $K_+$, $K_-$, and $K_0$ satisfy the formula ${\mathrm{Arf}}K_+-{\mathrm{Arf}}K_-=\{|bP_{4k+2}\cap I(K_0)|+1\}mod 2,$ where (1)$I()$ is the inertia group. and $I(K_0)$ is the inertia group of a smooth manifold which is orientation preserving diffeomorphic to $K_0$. (2)For a group $G$, $|G|$ denote the order of $G$. A local move formula is a relation of an invariant of a few knots related by a local move as above.
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.
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 $\tildeη$ invariants for $K$ is equivalent to that for $K'$.
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.
Let $n$ be any natural number. Let $K$ be any $n$-dimensional knot in $S^{n+2}$. We define a supersymmetric quantum system for $K$ with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for $K$. Our Witten indexes are topological invariants for $n$-dimensional knots. Our Witten indexes are not zero in general. If $K$ is equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten indexes restrict the Alexander polynomials of $n$-knots. If one of our Witten indexes for an $n$-knot $K$ is nonzero, then one of the Alexander polynomials of $K$ is nontrivial. Our Witten indexes are connected with homology with twisted coefficients. Roughly speaking, our Witten indexes have path integral representation by using a usual manner of supersymmetric theory.
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 σL=σL_1+σL_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 σK=σL_1+ σL_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}\{β^*(L)$+mod4 $\{{1/2}lk (L)\}\} =Arf L_1+Arf L_2+mod2 \{λ(L)\}, where β^*(L) is the Saito-Sato-Levine invariant and λ(L) is the Kirk-Livingston invariant.
Let K and K' be 2-knots. Suppose that K and K' are ribbon-move equivalent. Then the Farber-Levine pairing for K is equivalent to that for K' and the (Z-)torsion part of the first Alexander module of $K$ is isomorphic to that of K' as Z[Z] modules. Let K be a 2-knot which is ribbon-move equivalent to the trivial knot. Then the Atiyah-Patodi-Singer-Casson-Gordon-Ruberman Q/Z-valued \widetildeη-invariants of K for Z_d is zero. (d is a natural number. d>2.).
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 μ(L_1)=μ(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) μ(L)= μ(L_1) + μ(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 μ-invariant of 2-links is an order zero finite type invariant associated with ribbon-moves and there is a 2-knot whose μ-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.
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.
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.