33 research outputs found
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 -invariants of 2-knots
We discuss the ribbon-move for 2-knots, which is a local move. Let and
be 2-knots. Then we have: Suppose that and are ribbon-move
equivalent.
(1) Let (resp. ) be the -torsion submodule of the Alexander
module (resp. ). Then
is isomorphic to not only as -modules but also as
-modules.
(2) The Farber-Levine pairing for is equivalent to that for .
(3) The set of the values of the \Q/\Z-valued invariants for
is equivalent to that for .Comment: 8 pages 2 figure
Singularities of the projections of -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 -knots, local moves of -knots, and their associated invariants of -knots
Let be an integer. Let (respectively, ) be
the -sphere embedded in the -sphere . Let
and intersect transversely. Suppose that the smooth submanifold,
in is PL homeomophic to the -sphere.
Then and in is an -knot . We say
that the pair of n-knots is realizable.
We consider the following problem in this paper. Let and be
n-knots. Is the pair of -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 and be 2-knots. Let and 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 -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 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)\{{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