SOME RESULTS ON KNOTS AND LINKS IN ALL DIMENSIONS

Eiji Ogasa

Department of Mathematical Sciences University of Tokyo Komaba, Tokyo 153 Japan i33992@m-unix.cc.u-tokyo.ac.jp ogasa@ms513red.ms.u-tokyo.ac.jp

An (oriented) (ordered) m-component n-(dimensional) link is a smooth, oriented submanifold L = {K_{1}, …, K_{m}} of S^{n+2}, which is the ordered disjoint union of m manifolds, each PL homeomorphic to the standard n-sphere. (If m = 1, then L is called a knot.) We say that m-component n-dimensional links, L_{0} and L_{1}, are (link- )concordant or (link-)cobordant if there is a smooth oriented submanifold C̃={C_{1} , …,C_{m}} of S^{n+2} × [0, 1], which meets the boundary transversely in ∂C̃, is PL homeomorphic to L_{0} × [0, 1] and meets S^{n+2} × {l} in L_{l} (l = 0, 1).

We work in the smooth category.

§1

Let S_{1}^{3} and S_{2}^{3} be 3-spheres embedded in the 5-sphere S^{5} and intersect transversely. Then the intersection C is a disjoint collection of circles. Thus we obtain a pair of 1-links C in S_{i}^{3}, and a pair of 3-knots S_{i}^{3} in S^{5}.

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 (L_{1}, L_{2}) is obtained as the intersection of X_{1} and X_{2}.

In this paper we give a complete answer to the above question. Definition. (L_{1}, L_{2}, X_{1}, X_{2}) is called a quadruple of links if the following conditions (1), (2) and (3) hold.

(1) L_{i} = (K_{i1}, …, K_{im_{i}}) is an oriented ordered m_{i}-component 1-dimensional link (i = 1, 2). (2) m_{1} = m_{2}. (3) X_{i} is an oriented 3-knot. Definition.A quadruple of links (L_{1}, L_{2}, X_{1}, X_{2}) is said to be realizable if there exists a smooth transverse immersion f : S_{1}^{3} ∐ S_{2}^{3} ↬ S^{5} satisfying the following conditions.

(1) f|S_{i}^{3} is a smooth embedding and defines the 3-knot X_{i}(i = 1, 2) in S^{5}.

82 EIJI OGASA

(2) For C = f(S_{1}^{3}) ∩ f(S_{2}^{3}), the inverse image f^{−1}(C) in S_{i}^{3} defines the 1-link L_{i}(i = 1, 2). Here, the orientation of C is induced naturally from the preferred orientations of S_{1}^{3}, S_{2}^{3}, and S^{5}, and an arbitrary order is given to the components of C.

This research was partially suppported by Research Fellowships of the Promotion of Science for Young Scientists.

The following theorem characterizes the realizable quadruples of links.

Theorem 1.1. A quadruple of links (L_{1}, L_{2}, X_{1}, X_{2}) is realizable if and only if (L_{1}, L_{2}, X_{1}, X_{2}) satisfies one of the following conditions (i) and (ii).

(i) Both L_{1} and L_{2} are proper links, and

Arf(L_{1}) = Arf(L_{2}).

(ii) Neither L_{1} nor L_{2} is proper, and

lk(K_{1j}, L_{1} − K_{1j}) ≡ lk(K_{2j}, L_{2} − K_{2j}) mod 2 for all j.

Let f : S^{3} ↬ S^{5} be a smooth transverse immersion with a connected self-intersec-tion C in S^{5}. Then the inverse image f^{−1}(C) in S^{3} is a knot or a 2-component link. For a similar realization problem, we have:

Theorem 1.2.

(1) All 2-component links are realizable as above.

(2) All knots are realizable as above.

Remark. By Theorem 1.1 a quadruple of links (L_{1}, L_{2}, X_{1}, X_{2}) with K_{1} being the trivial knot and K_{2} being the trefoil knot is not realizable. But by Theorem 1.2, the two component split link of the trivial knot and the trefoil knot is realizable as the self-intersection of an immersed 3-sphere.

§2

We discuss the high dimensional analogue of §1. Definition. (K_{1}, K_{2}) is called a pair of n-knots if K_{1} and K_{2} are n-knots. (K_{1}, K_{2}, X_{1}, X_{2}) is called a quadruple of n-knots and (n + 2)-knots or a quadruple of (n, n + 2)-knots if K_{1} and K_{2} constitute a pair of n-knots (K_{1}, K_{2}) and X_{1} and X_{2} are diffeomorphic to the standard (n + 2)-sphere.

Definition. A quadruple of (n, n + 2)-knots (K_{1}, K_{2}, X_{1}, X_{2}) is said to be realizable if there exists a smooth transverse immersion f : S_{1}^{n+2} ∐ S_{2}^{n+2} ↬ S^{n+4} satisfying the following conditions.

(1) f|S_{i}^{n+2} defines X_{i} (i=1,2).

(2) The intersection Σ = f(S_{1}^{n+2})∩f(S_{2}^{n+2}) is PL homeomorphic to the standard sphere.

(3) f^{−1}(Σ) in S_{i}^{n+2} defines an n-knot K_{i} (i = 1, 2). A pair of n-knots (K_{1}, K_{2}) is said to be realizable if there is a quadruple of (n, n+2)- knots (K_{1}, K_{2}, X_{1}, X_{2}) which is realizable.

The following theorem characterizes the realizable pairs of n-knots.

Theorem 2.1. A pair of n-knots (K_{1}, K_{2}) is realizable if and only if (K_{1}, K_{2}) satisfies the condition that

\{ \begin{matrix}(K_{1}, K_{2}) is arbitrary & if n is even, \\ Arf(K_{1}) = Arf(K_{2}) & if n = 4m + 1, \\ σ(K_{1})=σ(K_{2}) & if n = 4m + 3, \end{matrix} (m ≧ 0,m ∈ ℤ).

There exists a mod 4 periodicity in dimension similar to the periodicity of highdimensional knot cobordism and surgery theory. ([CS1,2] and [L1,2].)

We have the following results on the realization of a quadruple of (n, n + 2)-knots.

Theorem 2.2. A quadruple of (n, n + 2)-knots T = (K_{1}, K_{2}, X_{1}, X_{2}) is realizable if K_{1} and K_{2} are slice.

Kervaire proved that all even dimensional knots are slice ([Ke]). Hence we have:

Corollary 2.3. If n is even, an arbitrary quadruple of (n, n+2)-knots T = (K_{1}, K_{2}, X_{1}, X_{2}) is realizable.

In order to prove Theorem 2.1, we introduce a new knotting operation for high dimensional knots, high dimensional pass-moves. The 1-dimensional case of Definition 2.1 is discussed on p.146 of [K1].

Definition. Let (2k + 1)-knot K be defined by a smooth embedding g : Σ^{2k+1} ↪ S^{2k+3}, where Σ^{2k+1} is PL homeomorphic to the standard (2k + 1)-sphere. (k ≧ 0.) Let D_{x}^{k+1}={(x_{1}, …, x_{k+1})| Σx_{i}^{2}< 1} and D_{y}^{k+1}={(y_{1}, …, y_{k+1})| Σy_{i}^{2}< 1}. Let D_{x}^{k+1}(r)={(x_{1} , …, x_{k+1})| Σx_{i}^{2}≦ r^{2}} and D_{y}^{k+1}(r)={(y_{1} , …, y_{k+1})| Σy_{i}^{2}≦ r^{2}}. A local chart (U, ϕ) of S^{2k+3} is called a pass-move-chart of K if it satisfies the following conditions.

(1) ϕ(U) ≅ ℝ^{2k+3}= (0, 1) × D_{x}^{k+1} × D_{y}^{k+1}

(2) ϕ(g(S^{2k+1}) ∩ U) = [\{\frac{1}{2}\} × D_{x}^{k+1} × ∂D_{y}^{k+1}(\frac{1}{3})] ⨿ [\{\frac{2}{3}\} × ∂D_{x}^{k+1}(\frac{1}{3}) × D_{y}^{k+1}]

Let g_{U} : Σ^{2k+1} ↪ S^{2k+3} be an embedding such that:

(1) g|{Σ^{2k+1} − g^{−1}(U)} = g_{U}|{Σ^{2k+1} − g^{−1}(U)}, and

(2) ϕ(g_{U}(Σ^{2k+1}) ∩ U) = [\{\frac{1}{2}\} × D_{x}^{k+1} × ∂D_{y}^{k+1}(\frac{1}{3})] ⨿

[\{\frac{2}{3}\} × ∂D_{x}^{k+1}(\frac{1}{3})× (D_{y}^{k+1} − D_{y}^{k+1}(\frac{1}{2}))]

∪ [[\frac{1}{3}, \frac{2}{3}] × ∂D_{x}^{k+1}(\frac{1}{3}) × ∂D_{y}^{k+1}(\frac{1}{2})]

∪ [\{\frac{1}{3}\} × ∂D_{x}^{k+1}(\frac{1}{3}) × D_{y}^{k+1}(\frac{1}{2})\}]

Let K_{U} be the (2k + 1)-knot defined by g_{U}. Then we say that K_{U} is obtained from K by the (high dimensional) pass-move in U. We say that (2k + 1)-knot K and K′ are (high dimensional) pass-move equivalent if there exist (2k + 1)-knots K_{1}, …, K_{q+1} and pass-move charts U_{i} (i = 1, …, q) of K_{1} such that (1) K_{1} = K, K_{q+1} = K′, and (2) K_{i+1} is obtained from K_{i} by the high dimensional pass-move in U_{i}.

High dimensional pass-moves have the following relation with the Arf invariant and the signature of knots.

Theorem 2.4. For simple (2k + 1)-knots K_{1} and K_{2}, the following two conditions are equivalent. (k ≧ 1.)

(1) K_{1} is pass-move equivalent to K_{2}.

(2) K_{1} and K_{2} satisfy the condition that \{ \begin{matrix}Arf(K_{1}) = Arf(K_{2}) & when k is even \\ σ(K_{1}) = σ(K_{2}) & when k is odd. \end{matrix}

See [L2] for simple knots.

Theorem 2.5. For (2k+1)-knots K_{1} and K_{2}, the following two conditions are equivalent. (k ≧ 0.)

(1) There exists a (2k + 1)-knot K_{3} which is pass-move equivalent to K_{1} and cobordant to K_{2}.

(2) K_{1} and K_{2} satisfy the condition that \{ \begin{matrix}Arf(K_{1}) = Arf(K_{2}) & when k is even \\ σ(K_{1}) = σ(K_{2}) & when k is odd. \end{matrix}

The case k = 0 of Theorem 2.5 follows from [K1],[K2].

§3 We discuss the case when three spheres intersect in a sphere. Let F_{i} be closed surfaces (i = 1, 2, …, μ). A surface-(F_{1}, F_{2}, …, F_{μ})-link is a

smooth submanifold L = (K_{1}, K_{2}, …, K_{μ}) of S^{4}, where K_{i} is diffeomorphic to F_{i}. If F_{i} is orientable we assume that F_{i} is oriented and K_{i} is an oriented submanifold which is orientation preserving diffeomorphic to F_{i}. If μ = 1, we call L a surface-F_{1}-knot.

An (F_{1}, F_{2})-link L = (K_{1}, K_{2}) is called a semi-boundary link if

[K_{i}] = 0 ∈ H_{2}(S^{4} − K_{j}; ℤ) (i ≠ j)

following [S].

An (F_{1}, F_{2})-link L = (K_{1}, K_{2}) is called a boundary link if there exist Seifert hypersurfaces V_{i} for K_{i} (i = 1, 2) such that V_{1} ∩ V_{2}=ϕ.

An (F_{1}, F_{2})-link (K_{1}, K_{2}) is called a split link if there exist B_{1}^{4} and B_{2}^{4} in S^{4} such that B_{1}^{4}∩B_{2}^{4} = ϕ and K_{i} ⊂ B_{i}^{4}.

Definition. Let L_{1} = (K_{12}, K_{13}), L_{2} = (K_{23}, K_{21}), and L_{3} = (K_{31}, K_{32}) be surfacelinks. (L_{1}, L_{2}, L_{3}) is called a triple of surface-links if K_{ij} is diffeomorphic to K_{ji}. ((i, j)=(1,2),(2,3),(3,1).) (Note that the knot type of K_{ij} is different from that of K_{ji}.)

Definition. Let L_{1} = (K_{12}, K_{13}), L_{2} = (K_{23}, K_{21}), and L_{3} = (K_{31}, K_{32}) be surfacelinks. A triple of surface-links (L_{1}, L_{2}, L_{3}) is said to be realizable if there exists a transverse immersion f : S_{1}^{4} ⨿ S_{2}^{4} ⨿ S_{3}^{4} ↬ S^{6} such that (1)f|S_{i}^{4} is an embedding(i=1,2,3), and (2) (f^{−1}(f(S_{i}^{4}) ∩ f(S_{j}^{4})), f^{−1}(f(S_{i}^{4}) ∩ f(S_{k}^{4})) ) in S_{i}^{4} is L_{i}. ( (i, j, k)=(1,2,3), (2,3,1), (3,1,2).)

Note. If (L_{1}, L_{2}, L_{3}) is realizable, then K_{ij} are orientable and are given an orientation naturally. From now on we assume that, when we say a triple of surface-links, the triple of surface-links consists of oriented surface-links.

We state the main theorem.

Theorem 3.1. Let L_{i} (i = 1, 2, 3) be semi-boundary surface-links. Suppose the triple of surface-links (L_{1}, L_{2}, L_{3}) is realizable. Then we have the equality

β(L_{1}) + β(L_{2}) + β(L_{3}) = 0,

where β(L_{i}) is the Sato-Levine invariant of L_{i}.

Refer to [S] for the Sato-Levine invariant. Since there exists a triple of surface-links (L_{1}, L_{2}, L_{3}) such that β(L_{1})=0, β(L_{2})=0 and β(L_{3})=1 ([R] and [S]), we have:

Corollary 3.2. Not all triples of oriented surface-links are realizable.

We have sufficient conditions for the realization.

Theorem 3.3. Let L_{i} (i = 1, 2, 3) be split surface-links. Then the triple of surfacelinks (L_{1}, L_{2}, L_{3}) is realizable.

Theorem 3.4. Suppose L_{i} are (S^{2}, S^{2})-links. If L_{i} are slice links(i = 1, 2, 3), then the triple of surface-links (L_{1}, L_{2}, L_{3}) is realizable.

It is well known that there exists a slice-link which is neither a boundary link nor a ribbon link. Hence we have:

Corollary 3.5. There exists a realizable triple of surface-links (L_{1}, L_{2}, L_{3}) such that neither L_{i} are boundary links and all L_{i} are semi-boundary links.

Besides the above results, we prove the following triple are realizable.

Theorem 3.6. There exists a realizable triple of surface-links (L_{1}, L_{2}, L_{3}) such that neither L_{i} are semi-boundary links.

Here we state:

Problem (1). Suppose β(L_{1})+β(L_{2})+β(L_{3})=0. Then is the triple of surface-links (L_{1}, L_{2}, L_{3}) realizable?

Using a result of [O], we can make another problem from Problem (1).

Problem (2). Is every triple of (S^{2}, S^{2})-links realizable?

Note. By Theorem 3.4, if the answer to Problem (2) is negative, then the answer to an outstanding problem: “Is every (S^{2}, S^{2})-link slice?” is “no.” (Refer [CO] to the slice problem.)

§4

An (oriented) n-(dimensional) knot K is a smooth oriented submanifold of ℝ^{n+1} × ℝ which is PL homeomorphic to the standard n-sphere. We say that n-knots K_{1} and K_{2} are equivalent if there exists an orientation preserving diffeomorphism f : ℝ^{n+1} × ℝ → ℝ^{n+1} × ℝ such that f(K_{1})=K_{2} and f|_{K_{1}} : K_{1} → K_{2} is an orientation preserving diffeomorphism. Let π: ℝ^{n+1} × ℝ → ℝ^{n+1} be the natural projection map. A subset P of ℝ^{n+1} is called the projection of an n-knot K if there exists an orientation preserving diffeomorphism β : X→ K and a smooth transverse immersion γ : X → ℝ^{n+1} such that π|_{K} ∘ β=γ and π|_{K} ∘ β(K)=γ(X)=P. The singular point set of the projection of an n-knot is the set { x ∈ π|_{K} (K) | ♯ (π|_{K}) ^{−1}(x)≧ 2. }.

It is well-known that the projection of any 1-dimensional knot is the projection of a 1-knot equivalent to the trivial knot. This fact is used in a way to define the Jones polynomial and in another way to define the Conway-Alexander polynomial.

We consider the following problem.

Problem. (1) Let K be an n-knot diffeomorphic to the standard sphere. Let P be the projection of K. Is P the projection of an n-knot equivalent to the trivial knot?

(2) Furthermore, suppose that the singular point set of P consists of double points. Is P the projection of an n-knot equivalent to the trivial knot?

The author proved that the answer to Problem (2) in the case of n ≧ 3 is negative and hence that the answer to Problem (1) in the case of n ≧ 3 is also negative. We prove:

Theorem 4.1. Let n be any integer greater than two. There exists an n-knot K such that the projection P has the following properties.

(1) P is not the projection of any knot equivalent to the trivial knot.

(2) The singular point set of P consists of double points.

(3) K is diffeomorphic to the standard sphere.

Note. Problem (1) in the case of n = 2 remains open. Dr. Taniyama has informed that the answer to Problem (2) in the case of n = 2 is positive, that the proof is easy, but that he has not published the proof.

§5 We have the following outstanding open problems.

Problem (1). Classify n-links up to link concordance for n ≧ 1.

Problem (2). Is every even dimensional link slice?

Problem (3). Is every odd dimensional link concordant to (a sublink of) a homology boundary link?

The author has modified Problem (2) to formulate the following Problem (4). We consider the case of 2-component links. Let L = (K_{1}, K_{2}) ⊂ S^{n+2} ⊂ B^{n+3} be a 2m-link ( 2m ≥ 2). By Kervaire’s theorem in [Ke] there exist D_{i}^{2m+1} (i = 1, 2) embedded in B^{2m+3} such that D_{i}^{2m+1} ∩ S^{2m+2}=∂D_{i}^{2m+1}=K_{i}. Then D_{1}^{2m+1} and D_{2}^{2m+1} intersect mutually in general. Furthermore D_{1}^{2m+1} ∩ D_{2}^{2m+1} in D_{i}^{2m+1} defines (2m − 1)-link.

Problem (4). Can we remove the above intersection D_{1}^{2m+1} ∩ D_{2}^{2m+1} by modifying embedding of D_{1}^{2m+1} and D_{2}^{2m+1}?

If the answer to Problem (4) is positive, the answer to Problem (2) is positive. Here we make another problem from Problem (4).

Problem (5). What is obtained as a pair of (2m − 1)-links (D_{1}^{2m+1} ∩ D_{2}^{2m+1} in D_{1}^{2m+1}, D_{1}^{2m+1} ∩ D_{2}^{2m+1} in D_{2}^{2m+1}) by modifying embedding of D_{1}^{2m+1} and D_{2}^{2m+1}?

We have the following theorem, which is an answer to Problem (5).

Theorem 5.1. For all 2-component 2m-link L = (K_{1}, K_{2}) (m ≥ 0), there exist D_{1}^{2m+1} and D_{2}^{2m+1} as above such that each of (2m − 1)-links D_{1}^{2m+1} ∩ D_{2}^{2m+1} in D_{1}^{2m+1} and D_{1}^{2m+1} ∩ D_{2}^{2m+1} in D_{2}^{2m+1} is the trivial knot.

We have the following Theorem 5.2. We say that n-dimensional knots, K and K′, are (link-)concordant or (link-)cobordant if there is a smooth oriented submanifold C of S^{n+2} ×[0, 1], which meets the boundary transversely in ∂C, is PL homeomorphic to L_{0}×[0, 1], and meets S^{n+2}×{l} in L_{l} (l = 0, 1). Then we call C a concordance-cylinder of K and K′.

Theorem 5.2. For all 2-component n-link L = (K_{1}, K_{2}) (n > 1), there exist a boundary link L′ = (K′_{1}, K′_{2}) satisfying that K′_{i} is concordant to K_{i} and a concordance-

cylinder \{ \begin{matrix}C_{1} \\ C_{2} \end{matrix} of \{ \begin{matrix}K_{1} \\ K′_{1} \end{matrix} and \{ \begin{matrix}K_{2} \\ K′_{2} \end{matrix} such that each of (n − 1)-links, C_{1} ∩ C_{2} in C_{1}

and C_{1} ∩ C_{2} in C_{2}, is the trivial knot.

When n is even, Theorem 5.2 is Theorem 5.1. Because all even dimensional boundary links are slice.

By the following exciting theorem of Cochran and Orr, when n is odd, Theorem 5.2 is best possible from a viewpoint.

Theorem. [CO] Not all 2-component odd dimensional links are concordant to boundary links.

§6

Let D_{1}^{n}, D_{2}^{n}, D_{3}^{n} be submanifolds of S^{n+2} diffeomorphic to the n-disc such that Int (D_{i}^{n}) ∩ Int (D_{j}^{n})=ϕ (for i ≠ j) and ∂D_{1}^{n}= ∂D_{2}^{n}=∂D_{3}^{n}. Then D_{1}^{n} ∪ D_{2}^{n} ∪ D_{3}^{n} is called an n-dimensional θ-curve in ℝ^{n+2}. The set of the constituent knots of an n-dimensional θ-curve θ in ℝ^{n+2} is a set of three n-knots in S^{n+2}, which are made from D_{1}^{n} ∪ D_{2}^{n}, D_{2}^{n} ∪ D_{3}^{n}, and D_{1}^{n} ∪ D_{3}^{n}.

The definitions in the case of the PL category are written in [Y].

Problem. Take any set of three n-knots. Is it the set of the constituent knots of an n-dimensional θ-curve?

In [Y] it is proved that if K_{1}, K_{2}, and K_{3} are ribbon n-knots, then the set (K_{1}, K_{2}, K_{3}) is the set of the constituent knots of an n-dimensional θ-curve.

We discuss the case of non-ribbon knots. The following theorems hold both in the smooth category and in the PL category.

We have the following theorems.

Theorem 6.1. Let n be any positive integer. Let K_{1} and K_{2} be trivial knots. There exist an n-dimensional θ-curve θ in ℝ^{n+2} and a non-ribbon knot K_{3} such that (K_{1}, K_{2}, K_{3}) is the set of the constituent knots of the n-dimensional θ-curve θ in ℝ^{n+2}.

Furthermore we have the following.

Theorem 6.2. Let m be any odd positive integer. Let K_{1} and K_{2} be trivial knots.

(1) There exist m-dimensional θ-curves θ in ℝ^{m+2} and a non-ribbon and non-slice knot K_{3} such that (K_{1}, K_{2}, K_{3}) is the set of the constituent knots of the m-dimensional θ-curve θ in ℝ^{m+2}.

(2) There exist m-dimensional θ-curves θ in ℝ^{m+2} and a non-ribbon and slice knot K_{3} such that (K_{1}, K_{2}, K_{3}) is the set of the constituent knots of the m-dimensional θ-curve θ in ℝ^{m+2}.

We have the following.

Theorem 6.3. When n = 2m + 1(m ≥ 1), there exists a set of three n-knots which is never the set of the constituent knots of any θ-curve.

The above problem remains open.

§7 We use Theorem 1.1 in §1 to give an answer to a problem of Fox. In [F] Fox submitted the following problem about 1-links. Here, note that “slice

link” in the following problem is now called “ordinary sense slice link,” and “slice link in the strong sense” in the following problem is now called “slice link” by knot theorists.

Problem 26 of [F]. Find a necessary condition for L to be a slice link; a slice link in the strong sense.

Our purpose is to give some answers to the former part of this problem. The latter half is not discussed here. The latter half seems discussed much more often than the former half. See e.g. [CO], [L3], etc.

We review the definition of ordinary sense slice links and that of slice links, which we now use.

We suppose m-component 1-links are oriented and ordered.

Let L = (K_{1}, …, K_{m}) be an m-component 1-link in S^{3} = ∂B^{4}. L is called a slice 1-link, which is “a slice link in the strong sense” in the sense of Fox, if there exist 2-discs D_{i}^{2}(i = 1, …, m) in B^{4} such that D_{i}^{2} ∩ ∂B^{4} =∂D_{i}^{2}, D_{i}^{2} ∩ D_{j}^{2}=ϕ(i ≠ j), and (∂D_{1}^{2}, …, ∂D_{m}^{2}) in ∂B^{4} defines L.

Take a 1-link L in S^{3}. Take S^{4} and regard S^{4} as (ℝ^{3} × ℝ) ∪ {∞}. Regard the 3-sphere S^{3} as ℝ^{3} ∪ {∞} in S^{4}. L is called an ordinary sense slice 1-link, which is “a slice link” in the sense of Fox, if there exists an embedding f : S^{2} ↪ ℝ^{3} ×ℝ such that f is transverse to ℝ^{3} × {0} and f(S^{2}) ∩ (ℝ^{3} × {0}) in ℝ^{3} × {0} defines L. Suppose f defines a 2-knot X. Then L is called a cross-section of the 2-knot X.

From now on we use the terms in the ordinary sense now current.

Ordinary sense slice 1-links have the following properties.

Theorem 7.1. Let L be a 1-dimensional ordinary sense slice link. Then the followings hold.

(1) L is a proper link.

(2) Arf(L) = 0.

§8

Let K_{1} and K_{2} be smooth submanifolds of S^{n+2} diffeomorphic to an n-dimensional closed smooth manifold M. The notion of cobordism between K_{1} and K_{2} is defined naturally. A Seifert surface of K_{i} and a Seifert matrix of K_{i} are defined naturally. The notion of matrix cobordism between two Seifert matrices is defined naturally.

It is also natural to ask the following problem.

Problem. Are K_{1} and K_{2} as above cobordant?

The author thinks that there exists a kind of surgery exact sequence.

The author obtained the following results.

Theorem 8.1. There exist a (2n + 1)-dimensional closed oriented smooth manifold M and smooth submanifolds K_{1} and K_{2} of S^{2n+3} diffeomorphic to M such that (1) K_{1} and K_{2} are not cobordant, and (2) the Seifert matrices of K_{1} and K_{2} are matrix cobordant.

Theorem 8.2. There exist a 2n-dimensional closed oriented smooth manifold M and smooth submanifolds K_{1} and K_{2} of S^{2n+2} diffeomorphic to M such that K_{1} and K_{2} are not cobordant

In the both cases the obstructions live in certain homotopy groups.

References

[CS1] Cappell, S. and Shaneson, J. L., The codimension two placement problem and homology equivalent manifolds, Ann. of Math. 99 (1974).

[CS2] Cappell, S. and Shaneson, J. L., Singular spaces, characteristic classes, and intersection homology, Ann. of Math. 134 (1991), 325–374.

[CO] Cochran, T. D. and Orr, K. E., Not all links are concordant to boundary links, Ann. of Math. 138 (1993), 519–554.

[F] Fox, R. H., Some problems in knot theory, Proc. 1961 Georgia Conf. on the Topology of 3-manifolds, Prentice-Hall, 1962, pp. 168–176.

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