---
abstract: 'Triple linking numbers were defined for $3$-component oriented surface-links in $4$-space using signed triple points on projections in $3$-space. In this paper we give an algebraic formulation using intersections of homology classes (or cup products on cohomology groups). We prove that spherical links have trivial triple linking numbers and that triple linking numbers are link homology invariants.'
author:
- |
    J. Scott Carter\
    University of South Alabama\
    Mobile, AL 36688\
    carter@mathstat.usouthal.edu
- |
    Seiichi Kamada\
    Osaka City University\
    Osaka 558-8585, JAPAN\
    kamada@sci.osaka-cu.ac.jp\
    skamada@mathstat.usouthal.edu
- |
    Masahico Saito\
    University of South Florida\
    Tampa, FL 33620\
    saito@math.usf.edu
- |
    Shin Satoh\
    Osaka City University\
    Osaka 558-8585, JAPAN\
    susato@sci.osaka-cu.ac.jp
title: 'Triple Linking of Surfaces in $4$-Space[[^1]]{}'
---

Introduction
============

A [*surface-link*]{} is a closed surface $F$ embedded in ${\bf R}^4$ locally flatly. In this paper, we always assume that $F$ is [*oriented*]{}, that is, each component of $F$ is orientable and given a fixed orientation. For a 3-component surface-link $F=K_1\cup K_2\cup K_3$, a linking number was defined in [@CJKLS] using its projection in ${\bf R}^3$ in a way that is analogous to the linking number in classical knot theory. In that paper, it was introduced as an example of non-triviality of the state-sum invariants of surface-links. In fact, the state-sum invariants in the classical link and surface-link case generalize linking number and Fox’s coloring number.

In the current paper, we give several alternative definitions of the triple linking number and some properties. The reader will find that this invariant is a quite natural generalization of the notion of classical linking number in contrast to a statement in Rolfsen [@Rolfsen] page 136: “There is, however no analogous notion of linking number to help us with codimension two link theory, for example, in higher dimensions”. We note, however, that Rolfsen himself with Massey [@MassayRolf] and with Fenn [@FennRolf] generalized classical linking numbers to higher dimensions using degrees of maps.

Let $F=K_1\cup K_2\cup K_3$ be a $3$-component surface-link in ${\bf R}^4$. It is known (see [@CS:book] for example) that a projection of $F$ into ${\bf R}^3$ can be assumed to have transverse double curves and isolated branch/triple points. At a triple point, three sheets intersect that have distinct relative heights with respect to the projection direction, and we call them top, middle, and bottom sheets, accordingly. If the orientation normals to the top, middle, bottom sheets at a triple point $\tau$ matches with this order the fixed orientation of ${\bf R}^3$, then the sign of $\tau$ is positive and $\varepsilon(\tau)=1$. Otherwise the sign is negative and $\varepsilon(\tau)=-1$. (See [@CJKLS; @CS:book].) It is also known that any closed oriented embedded surface $F$ in ${\bf R}^4$ bounds an oriented compact $3$-manifold $M$ embedded in ${\bf R}^4$, called a [*Seifert hypersurface*]{} of $F$, such that $\partial M=F$.

We give six methods for defining an integer (triple linking number) as follows.

-   Consider a surface diagram of $K_1\cup K_2\cup K_3$ in ${\bf R}^3$. A triple point is [*of type $(i,j,k)$*]{} if the top sheet comes from $K_i$, the middle comes from $K_j$, and the bottom comes from $K_k$. The sum of the signs of all the triple points of type $(1,2,3)$ is denoted by ${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)$. This is the definition given in [@CJKLS].

-   Let $M_i$ be a Seifert hypersurface for $K_i$ $(i=1,3)$. Assume that $M_i\cap K_2$ is a $1$-manifold in $K_2$ and that $M_1\cap K_2$ and $M_3\cap K_2$ intersect transversely. Count the intersections between them algebraically and denote the sum by ${\mbox{\rm Tlk}}_2(K_1,K_2,K_3)$.

-   Consider a Seifert hypersurface $M_1$ for $K_1$. Assume that $M_1\cap K_2$ is a $1$-manifold, which is disjoint from $K_3$. The linking number ${\mbox{\rm Link}}(M_1\cap K_2,K_3)$ is denoted by ${\mbox{\rm Tlk}}_3(K_1,K_2,K_3)$.

-   Let $M_i$ be a Seifert hypersurface for $K_i$ $(i=1,3)$ such that $M_1\cap M_3$ is a $2$-manifold which intersects $K_2$ transversely. Count the intersections between them algebraically and denote the sum by ${\mbox{\rm Tlk}}_4(K_1,K_2,K_3)$.

-   Let $M_i$ be a Seifert hypersurface for $K_i$ $(i=1,2,3)$ and let $N_2$ be a regular neighbourhood of $K_2$ in ${\bf R}^4$. We may assume that $M_i\cap \partial N_2$ is a $2$-manifold in $\partial N_2$ and that $M_1\cap \partial N_2$, $M_2\cap \partial N_2$ and $M_3\cap \partial N_2$ intersect transversely in a finite number of points. Count the intersections algebraically and denote the sum by ${\mbox{\rm Tlk}}_5(K_1,K_2,K_3)$.

-   Let $f:F_1 \cup F_2 \cup F_3 \rightarrow {\bf R}^4$ denote an embedding of the disjoint union of oriented surfaces $F_i$ representing $F=K_1\cup K_2\cup K_3$. Define a map $L:F_1 \times F_2 \times F_3 \rightarrow S^3 \times S^3$ by $$L(x_1,x_2,x_3) = \left( \frac {f(x_1)-f(x_2)}{||f(x_1)-f(x_2)||},
    \frac {f(x_2)-f(x_3)}{||f(x_2)-f(x_3)||} \right)$$ for $x_1\in F_1$, $x_2\in F_2$ and $x_3 \in F_3$, and denote the degree of $L$ by ${\mbox{\rm Tlk}}_6(K_1,K_2,K_3)$.

${\mbox{\rm Tlk}}_i(K_1,K_2,K_3)=\pm{\mbox{\rm Tlk}}_j(K_1,K_2,K_3)$ for any $i,j=1,\dots,6.$

[**Remark.**]{} In general, the [*triple linking number*]{} ${\mbox{\rm Tlk}}(K_i,K_j,K_k)$ for $i\ne j\ne k$ is defined to be the sum of the signs of all the triple points of type $(i,j,k)$ on a surface diagram of $F$;

$${\mbox{\rm Tlk}}(K_i,K_j,K_k)=
\sum_{{\scriptstyle \tau:\
\mbox{\footnotesize type}\ (i,j,k)}}
\ \varepsilon(\tau).$$ It is proved in [@CJKLS] that this number is an invariant of the surface-link $F$ (independent of a diagram in ${\bf R}^3$) by use of Roseman moves (Reidemeister moves for surface-link diagrams) [@Rose], and that this invariant vanishes in the case that $i=k$; that is, ${\mbox{\rm Tlk}}(K_i,K_j,K_i)=0$ for $i\neq j$. Hence throughout this paper, we always assume that $i, j, k$ are all distinct whenever we refer to ${\mbox{\rm Tlk}}(K_i,K_j,K_k)={\mbox{\rm Tlk}}_1(K_i,K_j,K_k)$.

We prove the following properties of triple linking by using the above interpretations.

-   ${\mbox{\rm Tlk}}(K_1,K_2,K_3)=-{\mbox{\rm Tlk}}(K_3,K_2,K_1)$.

-   ${\mbox{\rm Tlk}}(K_1,K_2,K_3)+{\mbox{\rm Tlk}}(K_2,K_3,K_1)+{\mbox{\rm Tlk}}(K_3,K_1,K_2)=0$.

<!-- -->

-   If $K_2$ is homeomorphic to a $2$-sphere, then ${\mbox{\rm Tlk}}(K_1,K_2,K_3)=0$.

-   If both of $K_1$ and $K_3$ are homeomorphic to a $2$-sphere, then ${\mbox{\rm Tlk}}(K_1,K_2,K_3)=0$.

In [@NSato] the asymmetric linking number ${\mbox{\rm Alk}}(K,K')$ for a two component oriented surface-link $F=K \cup K'$ was defined to be the non-negative generator of the image of $H_1(K) \rightarrow H_1(S^4 \backslash K') \cong {\bf Z}$.

If ${\mbox{\rm Alk}}(K_2, K_3)=0$, then ${\mbox{\rm Tlk}}(K_1, K_2,K_3)= {\mbox{\rm Tlk}}(K_3, K_2, K_1)= 0$.

Two $n$-component surface-links $F=K_1\cup \dots \cup K_n$ and $F'=K'_1\cup \dots \cup K'_n$ are [*link homologous*]{} if there is a compact oriented $3$-manifold $W$ properly embedded in ${\bf R}^4
\times [0,1]$ such that $W$ has $n$ components $W_1, \dots, W_n$ with $\partial W_i= K_i \times \{0\} \cup (-K'_i)\times
\{1\}$. This relation is sometimes called [*link-cobordism*]{}, but that term also denotes the concordance relation. Since link homotopy implies link homology, the following theorem implies that triple linking invariants are link homotopy invariants (this fact is also seen from the sixth definition of ${\mbox{\rm Tlk}}$). For related topics, refer to [@Tim1; @TO; @Kirk; @KK; @Koschorke; @MassayRolf; @Ruberman; @NSato].

Triple linking invariants are link homology invariants: If $F=K_1\cup K_2\cup K_3$ and $F'=K'_1\cup K'_2\cup K'_3$ are link homologous, then ${\mbox{\rm Tlk}}(K_i,K_j,K_k)={\mbox{\rm Tlk}}(K'_i,K'_j,K'_k)$.

See Remark 6.2 for further information about link homology.

By Theorem 1.2, for any 3-component surface-link $F=K_1\cup K_2\cup K_3$, there exists a pair of integers $a$ and $b$ such that $$(*) \left\{
\begin{array}{l}
{\mbox{\rm Tlk}}(K_1,K_2,K_3)=-{\mbox{\rm Tlk}}(K_3,K_2,K_1)=-(a+b),\\
{\mbox{\rm Tlk}}(K_2,K_3,K_1)=-{\mbox{\rm Tlk}}(K_1,K_3,K_2)=b, \\
{\mbox{\rm Tlk}}(K_3,K_1,K_2)=-{\mbox{\rm Tlk}}(K_2,K_1,K_3)=a.
\end{array} \right.$$ In [@CJKLS], it is shown that for any pair of integers $a$ and $b$, there exists a surface-link $F$ whose triple linking numbers satisfy the above equations. However, that paper does not treat any problem about genera of the components of $F$. By Theorem 1.3, we see that

\(1) if $a\ne 0$ and $b=0$, then $g(K_i)\geq 1$ $(i=1,2)$, and

\(2) if $a\ne 0$, $b\ne 0$ and $a+b\ne 0$, then $g(K_i)\geq 1$ $(i=1,2,3)$,

where $g(K_i)$ denotes the genus of $K_i$.

[(i)]{} For any integer $a\ne 0$, there exists a surface-link $F=K_1\cup K_2\cup K_3$ whose triple linking numbers satisfy the above equations $(*)$ with $b=0$ and $g(K_i)=1$ $(i=1,2)$ and $g(K_3)=0$.

[(ii)]{} For any pair of integers $a$ and $b$ with $a\ne 0$, $b\ne 0$ and $a+b\ne 0$, there exists a surface-link $F=K_1\cup K_2\cup K_3$ whose triple linking numbers satisfy the above equations $(*)$ and $g(K_i)=1$ $(i=1,2,3)$.

This paper is organized as follows: in Section 2, we interprete ${\mbox{\rm Tlk}}_1$ in terms of the decker curves of a surface diagram. In Section 3 we give precise definitions of triple linking numbers ${\mbox{\rm Tlk}}_i$ for $i=2,\dots,5$ (in terms of homology) and prove Theorem 1.1. Section 4 is devoted to proving Theorems 1.2–1.5. Proposition 1.6 is proved in Section 5.

Throughout this paper, all the homology and cohomology groups have the [**Z**]{}-coefficient.

Decker Curves and Triple Linking
================================

Let $F$ be a surface-link and $F^*$ a surface diagram of $F$ with respect to a projection $p:{\bf R}^4\rightarrow{\bf R}^3$. Let $\Gamma(F^*)$ denote the double point set of $F^*$; $$\{ p(x)\ |\  x \in F, \ p(x)=p(y) \ \mbox{for some} \
  y \in F, x \neq y \},$$ which consists of immersed curves, called [*double curves*]{}. A double curve $C^*$ is an immersed circle or an immersed arc in ${\bf R}^3$. If $C^*$ is an immersed circle, then $(p|_F)^{-1}(C^*)=C\cup C'$ for some pair of immersed circles $C$ and $C'$ in $F$. If $C^*$ is an immersed arc, then its endpoints are branch points of $F^*$ and $(p|_F)^{-1}(C^*)=C\cup C'$ for some pair of immersed arcs $C$ and $C'$ in $F$ with $\partial C= \partial C'$. The curves $C$ and $C'$ are called [*decker curves*]{} over $C^*$: one of them is in higher position than the other with respect to the projection direction, which is called an [*upper decker curve*]{} and the other is called an [*lower decker curve*]{}. We notice that the preimage of a triple point consists of three points of $F$ which are intersections of decker curves. See [@CS:book] for details.

Double curves and decker curves are oriented as follows: Let $x$ be a point of $F$ whose image $x^*=p(x)$ is not a branch point. There is a regular neighborhood $N$ of $x$ in $F$ such that $p|_N$ is an embedding. An [*orientation normal $\vec{n}$ to $N^*=p(N)$* ]{} in ${\bf R}^3$ at $x^*$ is specified in such a way that $(\vec{v}_1, \vec{v}_2, \vec{n})$ matches the orientation of ${\bf R}^3$, where the pair of tangents $(\vec{v}_1, \vec{v}_2)$ defines the orientation of $N^*$ that is induced from the orientation of $N \subset F$. If $y$ is a double point on a double curve $C^*$, then $C^*$ is locally an intersection of $N_1^*$ and $N_2^*$, where $N_1^*$ is upper and $N_2^*$ is lower. We assign a tangent vector $\vec{v}$ of $C^*$ at $y$ such that $(\vec{n}_1,
\vec{n}_2, \vec{v})$ matches the orientation of ${\bf R}^3$. This defines an orientation of $C^*$, cf. [@CJKLS; @CS:book]. We give an orientation to the lower decker curve over $C^*$ such that it inherits the orientation from $C^*$, and give the opposite orientation to the upper decker curve. Note that, if $C^*$ is an arc, then the orientations of $C$ and $C'$ are compatible (i.e., the union $C \cup C'$ forms an oriented immersed circle in $F$).

Let $F=K_1 \cup K_2 \cup K_3$ be a 3-component surface-link. A double curve $C^*$ is [*of type $(i,j)$*]{} if the upper decker curve lies in $K_i$ and the lower decker curve lies in $K_j$. A decker curve over $C^*$ is [*of type $(i,j)$*]{} if $C^*$ is so.

At a triple point $\tau$, if the orientation normals to the top, middle, and bottom sheets at $\tau$ matches with this order the fixed orientation of ${\bf R}^3$, then the sign of $\tau$ is positive and $\varepsilon(\tau)=+1$; otherwise the sign is negative and $\varepsilon(\tau)=-1$.

We interprete the triple linking ${\mbox{\rm Tlk}}_1$ in terms of double decker curves as follows. Let $D_{12}^{\ell}$ (resp. $D_{23}^u$) denote the union of lower decker curves of type $(1,2)$ (resp. upper decker curves of type $(2,3)$). Note that both $D_{12}^{\ell}$ and $D_{23}^u$ are contained in $K_2$.

${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)=-{\mbox{\rm Int}}_{K_2}(D_{12}^{\ell},D_{23}^u),
$ where ${\mbox{\rm Int}}_{K_2}(D_{12}^{\ell},D_{23}^u)$ is the intersection number in $K_2$.

[*Proof.*]{} Let $\tau$ be a triple point of type $(1,2,3)$. The preimage of $\tau$ consists of three points of $F$. Exactly one of them is on $K_2$ and that is a double point of $D_{12}^{\ell}$ and $D_{23}^u$. Conversely the image of a double point of $D_{12}^{\ell}$ and $D_{23}^u$ is a triple point of $F^*$ of type $(1,2,3)$. Hence there is a one-to-one correspondence between the set of triple points of type $(1,2,3)$ and double points of $D_{12}^{\ell}$ and $D_{23}^u$. If the sign of $\tau$ is positive (or negative, resp.) then the corresponding intersection of $D_{12}^{\ell}$ and $D_{23}^u$ is negative (resp. positive), see Figure 1. Thus we have the result.  ${\hbox{\rule{6pt}{6pt}}}$

Since $D_{12}^{\ell}$ is the union of circles in ${\bf R}^4$ disjoint from $K_3$, the linking number ${\mbox{\rm Link}}(D_{12}^{\ell}, K_3)$ is defined.

${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)={\mbox{\rm Link}}(D_{12}^{\ell}, K_3).$

[*Proof.*]{} Without loss of generality, we may assume that the projection $p$ is given by $p(w,x,y,z) = (x,y,z)$. For a real number $\lambda$ we denote by $t_\lambda: {\bf R}^4 \to {\bf R}^4$ the translation with $t_\lambda(w,x,y,z)= (w+\lambda, x,y,z)$. Let $M'_3$ be a 3-chain in ${\bf R}^4$ with $\partial M'_3 = K_3$. Take a sufficiently large number $R$ and consider a $3$-chain $$M_3 = \cup_{\lambda \in [0, R]} t_\lambda(K_3)
+ t_{R}(M'_3)$$ so that $\partial M_3= K_3$ and $D_{12}^{\ell} \cap M_3 = D_{12}^{\ell}
\cap (\cup_{\lambda \in [0, R]}
t_\lambda(K_3))$. The projection $p$ induces a one-to-one correspondence between the geometric intersection $D_{12}^{\ell} \cap (\cup_{\lambda \in [0, R]}
t_\lambda(K_3))$ and the subset of $D_{12}^{\ell *} \cap K_3^* = p(D_{12}^{\ell}) \cap p(K_3)$ consisting of points where $D_{12}^{l*}$ is higher than $K_3^*$ (in the over-under information of the surface diagram $F^*$), i.e., the set of triple points of $F^*$ of type $(1,2,3)$. Since the orientation of $D_{12}^{\ell}$ is parallel to the orientation of $D_{12}^{\ell *}$, the sign of an intersection of $D_{12}^{\ell}$ and $\cup_{\lambda \in [0, R]}
t_\lambda(K_3)$ coincides with the sign of the corresponding intersection of $D_{12}^{\ell*}$ and $K_3^*$, which is the sign of the triple point (see Figure 2). Thus we have the result.  

Proof of Theorem 1.1
====================

For a compact oriented $n$-manifold $M$ with $\{A,B\}= \{\partial M, \emptyset\}$, we denote by $$\cdot_M \quad : H_p(M,A) \times H_q(M,A) \rightarrow
H_{p+q-n}(M,A)$$ the intersection map, which is defined by $$x \cdot_M y = P_M( P_M^{-1}(x) \cup P_M^{-1}(y) )$$ where $P_M: H^{\ast}(M,B) \rightarrow H_{n-\ast}(M,A)$ is the Poincaré duality isomorphism (see [@Massey], page 391). We will use $\cdot$ and $P$ instead of $\cdot_M$ and $P_M$ when their meanings are obvious in context.

Let $F=K_1\cup K_2\cup K_3$ be a $3$-component surface-link. For simplicity of argument, we assume that $F$ is embedded in the $4$-sphere $S^4={\bf R}^4\cup\{\infty\}$. For a regular neighborhood $N_i$ of $K_i$ in $S^4$, we put $$E_i={\rm Cl}(S^4 \backslash N_i), \
E_{ij}={\rm Cl}(S^4 \backslash (N_i\cup N_j))
\mbox{ for $i \ne j$, and }
E= {\rm Cl}(S^4 \backslash (N_1\cup N_2\cup N_3)),$$ where Cl denotes the closure. We denote by $M_i$ a $3$-chain in $S^4$ with $\partial M_i = K_i$ for $i=1,2,3$ (the reader may suppose that it is a Seifert hypersurface for $K_i$, i.e., a compact oriented $3$-manifold embedded in $S^4$ with $\partial M_i=K_i$). We also denote by $M_i$ the homology class in $$H_3(S^4,K_i) {\cong} H_3(S^4,N_i) {\cong}
H_3(E_i,\partial E_i)$$ represented by $M_i$. By $u_i \in H^1(E_i)$ we denote the Poincaré dual of $M_i \in H_3(E_i,\partial E_i)$, i.e., $M_i= P(u_i) = u_i\cap[E_i]$. For a subset $X$ of $E_i$, we will denote by $u_i|_{X} \in H^1(X)$ the image of $u_i$ by the inclusion-induced homomorphism $H^1(E_i) \to H^1(X)$. Moreover, if $X$ is an $n$-manifold, we denote by $M_i|_{(X, \partial X)}$ (or $M_i|_{X}$ if $\partial X =\emptyset$) the Poincaré dual $P_X(u_i|_{X})= (u_i|_{X}) \cap [X] \in
H_{n-1}(X, \partial X)$ of $u_i|X$.

For $i \in \{1,3\}$, since $K_2 \subset E_i$, $M_i|_{K_2} \in H_1(K_2)$ is defined. (When we consider $M_i$ as a $3$-chain, the intersection of $M_i$ and $K_2$ (as a $1$-cycle in $K_2$) represents $M_i|_{K_2}$.) Let $${\mbox{\rm Tlk}}_2(K_1, K_2, K_3)=
\varepsilon_{K_2}(M_1|_{K_2} \cdot M_3|_{K_2}),$$ where $\varepsilon_{K_2}: H_0(K_2) \to {\bf Z}$ is the augmentation.

${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)={\mbox{\rm Tlk}}_2(K_1,K_2,K_3).$

[*Proof.*]{} We assume $F \subset {\bf R}^4 \subset S^4$ and continue the situation of the proof of Lemma 2.2. Let $M'_1$ be a 3-chain in ${\bf R}^4 (\subset S^4)$ with $\partial M'_1= K_1$ and consider a $3$-chain $M_1$ such that $$M_1 = -\cup_{\lambda \in [-R, 0]} t_\lambda(K_1)
+ t_{-R}(M'_1)$$ with $\partial M_1= K_1$. The intersection of $M_1$ and $K_2$ is equal to that of $-\cup_{\lambda \in [-R, 0]} t_\lambda(K_1)$ and $K_2$ which is the 1-chain $-D_{12}^{\ell}$ in $K_2$, and the intersection of $M_3$ and $K_2$ is equal to that of $\cup_{\lambda \in [0, R]} t_\lambda(K_3)$ and $K_2$ which is the 1-chain $D_{23}^u$ in $K_2$. Therefore, by Lemma 2.1, we have $$\begin{array}{ll}
{\mbox{\rm Tlk}}_1(K_1,K_2,K_3)
&= -{\mbox{\rm Int}}_{K_2}(D_{12}^{\ell},D_{23}^u) \\
&= -\varepsilon_{K_2}(D_{12}^{\ell} \cdot D_{23}^u) \\
&= -\varepsilon_{K_2}(-M_1|_{K_2} \cdot M_3|_{K_2})\\
&= {\mbox{\rm Tlk}}_2(K_1,K_2,K_3). \ {\hbox{\rule{6pt}{6pt}}}\end{array}$$

[**Remark.**]{} The argument in Lemmas 2.2 and 3.1 implies that for any Seifert hypersurface $M_1$ for $K_1$, the intersection of $M_1$ and $K_2$ (as a 1-cycle in $K_2$) is homologous to $-D_{12}^{\ell}$, and that for any Seifert hypersurface $M_3$ for $K_3$, the intersection of $M_3$ and $K_2$ (as a 1-cycle in $K_2$) is homologous to $D_{23}^u$.

We denote by $[M_1 \cap K_2]_{E_3} \in H_1(E_3)$ the homology class of the intersection $M_1 \cap K_2$ as a $1$-cycle in $E_3$. This is equal to the image of $M_1|_{K_2} \in H_1(K_2)$ under the inclusion-induced homomorphism $H_1(K_2) \to H_1(E_3)$ and also equal to the image of $M_1|_{(E_{13}, \partial E_{13})}
\cdot  [K_2]_{E_{13}} \in H_1(E_{13})$ under the inclusion-induced homomorphism $H_1(E_{13}) \to H_1(E_3)$, where $[K_2]_{E_{13}} \in H_2(E_{13})$ is represented by $K_2$. Let $$\begin{array}{ll}
{\mbox{\rm Tlk}}_3(K_1, K_2, K_3)
&=
{\mbox{\rm Link}}([M_1 \cap K_2]_{E_3}, K_3) \\
&=
\varepsilon_{E_3} ([M_1 \cap K_2]_{E_3} \cdot M_3),
\end{array}$$ where $M_3 \in H_3(E_3, \partial E_3)$ and $\varepsilon_{E_3}: H_0(E_3) \to {\bf Z}$ is the augmentation.

${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)=-{\mbox{\rm Tlk}}_3(K_1,K_2,K_3).$

[*Proof.*]{} In the situation of the proof of Lemma 2.2, $[M_1 \cap K_2]_{E_3} \in H_1(E_3)$ is represented by the $1$-cycle $-D_{12}^{\ell}$. Therefore, by Lemma 2.2, we have $$\begin{array}{ll}
{\mbox{\rm Tlk}}_3(K_1,K_2,K_3)
&= {\mbox{\rm Link}}([M_1 \cap K_2]_{E_3}, K_3) \\
&= {\mbox{\rm Link}}(-D_{12}^{\ell}, K_3) \\
&= -{\mbox{\rm Tlk}}_1(K_1,K_2,K_3). \ {\hbox{\rule{6pt}{6pt}}}\end{array}$$

We denote by $[M_1 \cap M_3]_{(E_{13}, \partial E_{13})}
\in H_1(E_{13}, \partial E_{13})$ the class of the intersection $M_1 \cap M_2$ as a $2$-cycle in $(E_{13}, \partial E_{13})$ when we regard $M_i$ as a $3$-chain. This is equal to the intersection product $M_1 |_{(E_{13}, \partial E_{13})}
\cdot M_3 |_{(E_{13}, \partial E_{13})}
\in H_2(E_{13}, \partial E_{13})$. Let $$\begin{array}{ll}
{\mbox{\rm Tlk}}_4(K_1, K_2, K_3)
&=
\varepsilon_{E_{13}} ([M_1 \cap M_3]_{(E_{13}, \partial E_{13})}
  \cdot [K_2]_{E_{13}}) \\
&=
\varepsilon_{E_{13}} (M_1 |_{(E_{13}, \partial E_{13})}
\cdot M_3 |_{(E_{13}, \partial E_{13})} \cdot [K_2]_{E_{13}}).
\end{array}$$

${\mbox{\rm Tlk}}_3(K_1,K_2,K_3)={\mbox{\rm Tlk}}_4(K_1,K_2,K_3).$

[*Proof.*]{} Let $i_*: H_*(E_{13}) \to H_*(E_3)$ and $i^*: H^*(E_3) \to H^*(E_{13})$ be the inclusion-induced homomorphisms. Recall that $[M_1 \cap K_2]_{E_3}= i_*(M_1|_{(E_{13}, \partial E_{13})}
\cdot  [K_2]_{E_{13}})$. Thus,

$$\begin{array}{ll}
{\mbox{\rm Tlk}}_4(K_1, K_2, K_3)
&=
\varepsilon_{E_{13}} (
M_1 |_{(E_{13}, \partial E_{13})}
\cdot M_3 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}}) \\
&=
\varepsilon_{E_{3}} \circ i_* (
M_1 |_{(E_{13}, \partial E_{13})}
\cdot M_3 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}})   \\
&= -
\varepsilon_{E_{3}} \circ i_* (
M_3 |_{(E_{13}, \partial E_{13})}
\cdot M_1 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}})  \\
&= -
\varepsilon_{E_{3}} \circ i_* (
u_3 |_{E_{13}}
\cap (M_1 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}}) )  \\
&= -
\varepsilon_{E_{3}} \circ i_* (
i^*(u_3)
\cap (M_1 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}}) )  \\
&= -
\varepsilon_{E_{3}}  (
u_3
\cap i_* (M_1 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}}) )  \\
&= -
\varepsilon_{E_{3}}  (
M_3
\cdot [M_1 \cap K_2]_{E_3} )  \\
&=
\varepsilon_{E_{3}}  (
[M_1 \cap K_2]_{E_3} \cdot M_3
)  \\
&=
{\mbox{\rm Tlk}}_3(K_1, K_2, K_3). \ {\hbox{\rule{6pt}{6pt}}}\end{array}$$

For $i \in \{1,2,3\}$, since $\partial N_2 \subset E_i$, $M_i|_{\partial N_2} \in H_2(\partial N_2)$ is defined. Let $$\begin{array}{ll}
{\mbox{\rm Tlk}}_5(K_1,K_2,K_3)
&=
\varepsilon_{\partial N_2}(
M_1|_{\partial N_2} \cdot
M_2|_{\partial N_2} \cdot
M_3|_{\partial N_2}) \\
&=
<u_1|_{\partial N_2} \cup
u_2|_{\partial N_2} \cup
u_3|_{\partial N_2}, [\partial N_2] >.
\end{array}$$

${\mbox{\rm Tlk}}_4(K_1,K_2,K_3)={\mbox{\rm Tlk}}_5(K_1,K_2,K_3).$

[*Proof.*]{} Let $i : \partial N_2 \to N_2$ be the inclusion map. In $H_0(N_2)$, we have $$\begin{array}{ll}
i_*(
M_1|_{\partial N_2} \cdot
M_2|_{\partial N_2} \cdot
M_3|_{\partial N_2})
&= -
i_*(
M_1|_{\partial N_2} \cdot
M_3|_{\partial N_2} \cdot
M_2|_{\partial N_2}) \\
&= -
i_*(
\partial_*(M_1|_{(N_2,\partial N_2)}) \cdot
\partial_*(M_3|_{(N_2,\partial N_2)}) \cdot
M_2|_{\partial N_2}) \\
&= -
i_*(
\partial_*(M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)}) \cdot
M_2|_{\partial N_2}) \\
&= -
(M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)}) \cdot
i_*(M_2|_{\partial N_2}) \\
&= -
M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)} \cdot
(-[K_2]_{N_2}) \\
&=
M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)} \cdot
[K_2]_{N_2}.
\end{array}$$ Thus $${\mbox{\rm Tlk}}_5(K_1,K_2,K_3) = \varepsilon_{N_2}(
M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)} \cdot
[K_2]_{N_2}
).$$ It is obvious that $$\varepsilon_{N_2}(
M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)} \cdot
[K_2]_{N_2} )
=
\varepsilon_{E_{13}}(
M_1|_{(E_{13},\partial E_{13})} \cdot
M_3|_{(E_{13},\partial E_{13})} \cdot
[K_2]_{E_{13}} )$$ and hence we have the result.  

${\rm Tlk}_6(K_1,K_2,K_3) =
\pm {\rm Tlk}_1(K_1,K_2,K_3).$

[*Proof.*]{} Since ${\rm Tlk}_6$ is an ambient isotopy invariant, we may assume that the surface-link $F= f(F_1) \cup f(F_2) \cup
f(F_3)$ is in general position with respect to the projection $p : {\bf R}^4 \to {\bf R}^3$ with $p(w,x,y,z) = (x,y,z)$. The preimage of a particular point $((1,0,0,0), (1,0,0,0))$ by $L$ consists of triples $(x_1,x_2,x_3) \in F_1 \times F_2 \times
F_3$ such that $p(f(x_1))= p(f(x_2)) = p(f(x_3))$ and $f(x_1)$ is the upper, $f(x_2)$ is the middle, $f(x_3)$ is the lower lift of the triple point $p(f(x_1))$. For each such triple $(x_1, x_2, x_3)$, let $D_T^2$, $D_M^2$, $D_B^2$ be regular neighborhoods of them in $F_1 \cup F_2 \cup F_3$, and let $\varepsilon \in \{+1, -1\}$ be the sign of the triple point $p(f(x_1))$. Let $(x_T,y_T)$, $(x_M, -\varepsilon z_M)$ and $(y_B,z_B)$ be coordinate systems of $D_T^2$, $D_M^2$ and $D_B^2$ around $x_1, x_2$ and $x_3$, respectively. Modifying $f$ up to ambient isotopy, we may assume that the restriction of $f$ to $D_T^2 \cup D_M^2 \cup D_B^2$ is given by defined by $$\begin{aligned}
(x_T,y_T) &
\mapsto & (0, x_0, y_0, z_0)+ (3,x_T,y_T,0) \\
(x_M,- \varepsilon z_M) &
\mapsto & (0, x_0, y_0, z_0)+ (2,x_M, 0, z_M) \\
(y_B,z_B) &
\mapsto & (0, x_0, y_0, z_0)+ (1,y_B, z_B) \end{aligned}$$ where $(x_0, y_0, z_0) \in {\bf R}^3$ is the triple point $p(f(x_1))$. In this situation, the restriction $$L': D_T^2 \times D_M^2 \times D_B^2 \rightarrow S^3 \times
S^3$$ is given by the formula $$\left(
\frac
{(1,x_T-x_M,y_T,-z_M)}
{\sqrt{1+ (x_T-x_M)^2+ y_T^2 + z_M^2}},
\frac
{(1,x_M,-y_B,z_M-z_B)}
{\sqrt{1+ x_M^2 +y_B^2 +(z_M-z_B)^2}}
\right).$$ The map $L'$ is injective and hence it is a homeomorphism onto its image. Its (local) degree is $+1$ or $-1$ which depends only on $\varepsilon$. Since the degree of $L$ is the sum of the (local) degrees of $L'$ for all triples $(x_1, x_2, x_3)$ in the preimage $L^{-1}((1,0,0,0),(1,0,0,0))$, this number agrees up to sign with the triple linking number ${\rm Tlk}_1(f(F_1), f(F_2), f(F_3)).$  

By Lemmas 3.1–3.5, we have Theorem 1.1.

Proof of Theorems 1.2–1.5
=========================

To prove Theorem 1.2, it is useful to change $\partial N_2$ in the definition of ${\mbox{\rm Tlk}}_5$ for $\partial E_2$.

${\mbox{\rm Tlk}}_5(K_1,K_2,K_3)
=- \varepsilon_{\partial E_2}(
M_1|_{\partial E_2} \cdot
M_2|_{\partial E_2} \cdot
M_3|_{\partial E_2})$, where the intersections are taken in $\partial E_2$.

[*Proof.*]{} Since $\partial N_2$ and $\partial E_2$ are the same $3$-submanifold of $S^4$ with opposite orientations, $[\partial N_2] = - [\partial E_2]$ in $H_3(\partial N_2)= H_3(\partial E_2)$. Thus, in $H_0(\partial N_2)=H_0(\partial E_2)$, $$\begin{array}{ll}
M_1|_{\partial N_2} \cdot
M_2|_{\partial N_2} \cdot
M_3|_{\partial N_2}
&=
(u_1|_{\partial N_2} \cup
u_2|_{\partial N_2} \cup
u_3|_{\partial N_2}) \cup [\partial N_2]  \\
&=
(u_1|_{\partial E_2} \cup
u_2|_{\partial E_2} \cup
u_3|_{\partial E_2}) \cup (-[\partial E_2])  \\
&= -
M_1|_{\partial E_2} \cdot M_2|_{\partial E_2} \cdot
M_3|_{\partial E_2}.   \ {\hbox{\rule{6pt}{6pt}}}\end{array}$$

[*Proof of Theorem $1.2$*]{} (i) $$\begin{array}{ll}
{\mbox{\rm Tlk}}_2(K_1, K_2, K_3)
&=
\varepsilon_{K_2}(M_1|_{K_2} \cdot M_3|_{K_2}) \\
&= -
\varepsilon_{K_2}(M_3|_{K_2} \cdot M_1|_{K_2}) \\
&= -
{\mbox{\rm Tlk}}_2(K_3, K_2, K_1).
\end{array}$$

\(ii) Note that $M_i|_{(E, \partial E)} = P_{E}(u_i|_{E})
\in H_3(E, \partial E)$ is the image of $M_i$ under $$H_3(E_i, \partial E_i) \to
H_3(E_i, \partial E_i \cup N_j \cup N_k)
\cong
H_3(E, \partial E),$$ and $M_i|_{\partial E_2}
\in H_2(\partial E_2)$ is the image of $M_i|_{(E, \partial E)}$ under $$H_3(E, \partial E) \to H_2(\partial E)
\cong
H_2(\partial E_1) \oplus H_2(\partial E_2) \oplus
H_2(\partial E_3) \to H_2(\partial E_2),$$ the boundary operator followed by the projection to $H_2(\partial E_2)$. We denote by $(M_i|_{\partial E_2})_{\partial E} \in H_2(\partial E)$ the image of $M_i|_{\partial E_2}$ under the inclusion-induced homomorphism $H_2(\partial E_2) \to H_2(\partial E)$. By Lemma 4.1, $${\mbox{\rm Tlk}}_5(K_1, K_2, K_3)= -
\varepsilon_{\partial E}(
(M_1|_{\partial E_2})_{\partial E} \cdot
(M_2|_{\partial E_2})_{\partial E} \cdot
(M_3|_{\partial E_2})_{\partial E}).$$ Thus, we have $$\begin{array} {rl}
{\mbox{\rm Tlk}}_5(K_1, K_2, K_3) + &
{\mbox{\rm Tlk}}_5(K_2, K_3, K_1) +
{\mbox{\rm Tlk}}_5(K_3, K_1, K_2) \\
=
& - \varepsilon_{\partial E}(
(M_1|_{\partial E_2})_{\partial E} \cdot
(M_2|_{\partial E_2})_{\partial E} \cdot
(M_3|_{\partial E_2})_{\partial E} \\
& +
(M_2|_{\partial E_3})_{\partial E} \cdot
(M_3|_{\partial E_3})_{\partial E} \cdot
(M_1|_{\partial E_3})_{\partial E} \\
& +
(M_3|_{\partial E_1})_{\partial E} \cdot
(M_1|_{\partial E_1})_{\partial E} \cdot
(M_2|_{\partial E_1})_{\partial E}
) \\
=
& - \varepsilon_{\partial E}(
(M_1|_{\partial E_2})_{\partial E} \cdot
(M_2|_{\partial E_2})_{\partial E} \cdot
(M_3|_{\partial E_2})_{\partial E} \\
& +
(M_1|_{\partial E_3})_{\partial E} \cdot
(M_2|_{\partial E_3})_{\partial E} \cdot
(M_3|_{\partial E_3})_{\partial E} \\
& +
(M_1|_{\partial E_1})_{\partial E} \cdot
(M_2|_{\partial E_1})_{\partial E} \cdot
(M_3|_{\partial E_1})_{\partial E}
) \\
=
& - \varepsilon_{\partial E}(
M_1|_{\partial E} \cdot
M_2|_{\partial E} \cdot
M_3|_{\partial E}) \\
=
& - \varepsilon_{\partial E}(
\partial_*(M_1|_{(E,\partial E)}) \cdot
\partial_*(M_2|_{(E,\partial E)}) \cdot
\partial_*(M_3|_{(E,\partial E)})) \\
=
& - \varepsilon_{\partial E}( \partial_* (
M_1|_{(E,\partial E)} \cdot
M_2|_{(E,\partial E)} \cdot
M_3|_{(E,\partial E)})) \\
=
& 0. \  {\hbox{\rule{6pt}{6pt}}}\end{array}$$ [*Proof of Theorem $1.3$*]{} (i) The intersection number between two oriented curves on a 2-sphere vanishes. By Lemma 2.1, we have ${\mbox{\rm Tlk}}(K_1,K_2,K_3)=0$.

\(ii) This is an immediate consequence of (i) and Theorem 1.2(ii).  ${\hbox{\rule{6pt}{6pt}}}$

[*Proof of Theorem $1.4$*]{} If ${\mbox{\rm Alk}}(K_2, K_3)=0$, then ${\mbox{\rm Tlk}}_3(K_1, K_2, K_3) ={\mbox{\rm Link}}([M_1 \cap K_2]_{E_3}, K_3) = 0$, for $[M_1 \cap K_2]_{E_3} \in H_1(E_3)$ is the image of $M_1|_{K_2} \in H_1(K_2)=0$. By Theorem 1.2, we have ${\mbox{\rm Tlk}}(K_3, K_2, K_1)=0$.  ${\hbox{\rule{6pt}{6pt}}}$

We consider surface-links $F=K_1\cup K_2\cup K_3$ in which each $K_i$ is not necessarily connected. Such a surface-link is called a [*$3$-partitioned*]{} surface-link. The definition of the triple linking of $F=K_1\cup K_2\cup K_3$ is generalized directly for 3-partitioned surface-links, and all results and proofs in Sections 2 and 3 are valid for 3-partitioned surface-links. Theorem 1.5 is a special case of the following:

If two $3$-partitioned surface-links $F=K_1\cup K_2\cup K_3$ and $F'=K'_1\cup K'_2\cup K'_3$ are link homologous, then ${\mbox{\rm Tlk}}(K_i,K_j,K_k)={\mbox{\rm Tlk}}(K'_i,K'_j,K'_k)$.

[*Proof.*]{} It is sufficient to prove ${\mbox{\rm Tlk}}(K_1, K_2, K_3) =
{\mbox{\rm Tlk}}(K'_1, K'_2, K'_3)$ in a special case that $K_i=K'_i$, $K_j= K'_j$ and $K_k$ is homologous to $K'_k$ in $S^4 \backslash (K_i \cup K_j)$, where $\{i,j,k\}=\{1,2,3\}$. If $k=2$, then ${\mbox{\rm Tlk}}_4(K_1, K_2, K_3)= {\mbox{\rm Tlk}}_4(K_1, K'_2, K_3)$ by definition. If $k=1$, then ${\mbox{\rm Tlk}}_3(K_1, K_2, K_3)= {\mbox{\rm Tlk}}_3(K'_1, K_2, K_3)$. (This is seen as follows: Let $M_1$ be a 3-chain with $\partial M_1=K_1$. Since $K'_1$ is homologous to $K_1$ in $S^4 \backslash (K_2\cup K_3)$, there is a 3-chain $B$ in $S^4 \backslash (K_2\cup K_3)$ with $\partial B = K'_1 - K_1$. Let $M'_1=
M_1 +B$, which is a 3-chain with $\partial M'_1= K'_1$. Then $[M_1 \cap K_2]_{E_3}= [M'_1 \cap K_2]_{E_3}$ in $H_1(E_3) \cong H_1(S^4 \backslash K_3)$. ) The case $k=3$ is reduced to the previous case ($k=1$) by use of Theorem 1.2(i).  

Proof of Proposition 1.6
========================

\(1) Let $\ell=k_1\cup k_2$ be a $(2,2a)$-torus link in a $3$-disk $D^3$ with ${\mbox{\rm Link}}(k_1,k_2)=a$. Let $\gamma$ be a simple loop in ${\bf R}^4$ which intersects a $3$-disk $B_0$ in ${\bf R}^4$ transversely at a single interior point of $B_0$ in the positive direction. Identify $D^3\times S^1$ with a regular neighborhood $N(\gamma)$ of $\gamma$ in ${\bf R}^4$ and let $T_1\cup T_2$ be the image of $\ell\times S^1=k_1\times S^1\cup k_2\times S^1$ in ${\bf R}^4$. Let $F=K_1\cup K_2\cup K_3$ be a surface-link with $K_1=T_1$, $K_2=T_2$ and $K_3=\partial B_0$. Then $F$ is the desired link.

\(2) Let $\ell= k_1 \cup k_2 \cup k_3$ be a pretzel link of type $(2a, -2b)$ in a 3-disk $D^3$ so that ${\mbox{\rm Link}}(k_1, k_2)=a$, ${\mbox{\rm Link}}(k_2, k_3)= -b$ and ${\mbox{\rm Link}}(k_1, k_3)=0$. Let $B_1, B_2, B_3$ be mutually disjoint 3-disks embedded in ${\bf R}^4$ and let $\gamma$ be a simple loop in ${\bf R}^4$ which intersects $B_i$ ($i=1,2,3$) transversely at a single interior point of $B_i$ in the positive direction. Identify $D^3 \times S^1$ with a regular neighborhood $N(\gamma)$ of $\gamma$ in ${\bf R}^4$ and let $T_1 \cup T_2 \cup T_3$ be the image of $\ell \times S^1 =
k_1 \times S^1 \cup k_2 \times S^1 \cup k_3 \times S^1$ in ${\bf R}^4$. Let $F = K_1 \cup K_2 \cup K_3$ be a surface-link obtained from $(T_1 \cup T_2 \cup T_3) \cup (\partial B_1 \cup
\partial B_2 \cup \partial B_3)$ by piping such that $F$ has a projection as in Figure 3. Then $F$ is the desired link.  

Remarks
=======

[**Remark 6.1**]{} The definition of ${\mbox{\rm Tlk}}_6$ can be seen as a direct analogue of (6) given in [@Rolfsen] page 133. See also [@FennRolf; @MassayRolf]. This generalizes the triple linking to all link maps, instead of embeddings. Moreover, it easily generalized to all dimensions. Let $M_i$ denote a closed connected $n$-manifold for $i=1,\ldots , n+1$. Let an embedding $f: \cup_{i=1}^{n+1} M_i
\rightarrow  {\bf R}^{n+2}$ be given. Define $L: \prod_{i=1}^{n+1} M_i \rightarrow
\prod_{j=1}^n S^{n+1}_j$ as follows. Let $x_i \in M_i$; for $i=1, \ldots, n$, let $\Delta_i = f(x_i)-f(x_{i+1})/ || f(x_i)-f(x_{i+1}) ||$. Then $$L(x_1,\ldots x_{n+1}) = ( \Delta_1, \Delta_2, \ldots , \Delta_n).$$ The general $(n+1)$-fold linking number, ${\mbox{\rm Glk}}$, is defined by $${\mbox{\rm Glk}}(f(M_1), \dots, f(M_{n+1}))= \deg(L).$$ We can generalize the notion of ${\mbox{\rm Tlk}}_1$ to a diagram in ${\bf R}^{n+1}$ of an $(n+1)$-component $n$-manifold-link $M_1 \cup \dots \cup M_{n+1}$ in ${\bf R}^{n+2}$; namely, a diagram has generic $(n+1)$-tuple points and we count the number of times $M_1$ is over $M_2$ is over ... is over $M_{n+1}$ with signs. It is difficult to show that this value is an invariant of the $n$-manifold-link in ${\bf R}^{n+2}$ directly, since we do not know Reidemeister moves for higher dimensions $(n \geq 3)$. However, the proof of Lemma 3.5 goes through to show that ${\mbox{\rm Glk}}$ is the same as this count (up to sign). Thus we have that this number (generalization of ${\mbox{\rm Tlk}}_1$) is an invariant of an $(n+1)$-component $n$-manifold-link.

[**Remark 6.2**]{} In classical link theory, the linking number determines the link homology classes completely. However, the triple linking of surface-links is [*not*]{} a complete invariant of the surface-link homology; there exists a pair of surface-links with the same triple linking invariants which are not link homologous. A classification of surface-link homology classes is discussed in a forthcoming paper.

[**Acknowledgments**]{} The authors would like to thank Akio Kawauchi for helpful suggestions. JSC is being supported by NSF grant DMS-9988107. SK and SS are being supported by Fellowships from the Japan Society for the Promotion of Science. MS is being supported by NSF grant DMS-9988101.

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[^1]: MRCN:57Q45
