Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The intersection of three spheres in a sphere and a new application of
  the Sato-Levine invariant   Take transverse immersions f from a disjoint unin of the three 4-spheres
$S^4_1$, $S^4_2$, and $S^4_3$ into $S^6$ with the following properties:
  (1) The restriction of $f$ to $S^4_i$ is an embedding,
  (2) The intersection of $f(S^4_i)$ and $f(S^4_j)$ is not empty and connected,
  (3)The intersection among $f(S^4_1)$, $f(S^4_2)$, and $f(S^4_3)$ is not
empty.
  Then we obtain three surface-links $L_i=(S^4_i\cap S^4_j, S^4_i\cap S^4_k)$
in $S^4_i$, where $(i,j,k)=(1,2,3), (2,3,1), (3,1,2).$ We prove that, we have
the equality $\beta(L_1)+\beta(L_2)+\beta(L_3)=0$, where $\beta(L_i)$ is the
Sato-Levine invariant of $L_i$, if all $L_i$ are semi-boundary links.

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The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat