  The exact (closed form) solutions of the equations of motion in the theory of
general relativity that describe motion of test particle and photon in Kerr and
Kerr-(anti) de Sitter spacetimes all involve the multivariable hypergeometric
function of Lauricella $F_D$: Kraniotis [Class. Quantum Grav. {\bf 21} 2004,
4743; Class. Quantum Grav. {\bf 22} 2005, 4391; Class. Quantum Grav. {\bf 24}
2007, 1775]. The domain of variables ${\cal D}_n$ of the corresponding function
depends on the first integrals of motion associated with the isometries of the
Kerr-(anti) de Sitter metric and Carter's constant $Q$ as well as on the
cosmological constant $\Lambda$ and the Kerr (rotation) parameter. In this work
we discuss the topological properties of the domain ${\cal D}_n$ and in
particular its fundamental connection with the theory of braids. An intrinsic
relationship of general relativity with the pure braids is established.
