  We study the crossing number of links that are formed by edges of a
triangulation T of the 3-sphere with n tetrahedra. We show that the crossing
number is bounded from above by an exponential function of n^2. In general,
this bound can not be replaced by a subexponential bound. However, if T is
polytopal (resp. shellable) then there is a quadratic (resp. biquadratic) upper
bound in n for the crossing number. In our proof, we use a numerical invariant
p(T), called polytopality, that we have introduced in math.GT/0009216.
