  We compute the $\delta$-invariant of a curve singularity parameterized by
generic sparse polynomials. We apply this to describe topological types of
generic singularities of sparse resultants and ``algebraic knot diagrams''
(i.e. generic algebraic spatial curve projections).
  Our approach is based on some new results on zero loci of Schur polynomials,
on transversality properties of maps defined by sparse polynomials, and on a
new refinement of the notion of tropicalization of a curve
(ultratropicalization), which may be of independent interest.
