  A classical result states that the determinant of an alternating link is
equal to the number of spanning trees in a checkerboard graph of an alternating
connected projection of the link.
  We generalize this result to show that the determinant is the alternating sum
of the number of quasi-trees of genus j of the dessin of a non-alternating
link.
  Furthermore, we obtain formulas for other link invariants by counting
quantities on dessins. In particular we will show that the $j$-th coefficient
of the Jones polynomial is given by sub-dessins of genus less or equal to $j$.
