Alternating sum formulae for the determinant and other link invariants
  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 $j$. 
Oliver T. Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin,   Neal W. Stoltzfus