  We propose two fixed-parameter tractable algorithms for the weighted Max-Cut
problem on embedded 1-planar graphs parameterized by the crossing number $k$ of
the given embedding. A graph is called 1-planar if it can be drawn in the plane
with at most one crossing per edge. Our algorithms recursively reduce a
1-planar graph to at most $3^k$ planar graphs, using edge removal and node
contraction. Our main algorithm then solves the Max-Cut problem for the planar
graphs using the FCE-MaxCut introduced by Liers and Pardella [23]. In the case
of non-negative edge weights, we suggest a variant that allows to solve the
planar instances with any planar Max-Cut algorithm. We show that a maximum cut
in the given 1-planar graph can be derived from the solutions for the planar
graphs. Our algorithms compute a maximum cut in an embedded weighted 1-planar
graph with $n$ nodes and $k$ edge crossings in time $O(3^k \cdot n^{3/2} \log
n)$.
