What is the fastest algorithm for Weighted Planar Max Cut
Shih, Wu, and Kuo's "unifying" paper reduces the planar maximum cut problem to two related problems in planar graphs: (1) maximum-weight bipartite matching and (2) minimum cycle (aka weighed girth).  Both problems are solved in $O(n^{3/2}\log n)$ time using divide-and-conquer algorithms that rely on Lipton and Tarjan's planar separator theorem.

After a long series of improvements, the minimum-weight cycle in a weighted planar graph can now be computed in $O(n \log\log n)$ time using a recent algorithm of Łącki and Sankowski (2011).  

However, I am unaware of any improvement on the time to find maximum-weight matchings, which means Shih, Wu, and Kuo's $O(n^{3/2}\log n)$ time algorithm is still the fastest algorithm known (at least to me) for weighted planar max-cut.