Permutations with Kazhdan-Lusztig polynomial P_{id,w}(q) = 1 + q^h
  Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial P_{id,w}(q)=1+q^h for some h. 
Alexander Woo, Sara Billey, Jonathan Weed