Inapproximability Ratios for Crossing Number

Assuming P≠NP, we show that if there is a constant-factor polynomial c-approximation algorithm for Crossing Number, then c≥2−1617≈1.058824. Adding the Unique Games Conjecture to the hypotheses, then c≥2−α≈1.121433, where α is the approximation ratio of the algorithm for Maximum Cut by Goemans and Williamson.