There are some C or Java programs available for computing crossing number, but not in wolfram language. But we can call it. One of them is QuickCross.
Import["!D://QuickCross//QC -g I~qkzXZLw", "Table"]
(*{{"CR", ":", 12}}*)
This is a bit like mathematica calling nauty-geng's results.
If one wishes to actually solve instances, Markus Chimani and Tilo Wiedera produced a mixed-integer linear program in 2016 that is able to compute crossing numbers for small graphs, complete with a proof file, and produced an online interface, Crossing Number Web Compute , where researchers can submit their owninstances.
The crossing number results of specific graphs can be referred to in the following review.
- Clancy K, Haythorpe M, Newcombe A. A survey of graphs with known or bounded crossing numbers[J]. arXiv preprint arXiv:1901.05155, 2019.