Is it yet known what genes make one likely to get asthma? Are some of the genetic mutations known and can be tested for?

Phenotypes of Asthma. There are particular chromosomes noted for being associated with the asthma genotype but this does not mean they are a part of the asthma phenotype (genes displaying the disease). There is no testing except for symptoms and pulmonary function testing (including direct and indirect provocation testing) to determine asthma.
It's coming! There are not genetic tests that i would recommend for asthma at this time, but there are several genetic markers for asthma that are used for research purposes.