Exponentially many nowhere-zero $Z_3$-, $Z_4$-, and $Z_6$-flows

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. These results may be seen as a counting alternative to the well-known proofs of existence of $Z_3$-, $Z_4$-, and $Z_6$-flows. In the dual setting, proving exponential number of 3-colorings of planar triangle-free graphs is a related open question due to Thomassen. As a part of the proof we obtain a new splitting lemma for 6-edge-connected graphs, that may be of independent interest.