  Computing unlinking number is usually very difficult and complex problem,
therefore we define BJ-unlinking number and recall Bernhard-Jablan conjecture
stating that the classical unknotting/unlinking number is equal to the
BJ-unlinking number. We compute BJ-unlinking number for various families of
knots and links for which the unlinking number is unknown. Furthermore, we
define BJ-unlinking gap and construct examples of links with arbitrarily large
BJ-unlinking gap. Experimental results for BJ-unlinking gap of rational links
up to 16 crossings, and all alternating links up to 12 crossings are obtained
using programs LinKnot and K2K. Moreover, we propose families of rational links
with arbitrarily large BJ-unlinking gap and polyhedral links with constant
non-trivial BJ-unlinking gap. Computational results suggest existence of
families of non-alternating links with arbitrarily large BJ-unlinking gap.
