On Virtual Crossing Number Estimates For Virtual Links
  We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in $R^{2}$.   We tackle this problem by the so-called $\xi$-polynomial whose leading (lowest) degree naturally estimates the virtual crossing number.   Several sufficient conditions for minimality together with infinite series of new examples are given.   We also state several open questions about $M$-diagrams, which are minimal according to our sufficient conditions. 
Denis Afanasiev and Vassily Manturov