For link diagrams on a connected sum of projective planes, rank(M_L) equals r minus n minus 1 plus rank(N_L), which counts equivalence classes under region crossing change.
Russell,Ineffective sets and the region crossing change operation, J
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Region crossing change on nonorientable surfaces
For link diagrams on a connected sum of projective planes, rank(M_L) equals r minus n minus 1 plus rank(N_L), which counts equivalence classes under region crossing change.