c(K;F) is zero only if t(K) <= delta(F), and otherwise c(K;F) >= 2(t(K) - delta(F)) + 1, with the bound sharp in linear order for connected-sum families.
Schultens,The Classification of Heegaard Splittings for (Compact Orientable Sur- face)×S 1, Proc
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Crossing Numbers of Knots on Closed Surfaces
c(K;F) is zero only if t(K) <= delta(F), and otherwise c(K;F) >= 2(t(K) - delta(F)) + 1, with the bound sharp in linear order for connected-sum families.