Comparing h-genera, bridge-1 genera and Heegaard genera of knots

Let h(K), g1(K), gH(K) and t(K) be the h-genus, bridge-1 genus, Heegaard genus and tunnel number of a knot K in the 3-sphere S³, respectively. It is known that gH(K) − 1 = t(K) ⩽ g1(K) ⩽ h(K) ⩽ gH(K). Then a natural question arises: under what conditions do the h-genus, bridge-1 genus, Heegaard genus and tunnel number of a knot become equal? We provide the necessary and sufficient conditions for those equalities and use these to show that for each integer n ⩾ 1, there are infinitely many knots in each of the following three families (Formula presented.) This resolves a conjecture by Morimoto (2005) that each of these families is nonempty.