On \(p\)-symmetric Heegaard splittings (Q2706863)

The notion of \(p\)-symmetric Heegaard splitting for a closed orientable 3-manifold \(M\) was defined by \textit{J. S. Birman} and \textit{H. M. Hilden} [Trans. Am. Math. Soc. 213, 315-352 (1975; Zbl 0312.55004)]. The \(p\)-symmetric Heegaard genus \(g_p(M)\) of \(M\) is the smallest integer \(g\) such that \(M\) admits a \(p\)-symmetric Heegaard splitting of genus \(g\). The author proves that every \(p\)-fold strictly-cyclic branched covering of a \(b\)-bridge link in the 3-sphere admits a \(p\)-symmetric Heegaard splitting of genus \(g=(b-1)(p-1)\). This gives a complete converse of one of the results proved in the quoted paper. The proof is based on the concept of special plat presentation of a link. Similar results are also obtained for the class of weakly \(p\)-symmetric Heegaard splittings.