On handle number of Seifert surfaces in \(S^ 3\)
For a Seifert surface \(R\) of a link \(L\) in \(S^ 3\), let \(M = \) the closure of the complement of \(R \times [0,1]\) in the exterior of \(L\) (say, \(E(L)\)), \(\gamma =\) the closure of \(\partial E(L) - \partial R \times [0,1]\), and \(R_ + = R \times \{1\}\), \(R_ - = R \times \{0\}\). This manifold pair \((M,\gamma)\) is called the complementary sutured manifold for \(R\) [\textit{D. Gabai}, Comment. Math. Helv. 61, 519-555 (1986; Zbl 0621.57003)]. A compression body \(W\) is a cobordism \(\text{rel }\partial\) between surfaces \(\partial_ + W\) and \(\partial_ - W\) obtained from \(\partial_ + W \times [0,1]\) by attaching 2-handles to \(\partial_ + W \times \{1\}\) and then 3-handles to cap off all resulting 2-spheres. Define \(h(W)\) to be the minimal number of 2-handles required in this construction of \(W\). A Heegard splitting \((W,W')\) of the complementary sutured manifold \((M,\gamma)\) for \(R\) is a pair of compression bodies such that \(M = W \cup W'\), \(W \cap W' = \partial_ + W = \partial_ + W'\), \(\partial_ - W = R_ +\), \(\partial_ - W' = R_ -\). Define the handle number of \(R\) by \(h(R) = \min\{h(W)\mid (W,W')\) is a Heegaard splitting of \((M,\gamma)\}\). Note that \(h(R) = 0\) if and only if \(R\) is a fiber surface. The author is primarily concerned with the case where \(R\) is obtained from a plumbing of two other Seifert surfaces \(R_ 1\) and \(R_ 2\). He proved in a previous paper [Osaka J. Math. 29, No. 1, 21-40 (1992; see the review above)] that for such a plumbing one has \(h(R_ 1) + h(R_ 2) - 1 \leq h(R) \leq h(R_ 1) + h(R_ 2)\). In this paper, the main theorem gives a sufficient condition to have \(h(R) = h(R_ 1) + h(R_ 2)\), and the handle numbers of all incompressible Seifert surfaces for prime knots of ten or fewer crossings are provided.
scientific article; zbMATH DE number 408427