Heegaard splittings of the pair of the solid torus and the core loop (Q5956920)

A Heegaard splitting of a compact connected orientable 3-manifold with boundary \(M\) is a homeomorphism \(M\cong C_1\cup_HC_2\), where \(C_1\), and \(C_2\) are compression bodies. In case of a manifold with empty boundary, the definition becomes the classical one. If \(T\subset M\) is a properly embedded 1-submanifold of \(M\), the Heegaard splitting \(M\cong C_1 \cup_H C_2\) is called a Heegaard splitting of the pair \((M,T)\), if \(T\) is transverse to the Heegaard surface \(H\) of the splitting and both \(T_1=T\cap C_1\), and \(T_2=T \cap C_2\) are sets of arcs properly and trivially embedded in \(C_1\), and \(C_2\) respectively. The authors showed in their previous paper [Thin position for 1-submanifold in 3-manifold, Pac. J. Math. 197, No. 2, 301-324 (2001; Zbl 1050.57016)] that every such pair \((M,T)\) admits Heegaard splittings. This paper shows that any Heegaard splitting of the pair \((D^2\times S^1,\{P\} \times S^1)\), where \(P\) is any point in the interior of \(D^2\), is standard (i.e., it is either stabilized or cancellable). In order to prove this, the notion of Heegaard splittings of pairs \((M,\Gamma)\), where \(\Gamma\) is a graph properly embedded in \(M\), has been introduced.