Constructing doubly-pointed Heegaard diagrams compatible with \((1,1)\) knots (Q2872793)

A way to obtain a \(3\)-manifold is identifying the boundaries of two genus \(g\) handlebodies by a homeomorphism. The homeomorphism is completely determined by two sets \(\alpha, \beta\) of \(g\) curves in the splitting surface \(\Sigma\). Every closed, orientable \(3\)-manifold can be encoded by a Heegaard diagram \((\Sigma, \alpha, \beta)\). By marking two points \(x, y \in \Sigma - \alpha - \beta\) in a given Heegaard diagram \((\Sigma, \alpha, \beta)\) for a \(3\)-manifold \(M\), a unique knot \(K\subset M\) is specified. The knot is composed of the union of the two trivial arcs on either side of \(\Sigma\) that join \(x\) and \(y\) while avoiding meridian disks bounded by \(\alpha, \beta\). It is said that \((\Sigma, \alpha, \beta, x, y)\) is a doubly-pointed Heegaard diagram compatible with \(K\).NEWLINENEWLINE The paper under review concerns the problem of constructing a genus \(1\) doubly-pointed Heegaard diagram compatible with a given knot in the three-sphere. The main result provides an algorithm to construct a Heegaard diagram for any \((1, 1)\) knot given a particular presentation, known as Schubert's normal form. The basic input for knot Floer homology is a Heegaard knot diagram, and Ozsváth and Szabó showed that, in the case of doubly-pointed genus \(1\) Heegaard diagrams, the computation of knot Floer homology is combinatorial. \textit{H. Goda} et al. [Geom. Dedicata 112, 197--214 (2005; Zbl 1081.57011)] constructed numerous examples of genus \(1\) doubly-pointed Heegaard diagrams compatible with \((1, 1)\) knots whose Floer homology was unknown, including several knots up to 10 crossings. Using Schubert's normal form which is a parametrization of \((1, 1)\) knots due to \textit{D. H. Choi} and \textit{K. H. Ko} [J. Knot Theory Ramifications 12, No. 4, 463--491 (2003; Zbl 1055.57003)] the author is able to generalize Goda, Matsuda, and Morifuji's examples.