Going between Heegaard diagrams and framed link diagrams

Question

Every closed, compact, orientable 3-manifold can be represented by a Heegaard diagram. Similarly every such 3-manifold can be represented by a framed link diagram. Is there any general procedure for going between these? Specifically, given a Heegaard diagram is there an algorithmic way to obtain a framed link diagram of the same manifold, and vice versa?

Answer 1

As PVAL already said in the comments Lickorish's proof of Lickerish-Wallace theorem somehow tells you how to go from Heegaard diagrams to surgery presentations.

To go from surgery diagrams to Heegaard decomposition there is an algorithm. I tell to you how it works for surgeries on knots, then you can easily generalise to the case of links.

• Step 1: arrange your knot $K \subset S^3$ so that the projection on the $xy$-plane is regular (Fig. 1)
• Step 2: pick a tubular neighbourhood of $K$ and denote by $T \subset S^3$ its boundary torus (Fig. 2)
• Step 3: picture on $T$ the surgery longitude (call it $\gamma$)
• Step 4: in a neighbourhood of each crossing add a pipe as pictured in Figure 3. The resulting surface is going to be the surface underlying the Heegaard diagram of the surgery, denote it by $\Sigma$.
• Step 5: take as $\beta$-curves the curve $\gamma$ together with the boundary of the compressing disks of the pipes we introduced at the crossings (the blu curve pictured in Fig. 3)
• Step 6: take as $\alpha$-curves the boundary of the bounded regions of the diagram as shown in Figure 4.

The reason why this algorithm produce the right answer is not that deep: after gluing three-dimensional 2-handles along the $\alpha$- and the $\beta$-curves $\not=\gamma$, we get a three-manifold $Y$ with $2$ boundary components (a torus and a sphere). If we fill the sphere boundary component of $Y$ with a three-ball we get the complement of $K$ in $S^3$, and the Dhen filling operation along $\gamma$ prescribes to attach a 2-handle along $\gamma$ and fill the only sphere boundary component of the resulting three-manifold with a three-ball. This is the same as attaching 2-handles along all the $\alpha$- and the $\beta$-curves and fill the two boundary components with three-balls as prescribed by the Heegaard diagram $(\Sigma, \alpha, \beta)$.