422 360 at 430 432 at 396 396 at 204 414 at 370 214 at 370 360 at 353 380 at 352 576 at 369 576 at 381 576 at 397 576 at 324 380 at 406 692 at 490 214 at 394 692 at 456 214 at 265 788 at 429 200 at 372 684 at 411 191 at 224 512 at 396 194 at 310 490 at 175 624 at 240 711 at 216 684 at 320 548 at 260 399 at 295 455 at 200 455 at 380 200 at 386 445 at 404 461 at 390 467 at 375 468 at 282 303 at 294 672 at 350 307 at 365 324 at 246 280 at 240 648 at 275 490 at 250 440

![The Pointed Heegaard triple](TrojniDiag "fig:") \[fig:triple\]

The intersections between the $\alpha $ and $\beta $ curves are denoted in the same way as before. New intersections between the $\alpha $, $\beta $ and $\gamma $ curves we will need are denoted by: $\alpha _{1}\cap \gamma _{1}=\{x_{1}',x_{2}',\ldots ,x_{m}'\}$, $\alpha _{2}\cap \gamma _{2}=\{s\}$, $\alpha _{3}\cap \gamma _{3}=\{f_{1}',f_{2}'\}$, $\beta _{1}\cap \gamma _{1}=\{t_{1}^{+},t_{1}^{-}\}$, $\beta _{2}\cap \gamma _{2}=\{r\}$, $\beta _{3}\cap \gamma _{3}=\{t_{2}^{+},t_{2}^{-}\}$ (see Figure \[fig:triple\]).

We express 