The indexed links of non-singular Morse-Smale flows on graph manifolds

Figure 1.  The shaded part is $ \partial_{-}R $

Figure 2.   

Figure 3.   

Figure 4.   

Figure 5.  $k_{2}$ and $k_{3}$ are two parallel $\left ( 3, 2 \right )$-cables of $S^{1}\times \left \{ 0 \right \}$

Figure 6.  $k_{2}$ is a $\left ( 2, 1 \right )$-cable of $S^{1}\times \left \{ 0 \right \}$

Figure 7.  $\beta' \cup \beta'' \simeq c(R_0)$, $\beta' \cup \alpha' \simeq k_1$, and $\beta'' \cup \alpha' \simeq k_2$

Figure 8.  $L_{v}$ is the star neighborhood of $v$

Figure 9.  $L_{v_{i}}$ is the block associated to $v_i$ for $i = 1, 2, 3$

Figure 10.  Let $L_{v_{i}}$ be the block associated to a saddle vertex $v_{i}$. It is easy to observe that $L' = (\partial _{-} L'\times I)+L_{v_{2}}+L_{v_{7}}+L_{v_{5}}+L_{v_{6}}+L_{v_{3}}+L_{v_{4}}+ L_{v_{1}}$, where $v_{3}$ and $v_{4}$ can be reached by $v_{2}$ through oriented paths. Obviously, $L = L_{v_{7}}+L_{v_{5}}+L_{v_{6}}+ L_{v_{1}}+L_{v_{2}}+L_{v_{3}}+L_{v_{4}}$

Figure 11.  These three generalized graphs are trees. There are two vertices that are adjacent to two degree-1 vertices in Figure (a), there is one vertex adjacent to two degree-1 vertices in Figure (b), and there is no vertex adjacent to two degree-1 vertices in Figure (c)

Figure 12.  This generalized graph supports $z_i$ degree-1 vertices and $b_i$ ends, and its first Betti number is $g_i$

Figure 13.  The number is the slope of the corresponding fiber. On the right side of each subgraph is the base orbifold of $ M $, where hollow circles correspond to index-1 knots and solid circles correspond to other knots of $ l $