In the paper "The structure and stability of persistence modules", Page 49, they use vineyard technique to visualize the 1-parameter family of persistence modules produced by three interpolation constructions described in the paper, generating the following figure:
Can someone explain to me how they come from? Thank you!
I'll explain the $2$-interleaving case; the $3$-interleaving case is left to you.
The interpolating families are constructed out of the maps in the sequence $$\begin{align} &\color{blue}{\mathbb{I}^{[0,4)}[q-3] \oplus \mathbb{I}^{[1,6)}[p-3]} \\ &\xrightarrow{\Omega'} \color{red}{\mathbb{I}^{[0,4)}[p-1] \oplus \mathbb{I}^{[1,6)}[q-1]} \\ &\xrightarrow{\Omega} \color{green}{\mathbb{I}^{[0,4)}[q+1] \oplus \mathbb{I}^{[1,6)}[p+1]} \\ &\xrightarrow{\Omega''} \color{orange}{\mathbb{I}^{[0,4)}[p+3] \oplus \mathbb{I}^{[1,6)}[q+3]}. \end{align}$$ Since we are using the canonical interleaving of interval modules, the entries of the maps in its standard matrix form are all $\pm 1$, if it is not necessarily zero because either the domain or codomain is zero.
Recall also that as persistence modules over the $(p,q)$-plane, $$\begin{align} \mathbb{I}^{[a,b)}[p-c] &= \mathbb{I}^{[a+c,b+c) \times (-\infty, \infty)} \\ \mathbb{I}^{[a,b)}[q-c] &= \mathbb{I}^{(-\infty, \infty) \times [a+c, b+c)}, \end{align}$$ so it is easy to read off the supports of these modules.
We shall visualize the interpolating families by sketching their supports.
Let us begin with $\mathrm{im}(\Omega)$. We want the image of $$\color{red}{\mathbb{I}^{[0,4)}[p-1] \oplus \mathbb{I}^{[1,6)}[q-1]} \xrightarrow{\Omega} \color{green}{\mathbb{I}^{[0,4)}[q+1] \oplus \mathbb{I}^{[1,6)}[p+1]}$$ inside the diagonal band $$\Delta_{[-1,1]} = \{(p,q) \mid -2 \leq q-p \leq 2\}.$$
The diagonal band is bordered by the brown lines above. The image, indicated in yellow, is the overlap between the red and green regions. We see that on $\Delta_{-1} = \{q-p=-2\}$ we recover the interval module $\mathbb{I}^{[0,4)}$, and on $\Delta_1 = \{q-p=2\}$ we get $\mathbb{I}^{[1,6)}$, so sweeping along diagonals of slope $1$ from the bottom right to the upper left, the yellow region describes an interpolating family between these two interval modules. In this case, we begin at $\mathbb{I}^{[0,4)}$, move linearly to $\mathbb{I}^{[0.5,5.5)}$, and finally move linearly to $\mathbb{I}^{[1,6)}$, as the vineyard confirms.
Now let's consider the kernel of $$\color{green}{\mathbb{I}^{[0,4)}[q+1] \oplus \mathbb{I}^{[1,6)}[p+1]} \xrightarrow{\Omega''} \color{orange}{\mathbb{I}^{[0,4)}[p+3] \oplus \mathbb{I}^{[1,6)}[q+3]}.$$
The kernel is the region in green that is not covered by orange, with the caveat that "double green" cancels out and is included regardless, based on the signs in the definition of $\Omega'$. So the $\ker(\Omega'')$ is the region in yellow, which describes an interpolating family that moves linearly from $\mathbb{I}^{[0,4)}$ to $\mathbb{I}^{[-0.5,4.5)}$ to $\mathbb{I}^{[1,6)}$.
Finally, let us look at the cokernel of $$\color{blue}{\mathbb{I}^{[0,4)}[q-3] \oplus \mathbb{I}^{[1,6)}[p-3]} \xrightarrow{\Omega'} \color{red}{\mathbb{I}^{[0,4)}[p-1] \oplus \mathbb{I}^{[1,6)}[q-1]}.$$
We want the red region that is not covered by the blue region, where the red region is counted with multiplicity (but not the blue, since $\Omega'$ has rank $\leq 1$). This is outlined in yellow. This time, in addition to the $\mathbb{I}^{[0,4)}$ to $\mathbb{I}^{[1.5,5.5)}$ to $\mathbb{I}^{[1,6)}$ family, this is also an extra family indicated by the cross-hatching that starts at $\mathbb{I}^{[3,3)}$, grows for a bit, then dies at $\mathbb{I}^{[2,2)}$ since the red region is counted with multiplicity $2$. This agrees with the vineyard shown.
As a word of encouragement, these calculations are not too difficult. You just need to have a clear understanding of the definitions, and exercise care with the linear algebra.