From a planar graph $\Gamma$, equipped with an integer-valued weight function $d:E(\Gamma) \sqcup V(\Gamma) \to \mathbb{Z}$, one can build a $3$-manifold $M_{\Gamma}$ as follows. For each vertex $v$, draw a small planar unknot centered at $v$. For each edge $e$ connecting vertices $v$ and $w$, add a series of $d(e)$ clasps between the corresponding unknots (with positive weights represented by right-handed clasps and negative weights represented by left-handed clasps). The result is a link with unknotted components - call it $L_{\Gamma}$. To get the $3$-manifold $M_{\Gamma}$, perform Dehn surgery on each component of $L_{\Gamma}$, with framings $$ f(v) = d(v) + \sum_{e \ni v} d(e). $$

In fact, *every 3-manifold $M$ is diffeomorphic to a manifold of the form $M_{\Gamma}$*. This can be proved using ideas from a paper of Matveev and Polyak (*A Geometrical Presentation of the Surface Mapping Class Group and Surgery*), as follows. Choose a Heegaard splitting for $M$, and write the gluing map as a composition of Lickorish twists. Then use the graphical calculus in sections 3 and 4 of that paper (omitting the twists denoted $\epsilon_i$) to produce a tangle whose plat closure is a framed link of the form $L_{\Gamma}$, such that the result of surgery on this link is $M$. This argument is given by Polyak in slides available on his website.[1][2]

There is another proof as well, which involves taking an arbitrary link surgery presentation and repeatedly simplifying it. There is a natural way to measure the complexity of a link diagram, such that links of minimal complexity are of the form $L_{\Gamma}$. It is always possible to reduce this complexity by adding cancelling unknotted components and doing handleslides.

One can take the idea further and show that there is a finite set of local moves which suffice to relate any two weighted planar graphs representing the same 3-manifold. Indeed, this follows abstractly from the fact that the mapping class group is finitely presented. However, the moves produced by Wajnryb's presentation (for example) are rather complicated and nasty-looking. It is therefore natural to ask if one can find a more appealing set of moves.

One must certainly include the following moves (please excuse the lack of pictures):

- Self-loops and edges with weight $0$ can be eliminated.
- Any two parallel edges can be combined, at the expense of adding their weights.
- Suppose that $v$ is a vertex incident to exactly one edge $e$, with $d(e) = \pm f(v) = \pm 1$. Then $v$ and $e$ can be deleted, at the expensing of changing the framing on the other endpoint of $e$, call it $w$. If $f(v) = 0$, then $w$ (and all of its incident edges) can be removed from the graph .
- Suppose that $v$ is a vertex incident to exactly two edges $e_1$ and $e_2$, with $d(e_1) = d(e_2) = - f(v) = \pm 1$, then the vertex $v$ can be replaced with a single edge joining the opposite endpoints of $e_1$ and $e_2$, call them $w_1$ and $w_2$. If $d(e_1) = - d(e_2)$ and $f(v) = 0$, then $e_1$ and $e_2$ can be contracted, with the resulting vertex having weight $d(w_1) + d(w_2)$.
- Suppose that $v$ is a vertex incident to exactly three edges, and suppose that $d(e_1) = d(e_2) = -d(e_3) = f(v)$. Then $v$, together with all $e_i$, can be eliminated at the expense of adding a triangle connecting the opposite endpoints of the edges $e_i$.

Let's call any of the above moves a "blowdown", and let's call their inverses "blowups". All of them can be easily deduced using Kirby calculus, or from relations in the mapping class group.

Question 1: Are blowups and blowdowns sufficient to relate any two planar graph presentations of a given $3$-manifold?

If the answer to this question is no, then there are additional non-local moves to consider:

- If two edges $e_1$ and $e_2$ connect the same pair of vertices (but are not necessarily parallel), then they can be combined, at the expense of adding their weights.
- If there is a vertex $v$ which divides $\Gamma$ into multiple components, those components can be "permuted around $v$". Any of these components which is connected to $v$ by a single edge $e$ with weight $\pm 1$ can also be "flipped over", at the expense of changing the weight on $e$.
- If there are two vertices $v$ and $w$ which separate the graph into multiple components, then any component which is joined to both $v$ and $w$ by a single pair of edges with opposite weights in $\{\pm 1\}$ can be "flipped over", at the expense of changing the weights on the edges.

Let's call any of the above moves a ``mutation''.

Question 2: Are blowups, blowdowns, and mutations sufficient to relate any two planar graph presentations of a given $3$-manifold?

If the answer to this question is also no, then it would be good to have a nice answer to the following (admittedly vague) question:

Question 3: What is the "simplest possible" set of moves which are sufficient to relate any two planar graph presentations of a given $3$-manifold?

One argument in favor of a positive answer to Questions 1 or 2, or at least a very nice answer to Question 3, is that Kirby calculus itself admits a finite set of simple local moves. One approach might be to find a *canonical* way of simplifying an arbitrary link surgery presentation to a planar graph presentation, and trace the effects of a Kirby move through the simplification process to see which graph moves are required to implement it.

There is also a relationship with double branched covers, which might be relevant. If all vertex weights $d(v) = 0$, then $M_{\Gamma}$ can be identified (after doing surgery on an essential 2-sphere) with the double cover branched over a link $Z \subset S^3$, whose "checkerboard graph" is $\Gamma$. In this picture, Reidemeister moves on $Z$ can be realized by blowdowns and blowups, and Conway mutations can be realized by graph mutations. Note that these moves do not suffice to relate links with diffeomorphic double branched covers - this might be viewed as evidence in favor of a negative answer to Questions 1 and 2.