From a flat graph. $ Gamma $, equipped with a whole value 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 flat unknot centered on $ v $. For each edge $ e $ connecting vertices $ v $ Y $ w $, add a series of $ d (e) $ clasps between the corresponding knots (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 him. $ L _ { Gamma} $. To get the $ 3 $-manifold $ M _ { Gamma} $, perform the Dehn surgery on each component of $ L _ { Gamma} $, with frames

$$ f (v) = d (v) + sum_ {e ni v} d (e). $$

In fact, *every 3-collector $ M $ It is diffeomorphic to a variety of the form $ M _ { Gamma} $*. This can be tested using ideas from an article by Matveev and Polyak (*A geometric presentation of the group of classes of surface mapping and surgery*), as follows. Choose a division of Heegaard for $ M $, and write the gluing map as a composition of Lickorish twists. Then use the graphical calculation in sections 3 and 4 of that document (omitting the indicated turns $ epsilon_i $) to produce a tangle whose platform closure is a framed link of the form $ L _ { Gamma} $, so that the result of the surgery in this link is $ M $. Polyak presents this argument in the slides available on its website.

There is also another test, which involves taking a presentation of arbitrary link surgery and simplifying it repeatedly. There is a natural way to measure the complexity of a link diagram, so that links of minimum complexity have the form $ L _ { Gamma} $. It is always possible to reduce this complexity by adding the cancellation of non-knotted components and making handles.

One can go deeper into the idea and show that there is a finite set of local movements that is enough to relate two weighted flat graphs that represent the same multiple of 3. In fact, this is deduced in an abstract way from the fact that the group of Mapping classes are presented finitely. However, the movements produced by Wajnryb's presentation (for example) are quite complicated and unpleasant. Therefore, it is natural to ask if one can find a more attractive set of movements.

Certainly we must include the following movements (sorry for the lack of photos):

- Self-loops and edges with weight. $ 0 $ It can be eliminated.
- Either of the two parallel edges can be combined, at the cost of adding their weights.
- Suppose that $ v $ It is an incident from vertex to exactly one edge. $ e $, with $ d (e) = f (v) = $ 1. So $ v $ Y $ e $ can be eliminated, as a consequence of changing the frame in the other endpoint of $ e $, call it $ w $. Yes $ f (v) = 0 $, so $ w $ (and all its incident edges) can be removed from the graph.
- Suppose that $ v $ It is a vertex incident exactly at two edges. $ e_1 $ Y $ e_2 $, with $ d (e_1) = d (e_2) = – f (v) = $ 1, then the vertex $ v $ can be replaced with a single edge that joins the opposite endpoints of $ e_1 $ Y $ e_2 $, call them $ w_1 $ Y $ w_2 $. Yes $ d (e_1) = – d (e_2) $ Y $ f (v) = 0 $, so $ e_1 $ Y $ e_2 $ It can be contracted, with the resulting vertex having weight. $ d (w_1) + d (w_2) $.
- Suppose that $ v $ is a vertex that falls exactly on three edges, and suppose that $ d (e_1) = d (e_2) = -d (e_3) = f (v) $. So $ v $, together with all $ e_i $, can be eliminated at the expense of adding a triangle that connects the opposite endpoints of the edges $ e_i $.

Let's call any of the above movements a "purge," and call their reverse "explosions." All of them can be easily deduced using the Kirby calculation, or the relationships in the group of mapping classes.

Question 1: Are explosions and explosions enough to relate them?

two presentations of planar graphics of a given $ 3 $-manifold?

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

- If two edges $ e_1 $ Y $ e_2 $ connects the same pair of vertices (but not necessarily parallel), then can be combined, at the expense of adding their weights.
- If there is a vertex $ v $ what divides $ Gamma $ in multiple components, those components can be "swapped around $ v $"Any of these components that is connected to $ v $ by a single edge $ e $ with weight $ ± 1 $ You can also "flip", at the cost of changing the weight in $ e $.
- If there are two vertices. $ v $ Y $ w $ that separate the graph into multiple components, then any component that joins both $ v $ Y $ w $ by a single pair of edges with opposite weights in $ { pm 1 } $ you can "flip", at the expense of changing the weights at the edges.

Let's call any of the above movements a “ mutation. & # 39; & # 39 ;.

Question 2: Are explosions, explosions and mutations enough to relate two of them?

Presentations of planar graphics of a certain. $ 3 $-manifold?

If the answer to this question is also no, it would be good to have a good answer to the following question (it is true that it wanders):

Question 3: What is the set of "simplest possible" movements that is enough to relate two presentations of flat graphics of a given $ 3 $-manifold?

An argument in favor of a positive response to Questions 1 or 2, or at least a very good answer to Question 3, is that Kirby's calculation admits a finite set of simple local movements. One approach could be to find a *canonical* way to simplify an arbitrary link surgery presentation to a planar chart presentation, and to track the effects of a Kirby movement through the simplification process to see what chart movements are required to implement it.

There is also a relationship with double branching tires, which could be relevant. If all the vertex weights $ d (v) = 0 $, so $ M _ { Gamma} $ can be identified (after performing surgery on an essential 2 sphere) with the double branched cover on a link $ Z subset S ^ 3 $, whose "checkerboard graphic" is $ Gamma $. In this photo, Reidemeister goes ahead. $ Z $ it can be done by explosions and explosions, and Conway's mutations can be made by graphic mutations. Note that these movements are not enough to link links with branched double diffeomorphic covers; this could be considered evidence in favor of a negative response to Questions 1 and 2.