
BRAID VARIETY CLUSTER STRUCTURES, I: 3D PLABIC GRAPHS 5
2.2. 3D plabic graphs. Let w= (i1, i2, . . . , im) be a reduced word for w. Consider the
unique rightmost subexpression ufor uinside w, and let Ju,w⊂[m] := {1,2, . . . , m}be the
set of indices not used in u. The 3D plabic graph Gu,wis obtained from the wiring diagram
for wby replacing all crossings in [m]\Ju,wby overcrossings and replacing each crossing
c∈Ju,wby a black-white bridge edge bc; see Figure 1. We place a marked point on each of
the nleftmost boundary vertices of Gu,w, and denote by Mthe set of these marked points.
The number of bridges in Gu,wis |Ju,w|=ℓ(w)−ℓ(u), which is the dimension of ◦
Ru,w.
To each index c∈Ju,wwe will associate an (oriented) relative cycle Ccin Gu,w, which by
definition is either a cycle in Gu,wor a union of oriented paths in Gu,wwith endpoints in M.
Each relative cycle Ccwill naturally bound a disk Dc.1For instance, in Figure 3, the
vertical sections of Dcare shown in wavy pink lines. We indicate the relative position of Dc
in R3with respect to the edges of Gu,wby over/under-crossings. We will compute Dc, and
therefore its boundary Cc, starting from the bridge bcand proceeding to the left using the
propagation rules in Figure 2. We choose the counterclockwise orientation of Cc, so that as
one traverses Cc, the disk Dcis to the left. See Section 3.4 for a description of relative cycles
in the case of double braid varieties.
2.3. The quiver. Aquiver Qis a directed graph without directed cycles of length 1 and 2.
An ice quiver e
Qis a quiver whose vertex set e
V=V(e
Q) is partitioned into frozen and mutable
vertices: e
V=Vfro ⊔Vmut. The arrows between pairs of frozen vertices are automatically
omitted.
The procedure in Section 2.2 yields a bicolored graph Gu,wdecorated with a family
(Cc)c∈Ju,wof relative cycles. To this data, we associate an ice quiver e
Qu,w. Our construction
will rely on the results of [FG09, GK13]. The vertex set V(e
Qu,w) := Ju,wis in bijection with
the set of relative cycles. If a relative cycle Ccis actually a cycle in Gu,wthen cis a mutable
vertex of e
Qu,w; otherwise, if Ccis a union of paths with endpoints in M,cis a frozen vertex
of e
Qu,w.
To compute the arrows of e
Qu,w, we consider Gu,was a ribbon graph, with counterclockwise
half-edge orientations around white vertices and clockwise half-edge orientations around
black vertices. Let Su,wbe the surface with boundary obtained by replacing every edge of
Gu,wby a thin ribbon and gluing the ribbons together according to the local orientations at
the vertices of Gu,w. See Figure 7(d) for an example of Su,w. The nmarked points of Gu,w
give rise to nmarked points on ∂Su,w, the set of which is also denoted by M.
We view each relative cycle Ccas an element of the relative homology Λu,w:= H1(Su,w,M).
It turns out that each mutable relative cycle can be also viewed as an element of the dual
lattice Λ∗
u,w; see Section 3.3. The (signed) number of arrows between two vertices c, d ∈Ju,w
in e
Qu,w, where dis mutable, is defined to be the (signed) intersection number ⟨Cc, Cd⟩of
the relative cycles Ccand Cd. These intersection numbers can be computed explicitly using
simple pictorial rules; see Algorithm 3.9. Alternatively, one can compute the quiver e
Qu,wby
summing up half-arrow contributions; see Section 3.7. One can check that in the case of our
running example (Figure 1), both descriptions yield the quiver shown in Figure 4. See also
Example 3.11.
1More precisely, when Ccis a cycle, ∂Dc=Cc, and when Ccis a union of paths with marked endpoints,
∂Dcis the union of Cctogether with several straight line segments connecting pairs of marked points.