Let $e$ be a regular edge or a bridge with end vertices $v_{1}$ and $v_{2}$ of
any representative of a rank $3$ w-colored graph. Contracting $e$ in the
representative yields a new vertex $v$ which is connected.

###### Proof.

Consider $e$ which is not a self-loop, $e$ is common to all $\mathcal{G}$
representative of a rank 3 w-colored graph. Its strands join in $v_{1}$ and in
$v_{2}$ a collection of connected families of pre-flags. The initial vertices
$v_{1}$ and $v_{2}$ being connected, two different cases may occur:

(i) All strands of $e$ are connected to a unique pre-flag either in $v_{1}$ or
$v_{2}$. This implies that there are just two pre-flags in $v_{1}$ or in
$v_{2}$. Contracting $e$, the vertex $v$ obtained is connected.

(ii) Assume now that attached to each strand of $e$, we have $d$ families of
connected pre-flags in $v_{1}$ or in $v_{2}$, $2\leq d\leq 3$ (see an
illustration in Figure 28). This is enforced by the coloring which prevents
faces to pass through $e$ more than once. If contracting $e$ gives a
disconnected vertex then, in each $v_{1}$ and $v_{2}$, the connected families
of pre-flags do not have any bow between them888Note that this is only a
necessary condition for obtaining a disconnected vertex.. Notice that there is
necessarily at least one connected family of pre-flags joining $e$ by a unique
bow $b$. For such a family $\xi$, all pre-flags are fully connected via bows
and there is a unique pre-flag point, the one related to $e$ via $b$, not
linked to any other pre-flag in $\xi$. By an argument of parity, we aim at
showing that it is impossible.

Figure 28.  Families $F_{1},F_{2}$ and $F_{3}$ of connected pre-flags attached
to each strand of a non self-loop edge.

$F_{1}$$F_{2}$$F_{3}$$e$

For simplicity, let us fix that $e$ is of color $0$ and that the strand $s$ of
color pair $(01)$ in $e$ touches the connected and separate family $\xi$ a
pre-flag $f$. Two cases may occur: either $f$ is of color $1$ or of color
$\bar{0}$.

(A) If $f$ of color $1$: Denote $N_{i}$ and $\bar{N}_{i}$ the number of pre-
flags in $\xi$ of color $i=0,1,2,3$ and
$\bar{i}=\bar{0},\bar{1},\bar{2},\bar{3}$, respectively. There are exactly
$N_{i}+\bar{N}_{i}$ pre-flag points labeled by color pairs $(ij)$, $i=0,2,3$,
$j\in\\{0,1,2,3\\}$ but $j\neq i$. These points come from pre-flags of colors
$0,2,3$ and $\bar{0},\bar{2},\bar{3}$. Meanwhile, there are
$(N_{1}-1)+\bar{N}_{1}$ pre-flag points of color pairs $(01)$, and
$N_{1}+\bar{N}_{1}$ of color pairs $(1j)$, $j\in{2,3}$. These last points come
uniquely from pre-flags with color $1$ and $\bar{1}$. Assuming that all pre-
flag points but one in $\xi$ should be totally connected by bows implies the
following equations:

$\displaystyle N_{0}+\bar{N}_{0}+(N_{1}-1)+\bar{N}_{1}=2k^{01}\,,$ (72)
$\displaystyle N_{0}+\bar{N}_{0}+N_{2}+\bar{N}_{2}=2k^{02}\,,$ (73)
$\displaystyle N_{0}+\bar{N}_{0}+N_{3}+\bar{N}_{3}=2k^{03}\,,$ (74)
$\displaystyle N_{1}+\bar{N}_{1}+N_{2}+\bar{N}_{2}=2k^{12}\,,$ (75)
$\displaystyle N_{1}+\bar{N}_{1}+N_{3}+\bar{N}_{3}=2k^{13}\,,$ (76)
$\displaystyle N_{2}+\bar{N}_{2}+N_{3}+\bar{N}_{3}=2k^{23}\,,$ (77)

for some $k^{ij}$ positive integers counting the number of bow with colors
$(ij)$. It is simple to show that this system leads to the inconsistent
equation

$\displaystyle 2(N_{0}+\bar{N}_{0})+2(k^{12}-k^{02})-1=2k^{01}\,.$ (78)

(B) If $f$ is of color $\bar{0}$: $N_{0}+(\bar{N}_{0}-1)$ becomes the number
of pre-flag points of color pairs $(01)$ in $\xi$ induced by pre-flags of
color $0$ and $\bar{0}$ and $N_{1}+\bar{N}_{1}$ is the number of pre-flags
points of colors $01$ induced by pre-flags of color $1$ and $\bar{1}$. This is
the same as switching the role of $1$ and $\bar{0}$ in the above case. The
above system holds again as constraints on the number of pre-flag points and
yields the same conclusion.

∎

The above lemma should clearly admit an extension to any dimension $D$ by
partitioning $D$ and analyzing the connection of the strands of $e$ and the
connected pre-flag families. We do not need however such a stronger and much
technical result for the following.

In contrast, the contraction of a self-loop may disconnect the vertex. In any
case, since self-loop graphs are terminal forms, a special treatment for them
is required.

The whole construction above is mainly made in order to maintain the
consistency of the $p$-bubble definition. For instance, edges are still
1-colored objects, (closed and open) faces are bicolored connected objects,
(closed and open) 3-bubbles are connected objects with 3 colors. However,
3-bubble are no longer built uniquely with three valent vertices. They can be
made with vertices with lower or greater valence as illustrated in Figure 29.
In any case, Definition 31 is still valid and we shall focus on this.

Figure 29.  Two bubbles graphs $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ of
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$, respectively: $\mathbf{b}_{1}$ has a
vertex with valence 2 and $\mathbf{b}_{2}$ a vertex with valence 4.

$\mathcal{G}_{1}$$\mathbf{b}_{1}$$0$$\bar{0}$$\mathcal{G}_{2}$$\mathbf{b}_{2}$$0$$\bar{0}$

######  Lemma 5.

Let $\mathcal{G}$ be a representative of $\mathfrak{G}$ a rank $D$ w-colored
graph. $\bullet$ Cutting an edge $e$ in $\mathcal{G}$ yields a representative
$\mathcal{G}\vee e$ of another rank $D$ w-colored graph denoted
$\mathfrak{G}\vee e$. $\bullet$ Contracting an edge $e$ in $\mathcal{G}$
yields a representative $\mathcal{G}/e$ of another rank $D$ w-colored graph
denoted $\mathfrak{G}/e$.

###### Proof.

Let $e$ be an edge in $\mathcal{G}^{\prime}$ representative of some w-colored
graph $\mathfrak{G}$ and $\mathcal{G}$ the representative of $\mathfrak{G}$
resulting from a contraction of some colored graph
$\mathcal{G}_{\text{color}}$.

The cut graph $\mathcal{G}^{\prime}\vee e$ is equivalent to $\mathcal{G}\vee
e$. We use again here the expansion and contraction move: starting from
$\mathcal{G}\vee e$, we expand this graph in $\mathcal{G}_{\text{color}}\vee
e$ and contract it back to nothing but itself. One finds that $\mathcal{G}\vee
e$ is therefore a graph resulting from the contraction of a colored graph. Its
equivalence class is the w-colored graph desired.

Contracting now $e$ in $\mathcal{G}^{\prime}$, one gets a graph
$\mathcal{G}^{\prime}/e$ equivalent to $\mathcal{G}/e$. There exists a colored
graph $\mathcal{G}^{0}_{\text{color}}$ contracting to $\mathcal{G}/e$. This is
nothing but $\mathcal{G}_{\text{color}}$ on which after all contraction
yielding $\mathcal{G}$, one perform another contraction on $e$. Note that we
could have defined
$\mathcal{G}^{0}_{\text{color}}=\mathcal{G}_{\text{color}}/e$ and perform the
same series of contraction bringing initially $\mathcal{G}$. The end result is
$\mathcal{G}/e$. Thus the equivalence class of $\mathcal{G}/e$ is the
w-colored graph that we seek.

∎

We henceforth restrict the rank of graphs to $D=3$ and aim at studying the
universal invariant polynomial extending BR polynomial for this new category
of graphs. The extension for any $D$ should require more work but could be
almost inferred from the subsequent analysis. From now rank 3 w-colored graphs
will be called more simply w-colored graphs.

Mostly, we have studied non self-loop edges so far and their main properties
under contraction and cut can be guessed in any dimension. Self-loops are more
subtle and their behavior under contraction is much more involved. Restricting
to dimension 3 will simplify the analysis.

Self-loops find also a meaning in the present situation. A self-loop can be at
most a 3–inner edge. A 3–inner edge determines by itself a graph, see Figure
30O. If it is a 2–inner self-loop then $e$ should be of the form illustrated
in Figure 30A. Otherwise, $e$ may be a 1–inner self-loop (Figure 30B) or a
0–inner (Figure 30C).

Figure 30.  $p$–inner self-loops: a 3–inner (O) and a 2–inner (A) self-loop
with their unique possible configuration; a 1–inner self-loop with possible
two sectors $v_{1}$ and $v_{2}$ (B) and a 0–inner self-loop with three
possible sectors $v_{1},v_{2}$ and $v_{3}$ (C).

OABC$v_{1}$$v_{2}$$v_{1}$$v_{2}$$v_{3}$

For a 2–inner self-loop $e$, the outer strand of $e$ should be in contact with
other flags or edges, otherwise $e$ would be a 3–inner edge. We represent the
rest of the vertex with possible edges and flags attached by a black diagram
in Figure 30A.

A 1–inner self-loop $e$ has two outer strands which should be in contact with
two sectors of the vertex, $v_{1}$ and $v_{2}$ as illustrated in Figure 30B,
such that $v_{1}$ and $v_{2}$ have at least two flags or edges. Remark that
nothing prevents $v_{1}$ and $v_{2}$ to have pre-flags in common or to be
linked by bows. In this situation, we shall say that these sectors coincide
otherwise we call them separate. A 2–inner self-loop $e$ has three such
sectors, $v_{1},v_{2}$ and $v_{3}$, which can pairwise coincide or can be
pairwise separate. Separate sectors should contain each at least two flags or
edges.

The notion of trivial self-loop can be addressed at this point for this
category of graphs. We call a self-loop $e$ trivial if it is a 3–inner or
2–inner self-loop or if it is a 1–inner or 0–inner self-loop such that there
is no edge between separate sectors $v_{i}$.

For a 3–inner self-loop, the contraction gives three trivial discs, see Figure
31O. For a trivial 2–inner self-loop the contraction is still straightforward
and yields Figure 31A. For a 1–inner self loop contraction (see Figure 31B),
one gets one extra disc and has two possible configurations: either the vertex
remains connected or it gets disconnected with two (possibly non trivial)
vertices in both situations. Concerning the contraction of a trivial 1–inner
self-loop, it is immediate that the vertex gets disconnected in two non
trivial vertices. For a 0–inner self-loop contraction (see Figure 31C), we
have no additional disc but three types of configurations with up to three
disconnected (and possibly non trivial) vertices. The contraction of a trivial
0–inner self-loop yields directly three disconnected and non trivial vertices.

Figure 31.  $p$–inner self-loop contractions corresponding to O, A, B and C of
Figure 30, respectively.

OABC$v_{1}$$v_{2}$$v_{1}$$v_{2}$$v_{3}$

######  Lemma 6 (Rank 3 trivial self-loop contraction).

Let $\mathfrak{G}$ a w-colored graph and $\mathcal{G}$ any of its
representative with boundary $\partial{\mathcal{G}}$, $e$ be a trivial self-
loop of $\mathcal{G}$, and $\mathcal{G}^{\prime}=\mathcal{G}/e$ the result of
the contraction of $\mathcal{G}$ by $e$ with boundary denoted by
$\partial{\mathcal{G}}^{\prime}$. Let $k$ denote the number of connected
components of $\mathcal{G}$, $V$ its number of vertices, $E$ its number of
edges, $F_{{\rm int}}$ its number of faces, $B_{\rm int}$ its number of closed
bubbles and $B_{\rm ext}$ its number of open bubbles, $C_{\partial}$ the
number of connected component of $\partial{\mathcal{G}}$, $f=V_{\partial}$ its
number of flags, $E_{\partial}=F_{{\rm ext}}$ the number of edges and
$F_{\partial}$ the number of faces of $\partial{\mathcal{G}}$, and let
$k^{\prime},V^{\prime},E^{\prime},F_{{\rm
int}},C^{\prime}_{\partial},B^{\prime}_{{\rm int}},B^{\prime}_{\rm
ext},C^{\prime}_{\partial},f^{\prime},E^{\prime}_{\partial}$, and
$F^{\prime}_{\partial}$ denote the similar numbers for $\mathcal{G}^{\prime}$
and its boundary $\partial{\mathcal{G}}^{\prime}$.

$\bullet$ If $e$ is a 3–inner self-loop, then

$\displaystyle k^{\prime}=k+2\,,\quad V^{\prime}=V+2\,,\quad
E^{\prime}=E-1\,,\quad F^{\prime}_{{\rm int}}=F_{{\rm int}}\,,$ (79)
$\displaystyle C^{\prime}_{\partial}=C_{\partial}\,,\quad f^{\prime}=f\,,\quad
E^{\prime}_{\partial}=E_{\partial}\,,\quad
F^{\prime}_{\partial}=F_{\partial}\,,$ (80) $\displaystyle B^{\prime}_{\rm
int}=B_{{\rm int}}-3\,,\qquad B^{\prime}_{\rm ext}=B_{\rm ext}\,.$ (81)

$\bullet$ If $e$ is a 2–inner self-loop, then

$\displaystyle k^{\prime}=k+2\,,\quad V^{\prime}=V+2\,,\quad
E^{\prime}=E-1\,,\quad F^{\prime}_{{\rm int}}=F_{{\rm int}}\,,$ (82)
$\displaystyle C^{\prime}_{\partial}=C_{\partial}\,,\quad f^{\prime}=f\,,\quad
E^{\prime}_{\partial}=E_{\partial}\,,\quad
F^{\prime}_{\partial}=F_{\partial}\,,$ (83) $\displaystyle B^{\prime}_{\rm
int}=B_{{\rm int}}-1\,,\qquad B^{\prime}_{\rm ext}=B_{\rm ext}\,.$ (84)

$\bullet$ If $e$ is a trivial $p$–inner self-loop such that $p=0,1$, then

$\displaystyle k^{\prime}=k+2\,,\quad V^{\prime}=V+2\,,\quad
E^{\prime}=E-1\,,\quad F^{\prime}_{{\rm int}}=F_{{\rm int}}\,,$ (85)
$\displaystyle C^{\prime}_{\partial}=C_{\partial}\,,\quad f^{\prime}=f\,,\quad
E^{\prime}_{\partial}=E_{\partial}\,,\quad
F^{\prime}_{\partial}=F_{\partial}\,,$ (86) $\displaystyle B^{\prime}_{\rm
int}+B^{\prime}_{\rm ext}=B_{{\rm int}}+B_{\rm ext}+\alpha_{p}\,,$ (87)

where $\alpha_{p}=3-2p$.

###### Proof.

$\bullet$ The first situation of a 3–inner self-loop defines a particular
closed graph made of a single vertex with a unique self-loop. The contraction
destroys the vertex and the three closed bubbles and generate three discs. The
result follows.

For $p$–inner self loops, $p\leq 2$, there should be other flags or edges on
the same end vertex. The first lines of (84)-(87) should not cause any
difficulty using the definition of the contraction which preserve (open and
closed) strands. Let us focus on the number of bubbles of the graph.

$\bullet$ Consider that $e$ is 2–inner. One first should notice that there is
a bubble generated by the two inner faces of $e$ and that contracting $e$
destroys this closed bubble. Besides, the other (open or closed) bubbles which
passed through $e$ and which were defined by one of the two inner faces are
simply deformed. They loose one face (the inner) but remain in the contracted
graph. The second line of (84) follows.

Let us assume now that $e$ is trivial and 1– or 0–inner with color $a$.
Consider in the initial vertex $v$, the two or three separate sectors of pre-
flags $v_{1}$, $v_{2}$ and $v_{3}$ determined by the strands of $e$. We remind
that $v_{1}$, $v_{2}$ and $v_{3}$ should contain each at least two flags or
edges. Consider as well the three bubbles labeled by colors $(aa_{1}a_{2})$,
$(aa_{1}a_{3})$ and $(aa_{2}a_{3})$ passing through $e$.

$\bullet$ If $e$ is 1–inner, let us assume that the face $(aa_{3})$ is the one
which is inner, without loss of generality. The edge $e$ decomposes in three
ribbon edges each determined by a couple of pairs $(aa_{i};aa_{j})$, $i<j$,
such that the strand $(aa_{1})$ has an end point in the sector $v_{1}$ and
$(aa_{2})$ an end point in $v_{2}$. If $e$ is trivial and 1–inner, its
contraction yields two connected vertices plus one disc. Note that only the
bubbles $\mathbf{b}_{aa_{1}a_{3}}$ and $\mathbf{b}_{aa_{2}a_{3}}$ (in the
whole graph) include the face $(aa_{3})$. Contracting $e$, the bubbles
$\mathbf{b}_{aa_{1}a_{3}}$ and $\mathbf{b}_{aa_{2}a_{3}}$ just loose that face
but are still present since in $v_{1}$ and $v_{2}$, there exist still faces
where $\mathbf{b}_{aa_{1}a_{3}}$ and $\mathbf{b}_{aa_{2}a_{3}}$ passed before.
Now the last bubble $\mathbf{b}_{aa_{1}a_{2}}$ gets split in two parts: these
parts are disjoint bubbles with the same triple colors since there is no edge
in the graph relating them. These bubbles can be open or closed depending on
the nature of $\mathbf{b}_{aa_{1}a_{3}}$ in $v_{1}$ and $v_{2}$. Hence,
$B^{\prime}_{{\rm int}}+B^{\prime}_{\rm ext}=B_{{\rm int}}+B_{\rm ext}+1$.

$\bullet$ Now let us discuss the 0–inner self-loop situation with its three
separate sectors $v_{1}$, $v_{2}$ and $v_{3}$. The discussion is somehow
similar to the 1–inner case. The contraction of $e$ yields three connected
vertices. Using the similar routine explained above, one finds that each of
the bubbles $\mathbf{b}_{aa_{i}a_{j}}$, $i<j$ gets split into two parts after
the contraction. Each of these pairs of parts are not related at all by any
edge or bow. Thus each bubble produces two bubbles. The resulting bubbles may
be open or closed depending on the nature of $\mathbf{b}_{aa_{i}a_{j}}$ in
each sector. We have $B^{\prime}_{{\rm int}}+B^{\prime}_{\rm ext}=B_{{\rm
int}}+B_{\rm ext}+3$.

∎

It is clear that the contraction of a 2–inner self-loop also obeys (87) as
well for $p=2$. In fact (84) is a stronger result because it mainly separates
the variation of the numbers of closed and open bubbles. For a general 1–inner
self-loop contraction, it turns out that the number $B_{\rm int}+B_{\rm ext}$
may vary of 1 or not. Similarly, for a non necessarily trivial 0–inner self-
loop contraction, the same number of bubbles may vary from 0,1,2 up to 3.

######  Lemma 7 (Faces of a bridge).

$\bullet$ The faces passing through a bridge of any representative of a rank 3
w-colored graph are necessarily open. $\bullet$ These faces belong to the same
connected component of the boundary graph of the representative.

###### Proof.

A bridge $e$ with color $a$ has three strands of color pairs $(aa_{1})$,
$(aa_{2})$ and $(aa_{3})$ defining three faces passing through $e$.

The first point is quite clear: following a strand inside the graph, for
instance, on the left side of the bridge, before closing that face one has to
go on the other side of the bridge without passing by $e$ (we recall that by
the presence of strands with different color pair, we cannot pass twice by a
bridge). This is impossible by definition of a bridge. Hence each strand
should terminate inside of each side of the bridge hence are open.

Assume that one the faces, say $(aa_{1})$, does not belong to the same
connected component of the boundary of the others. Consider in its own
connected component all boundary vertices. Each of these vertices are of
valence 3 equipped with edges with different color pairs $(a_{i}a_{j})$. Since
$(aa_{1})$ should connect one of these boundary vertices, there exists at
least one vertex with half edges of color $a$ or $a_{1}$. Consider all
possible vertices where the strand $(aa_{1})$ could connect. There are $N_{a}$
and $N_{a_{1}}$ such vertices whereas there are $N_{a_{2}}$ and $N_{a_{3}}$
vertices which cannot be joined by $(aa_{1})$. Each of these vertices have
$N_{a_{\alpha}}$, $\alpha=\emptyset,1,2,3$, half edges of each color pair
$(a_{\alpha}a_{j})$, $j\neq\alpha$. Since all these half-edges should be fully
paired in order to have a connected component therefore each of the following
sums should give an even positive integer

$\displaystyle N_{a}+N_{a_{1}}+1\,,\qquad N_{a}+N_{a_{j}}\,,\qquad
N_{a_{1}}+N_{a_{j}}\,,\quad j=2,3\,.$ (88)

This system is clearly inconsistent.

∎

######  Lemma 8 (Cut/contraction of special edges).

Let $\mathcal{G}$ be a representative of $\mathfrak{G}$ a rank 3 w-colored
graph and $e$ an edge in $\mathcal{G}$. Then,

$\bullet$ if $e$ is a bridge, in the above notations, we have

$\displaystyle k(\mathcal{G}\vee e)=k(\mathcal{G}/e)+1,\;V(\mathcal{G}\vee
e)=V(\mathcal{G}/e)+1,\;E(\mathcal{G}\vee
e)=E(\mathcal{G}/e),\;f(\mathcal{G}\vee e)=f(\mathcal{G}/e)+2,$ (89) (90)
$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G}/e)\,,\quad B_{{\rm int}}(\mathcal{G}\vee e)=B_{{\rm
int}}(\mathcal{G}/e)\,,$ (91) $\displaystyle C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G}/e)+1\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G}/e)+3\,,\quad F_{\partial}(\mathcal{G}\vee
e)=F_{\partial}(\mathcal{G}/e)+3\,,$ (92) $\displaystyle B_{{\rm
ext}}(\mathcal{G}\vee e)=B_{{\rm ext}}(\mathcal{G}/e)+3\,;$ (93)

$\bullet$ if $e$ is a trivial $p$–inner self-loop, $p=0,1,2$, we have

$\displaystyle k(\mathcal{G}\vee e)=k(\mathcal{G}/e)-2,\;V(\mathcal{G}\vee
e)=V(\mathcal{G}/e)-2,\;E(\mathcal{G}\vee
e)=E(\mathcal{G}/e),\;f(\mathcal{G}\vee e)=f(\mathcal{G}/e)+2,$ (94) (95)
$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)+C_{\partial}(\mathcal{G}\vee
e)=F_{{\rm int}}(\mathcal{G}/e)+C_{\partial}(\mathcal{G}/e)-2\,,\quad
E_{\partial}(\mathcal{G}\vee e)=E_{\partial}(\mathcal{G}/e)+3\,,$ (96)
$\displaystyle B_{{\rm int}}(\mathcal{G}\vee e)+B_{{\rm ext}}(\mathcal{G}\vee
e)=B_{{\rm int}}(\mathcal{G}/e)+B_{{\rm ext}}(\mathcal{G}/e)-(3-2p)\,.$ (97)

Moreover, given $\partial\mathcal{G}$ the boundary of $\mathcal{G}$ and a
bridge or trivial $p$–inner self-loop $e$, $p=0,1,2,3$,

$2C_{\partial}(\mathcal{G}\vee e)-\chi(\partial(\mathcal{G}\vee
e))=2C_{\partial}(\mathcal{G})-\chi(\mathcal{G})=2C_{\partial}(\mathcal{G}/e)-\chi(\partial(\mathcal{G}/e))\,,$
(98)

where $\chi(\partial{\mathcal{G}})$ denote the Euler characteristics of the
boundary of $\mathcal{G}$.

###### Proof.

We start by the bridge case. The equations in (90) are easily found. Let us
focus on (91). By Lemma 7, we know that necessarily the faces passing through
$e$ are open. All closed faces on each side of the bridge are conserved after
cutting $e$. The same are still conserved after edge contraction. Hence
$F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm int}}(\mathcal{G}/e)$ and $B_{{\rm
int}}(\mathcal{G}\vee e)=B_{{\rm int}}(\mathcal{G}/e)$. We now prove (92).
Still by the second point of Lemma 7, the three external faces belong to the
same boundary component. After cutting $e$, this unique component yields two
boundary components. It is direct to get $C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G}/e)+1$, $E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G}/e)+3$ (the cut of $e$ divides each external face
into two different external strands) and $F_{\partial}(\mathcal{G}\vee
e)=F_{\partial}(\mathcal{G}/e)+3$ because
$C_{\partial}(\mathcal{G}/e)=C_{\partial}(\mathcal{G})$,
$E_{\partial}(\mathcal{G}/e)=E_{\partial}(\mathcal{G})$ and
$F_{\partial}(\mathcal{G}/e)=F_{\partial}(\mathcal{G})$ which are immediate
from Lemma 1. Concerning the number of external bubbles, there are three
bubbles in $\mathcal{G}$ passing through the bridge. Each of these is
associated with two color pairs $(aa_{i};aa_{j})$, $i<j$. These bubbles are
clearly in $\mathcal{G}/e$ and cutting the bridge each of these bubble splits
in two. This yields (93).

We focus now on a trivial $p$–inner self-loop $e$. The relations (95) can be
determined without difficulty. We concentrate on the rest of the equations.
Consider the faces $f_{i}$ in $e$ made with outer strands. For $p=0,1,2$, we
have $f_{i}$, $1\leq i\leq 3-p$. These faces can be open or closed. We do a
case by case study according to the number of open or closed faces among the
$f_{i}$’s.

\- Assume that $3-p$ of $f_{i}$’s are closed. Cutting $e$ entails

$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G})-3\,,\quad C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})+1\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3\,,$ (99) $\displaystyle B_{{\rm
int}}(\mathcal{G}\vee e)+B_{{\rm ext}}(\mathcal{G}\vee e)=B_{{\rm
int}}(\mathcal{G})+B_{{\rm ext}}(\mathcal{G})\,,$ (100) $\displaystyle
F_{\partial}(\mathcal{G}\vee e)=F_{\partial}(\mathcal{G})+3\,.$ (101)

Note that in this situation only the variation of the total number of bubbles
can be known.

\- Assume that $3-p-1$ of the $f_{i}$’s are closed and one is open. Cutting
$e$ gives

$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G})-2\,,\quad C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3\,,$ (102) $\displaystyle B_{{\rm
int}}(\mathcal{G}\vee e)=B_{{\rm int}}(\mathcal{G})-1\,,\quad B_{{\rm
ext}}(\mathcal{G}\vee e)=B_{{\rm ext}}(\mathcal{G})+1\,,$ (103) $\displaystyle
F_{\partial}(\mathcal{G}\vee e)=F_{\partial}(\mathcal{G})+1\,.$ (104)

\- Assume that $3-p-2$ of the $f_{i}$’s are closed and two are open. Cutting
$e$ gives

$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G})-1\,,\quad C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})-1\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3\,,$ (105) $\displaystyle B_{{\rm
int}}(\mathcal{G}\vee e)=B_{{\rm int}}(\mathcal{G})\,,\quad B_{{\rm
ext}}(\mathcal{G}\vee e)=B_{{\rm ext}}(\mathcal{G})\,,$ (106) $\displaystyle
F_{\partial}(\mathcal{G}\vee e)=F_{\partial}(\mathcal{G})-1\,.$ (107)

Note that this case does not apply for $p=2$.

\- For $p=0$ an additional situation applies: assume that all three $f_{i}$’s
are open. Cutting $e$ gives

$\displaystyle F_{{\rm int}}(\mathcal{G}\vee e)=F_{{\rm
int}}(\mathcal{G})\,,\quad C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})-2\,,\quad E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3\,,$ (108) $\displaystyle B_{{\rm
int}}(\mathcal{G}\vee e)=B_{{\rm int}}(\mathcal{G})\,,\quad B_{{\rm
ext}}(\mathcal{G}\vee e)=B_{{\rm ext}}(\mathcal{G})\,,$ (109) $\displaystyle
F_{\partial}(\mathcal{G}\vee e)=F_{\partial}(\mathcal{G})-3\,.$ (110)

Lemma 6 relates the same numbers for $\mathcal{G}/e$ and $\mathcal{G}$ from
which one is able to prove (96) and (97).

Last, we prove the relation (98) on the Euler characteristics of the different
boundaries. We first note that, from (90), (92), (95) and (96), for any
special (bridge or trivial $p$–inner, $p=0,1,2$) edge,

$f(\mathcal{G}\vee e)=f(\mathcal{G})+2=f(\mathcal{G}/e)+2\,,\qquad
E_{\partial}(\mathcal{G}\vee
e)=E_{\partial}(\mathcal{G})+3=E_{\partial}(\mathcal{G}/e)+3\,.$ (111)

For the bridge case, (98) follows from the relations
$F_{\partial}(\mathcal{G}\vee
e)=F_{\partial}(\mathcal{G})+3=F_{\partial}(\mathcal{G}/e)+3$ and
$C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})+1=C_{\partial}(\mathcal{G}/e)+1$ in (92). The
result holds also for trivial $0,1,2$–inner self-loops, after the case by case
study giving (100)–(110). Last, for the 3–inner self-loop, in addition to
(111) which still holds, the following relations are valid

$\displaystyle C_{\partial}(\mathcal{G}\vee
e)=C_{\partial}(\mathcal{G})+1=C_{\partial}(\mathcal{G}/e)+1\,,\quad
F_{\partial}(\mathcal{G}\vee
e)=F_{\partial}(\mathcal{G})+3=F_{\partial}(\mathcal{G}/e)+3\,,$ (112)

and allow us to conclude.

∎

###  4.3. Polynomial invariant for $3D$ w-colored tensor graphs

We shall define first an invariant for rank 3 w-colored graphs, check its
consistency and then state our main result.

######  Definition 36 (Topological invariant for rank 3 w-colored graph).

Let $\mathfrak{G}(\mathcal{V},\mathcal{E},\mathfrak{f}^{0})$ be a rank 3
w-colored stranded graph with flags. The generalized topological invariant
associated with $\mathfrak{G}$ is given by the following function associated
with any of its representatives $\mathcal{G}$ (using the above notations)

$\displaystyle\mathfrak{T}_{\mathfrak{G}}(x,y,z,s,w,q,t)=\mathfrak{T}_{\mathcal{G};a}(x,y,z,s,w,q,t)=$
(113) $\displaystyle\sum_{A\Subset\mathcal{G}}(x-1)^{{\rm
r}\,(\mathcal{G})-{\rm r}\,(A)}(y-1)^{n(A)}z^{5k(A)-[3(V-E(A))+2(F_{{\rm
int}}(A)-B_{{\rm int}}(A)-B_{{\rm ext}}(A))]}$ (114)
$\displaystyle\qquad\times\quad
s^{C_{\partial}(A)}\,w^{F_{\partial}(A)}q^{E_{\partial}(A)}t^{f(A)}\,.$

Our initial task is to show that the above definition does not actually depend
on the representative $\mathcal{G}$ of $\mathfrak{G}$. Given any other
representative $\mathcal{G}^{\prime}$ of $\mathfrak{G}$, there is a one to one
map between spanning c-subgraphs of $\mathcal{G}$ and those of
$\mathcal{G}^{\prime}$. This can be achieved by mapping any
$A\Subset\mathcal{G}$ onto $A^{\prime}\Subset\mathcal{G}^{\prime}$ such that
$A^{\prime}=(A\setminus D_{\mathcal{G}})\cup D_{\mathcal{G}^{\prime}}$ which
can be easily inverted. Now, one checks that ${\rm r}\,(\mathcal{G})$, ${\rm
r}\,(A)$, $n(A)$ and $5k(A)-(3V+2F_{\rm int}(A))$ do not depend of the
representative. The remaining exponents $B_{\rm int}(A)$, $B_{\rm ext}(A)$,
$C_{\partial}(A)$, $f(A)$, $E_{\partial}(A)$ and $F_{\partial}(A)$ depend only
on $A\setminus D_{\mathcal{G}}=A^{\prime}\setminus D_{\mathcal{G}^{\prime}}$
hence do not depend on the representative.

One notices that the above consistency checking establishes that only the
class of the spanning c-subgraphs of some representative matters in the
definition of $\mathfrak{T}_{\mathfrak{G}}$. Thus, we could have introduced a
state sum on $\mathfrak{g}\Subset\mathfrak{G}$ in order to define this
invariant directly from the equivalent classes point of view using Proposition
11. We will keep however the more explicit formulation in terms of the
spanning c-subgraphs of representatives.

Another crucial point is to show that the exponent of $z$ in (113) is always a
non negative integer.

######  Proposition 12.

Let $\mathcal{G}$ be any representative of $\mathfrak{G}$ a rank 3 w-colored
graph. Then

$\zeta(\mathcal{G})=3(E(\mathcal{G})-V(\mathcal{G}))+2[B_{{\rm
int}}(\mathcal{G})+B_{{\rm ext}}(\mathcal{G})-F_{{\rm int}}(\mathcal{G})]\geq
0\,.$ (115)

###### Proof.

Let us consider the set of closed and open bubbles $\mathcal{B}_{{\rm int}}$
and $\mathcal{B}_{{\rm ext}}$ of $\mathcal{G}$, respectively (in this proof,
we drop the dependence in the graph $\mathcal{G}$ in all quantities). Let
$B_{{\rm int}}$ and $B_{{\rm ext}}$ be their respective cardinal. Each open or
closed bubble $\mathbf{b}$ is an open or a closed ribbon graph with
$V_{\mathbf{b}}$ number of vertices, $E_{\mathbf{b}}$ number of edges,
$F_{{\rm int};\mathbf{b}}$ number of closed faces and
$C_{\partial}(\mathbf{b})$ number of closed faces obtained uniquely from open
strands after pinching (coinciding with the number of connected component of
the boundary of the open bubble $\mathbf{b}$).

Any $\mathbf{b}_{i}\in\mathcal{B}_{{\rm int}}$ being a closed ribbon graph,
its Euler characteristics writes

$2-\kappa_{\mathbf{b}_{i}}=V_{\mathbf{b}_{i}}-E_{\mathbf{b}_{i}}+F_{{\rm
int};\mathbf{b}_{i}}\,,$ (116)

where $\kappa_{\mathbf{b}_{i}}$ refers to the genus of $\mathbf{b}_{i}$ or
twice its genus if $\mathbf{b}_{i}$ is oriented. Summing over all internal
bubbles, we get

$2B_{{\rm int}}-\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\kappa_{\mathbf{b}_{i}}=\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\left[V_{\mathbf{b}_{i}}-E_{\mathbf{b}_{i}}+F_{{\rm
int};\mathbf{b}_{i}}\right].$ (117)

Using the colors, one observes that each edge of $\mathcal{G}$ splits into
three ribbon edges belonging either to an open or a closed bubble, and each
internal face of $\mathcal{G}$ belongs to two bubbles which might be open or
closed. Thus we have

$\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}E_{\mathbf{b}_{i}}+\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}E_{\mathbf{b}_{x}}=3E\,,\qquad\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}F_{{\rm int};\mathbf{b}_{i}}+\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}F_{{\rm int};\mathbf{b}_{x}}=2F_{{\rm int}}\,.$ (118)

In addition, each vertex of the graph can be decomposed, at least, in three
vertices (3 vertices is the minimum given by the simplest vertex of the form
$\mathcal{G}_{1}$ in Figure 29) which could belong to an open or closed
bubble, we have

$\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}V_{\mathbf{b}_{i}}+\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}V_{\mathbf{b}_{x}}\geq 3V\,.$ (119)

Combining (118) and (119), we re-write (117) as

$3V-3E+2F_{{\rm int}}-2B_{{\rm int}}-\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}\left[V_{\mathbf{b}_{x}}-E_{\mathbf{b}_{x}}+F_{{\rm
int};\mathbf{b}_{x}}\right]\leq-\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\kappa_{\mathbf{b}_{i}}\,.$ (120)

We complete the last sum involving $\mathcal{B}_{{\rm ext}}$ by adding
$C_{\partial}(\mathbf{b}_{x})$ in order to get

$\displaystyle\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}\left[V_{\mathbf{b}_{x}}-E_{\mathbf{b}_{x}}+F_{{\rm
int};\mathbf{b}_{x}}+C_{\partial}(\mathbf{b}_{x})\right]=\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}(2-\kappa_{\widetilde{\mathbf{b}}_{x}})\,,$ (121)

which, substituted in (120), leads us to

$\displaystyle 3V-3E+2F_{{\rm int}}-2B_{{\rm int}}-2B_{{\rm
ext}}\leq-\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\kappa_{\mathbf{b}_{i}}-\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}\big{(}C_{\partial}(\mathbf{b}_{x})+\kappa_{\widetilde{\mathbf{b}}_{x}}\big{)}$
(122)

from which the lemma results. ∎

In fact, $\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}C_{\partial}(\mathbf{b}_{x})=F_{\partial}(\mathcal{G})$ is the number of
faces of the boundary graph $\mathcal{G}$ (see Remark 1). The above bound can
be refined since $F_{\partial}(\mathcal{G})\geq B_{{\rm ext}}(\mathcal{G})$
which merely follows from the fact that each
$\mathbf{b}_{x}\in\mathcal{B}_{{\rm ext}}$ has at least a boundary
contributing to $F_{\partial}(\mathcal{G})$ such that

$\displaystyle B_{{\rm
ext}}(\mathcal{G})\leq\sum_{\mathbf{b}_{x}}C_{\partial}(\mathbf{b}_{x})=F_{\partial}(\mathcal{G})\,.$
(123)

Thus we also have

$3V-3E+2F_{{\rm int}}-2B_{{\rm int}}-B_{{\rm
ext}}\leq-\sum_{\mathbf{b}_{i}\in\mathcal{B}_{{\rm
int}}}\kappa_{\mathbf{b}_{i}}-\sum_{\mathbf{b}_{x}\in\mathcal{B}_{{\rm
ext}}}\kappa_{\widetilde{\mathbf{b}}_{x}}\,.$ (124)

This may lead as well to yet another well defined invariant. However, we will
not use this relation in the following due to some rich relations that
$-\zeta(A)\geq 0$ entails. In particular, we have other positive combinations.

######  Proposition 13.

Let $\mathcal{G}$ be any representative of $\mathfrak{G}$ a rank 3 w-colored
graph. Then

$\displaystyle\zeta^{\prime}(\mathcal{G})=3[E(\mathcal{G})-V(\mathcal{G})]+2[B_{{\rm
int}}(\mathcal{G})+B_{{\rm ext}}(\mathcal{G})-F_{{\rm
int}}(\mathcal{G})-C_{\partial}(\mathcal{G})]\geq 0\,,$ (125)
$\displaystyle\zeta^{\prime\prime}(\mathcal{G})=3[E(\mathcal{G})-V(\mathcal{G})-C_{\partial}(\mathcal{G})]+2[B_{{\rm
int}}(\mathcal{G})+B_{{\rm ext}}(\mathcal{G})-F_{{\rm int}}(\mathcal{G})]\geq
0\,.$ (126)

###### Proof.

By the color prescription, each connected component of the boundary of
$\mathcal{G}$ has at least three faces. Therefore

$\displaystyle 3C_{\partial}\leq F_{\partial}\,.$ (127)

Then we also have $-F_{\partial}+2C_{\partial}\leq 0$. The proposition follows
from (122) and the fact that
$F_{\partial}=\sum_{\mathbf{b}_{x}}C_{\partial}(\mathbf{b}_{x})$ in the proof
of Proposition 12.

∎

Proposition 12 ensures $\zeta(A)\geq 0$, for any $A\Subset\mathcal{G}$, hence
the following statement.

######  Proposition 14 (Polynomial invariant).

$\mathfrak{T}_{\mathfrak{G}}=\mathfrak{T}_{\mathcal{G}}$ is a polynomial.

The quantity $V-E(A)+F_{{\rm int}}(A)-B_{{\rm int}}(A)$ is nothing but the
Euler characteristics for a closed colored graph $A$ understood as a cellular
complex [10]. This quantity always proves to be bounded by
$-\sum_{\mathbf{b}_{i}}\kappa_{\mathbf{b}_{i}}$. If the graph is not closed,
the same cellular complex has a boundary bearing itself a cellular
decomposition. The quantity $-\zeta(A)$ represents, in the present specific
instance, a weighted notion for Euler characteristics of the cellular complex
corresponding to $A$ which also take into account the boundary due to $B_{{\rm
ext}}$. The quantity $5k(A)-\zeta(A)$ will be however the one of interest in
the following.

Let us call $\mathcal{G}^{0}$ a representative graph made only with a finite
family of vertices with flags and without any edges possibly with discs. Then
$E(\mathcal{G}^{0})=B_{{\rm int}}(\mathcal{G}^{0})=0$, $F_{{\rm
int}}(\mathcal{G}^{0})=D$, $D$ being the number of trivial discs in
$\mathcal{G}^{0}$, $k(\mathcal{G}^{0})=V(\mathcal{G}^{0})$,
$k(\mathcal{G}^{0})-D=C_{\partial}(\mathcal{G}^{0})\geq 0$, we can check the
consistency of (113) as the following makes still a sense:

$\mathfrak{T}_{\mathcal{G}^{0}}(x,y,z,s,w,q,t)=z^{2[k(\mathcal{G}^{0})-D+B_{{\rm
ext}}(\mathcal{G}^{0})]}s^{k(\mathcal{G}^{0})-D}w^{F_{\partial}(\mathcal{G}^{0})}q^{E_{\partial}(\mathcal{G}^{0})}\,t^{|\mathfrak{f}^{0}(\mathcal{G}^{0})|}.$
(128)

It may exist several possible reductions of the above polynomial. We will
however focus on the most prominent ones given by

$\displaystyle\mathfrak{T}_{\mathcal{G}}(x,y,z,z^{-2},w,q,t)=\mathfrak{T}^{\prime}_{\mathcal{G}}(x,y,z,w,q,t)\,,\quad\mathfrak{T}_{\mathcal{G}}(x,y,z,z^{-2}s^{2},s^{-1},s,s^{-1})=\mathfrak{T}^{\prime\prime}_{\mathcal{G}}(x,y,z,s)\,,$
(129)
$\displaystyle\mathfrak{T}_{\mathcal{G}}(x,y,z,z^{2}z^{-2},z^{-1},z,z^{-1})=\mathfrak{T}^{\prime\prime\prime}_{\mathcal{G}}(x,y,z)\,.$
(130)

Proposition 13 ensures that $\mathfrak{T}^{\prime}_{\mathcal{G}}$ is a
polynomial. Meanwhile, $\mathfrak{T}^{\prime\prime}_{\mathcal{G}}$ is also a
polynomial with exponent of $s$ the Euler characteristics of the boundary
$\partial{\mathcal{G}}$.
$\mathfrak{T}^{\prime\prime\prime}_{\mathcal{G}}(x,y,z)$ combine both
invariants in a single exponent. These polynomials will be relevant in the
next analysis. Note that we could have introduced another polynomial between
$\mathfrak{T}^{\prime}$ and $\mathfrak{T}^{\prime\prime}$ which expresses as
$\mathfrak{T}_{\mathcal{G}}(x,y,z,s^{2},s^{-1},s,s^{-1})=\mathfrak{T}^{0}_{\mathcal{G}}(x,y,z,s)$.
But this latter turns out to satisfy the same properties as $\mathfrak{T}$ and
thus does not provide anything new.

We are now in position to prove our main theorem.

######  Theorem 4 (Contraction/cut rule for w-colored graphs).

Let $\mathfrak{G}$ be a rank 3 w-colored graph with flags. Then, for a regular
edge $e$ of any of the representative $\mathcal{G}$ of $\mathfrak{G}$, we have

$\mathfrak{T}_{\mathcal{G}}=\mathfrak{T}_{\mathcal{G}\vee
e}+\mathfrak{T}_{\mathcal{G}/e}\,,$ (132)

for a bridge $e$, we have $\mathfrak{T}_{\mathcal{G}\vee
e}=z^{8}s(wq)^{3}t^{2}\mathfrak{T}_{\mathcal{G}/e}$ and

$\mathfrak{T}_{\mathcal{G}}=[(x-1)z^{8}s(wq)^{3}t^{2}+1]\mathfrak{T}_{\mathcal{G}/e}\,;$
(133)

for a trivial p-inner self-loop $e$, $p=0,1,2$, we have

$\mathfrak{T}_{\mathcal{G}}=\mathfrak{T}_{\mathcal{G}\vee
e}+(y-1)z^{4p-7}\,\mathfrak{T}_{\mathcal{G}/e}\,.$ (134)

###### Proof.

Let $\mathcal{G}$ be any representative of the class $\mathfrak{G}$. Same
initial remarks as stated in the proof of Theorem 3 hold here for
$\mathcal{G}$. Our main concern is the change in the number of internal and
external bubbles and the independence of the final result on the
representative $\mathcal{G}$.

We concentrate first on (132). Consider an ordinary edge $e$ of $\mathcal{G}$,
the set of spanning c-subgraphs which do not contain $e$ being the same as the
set of spanning c-subgraphs of $\mathcal{G}\vee e$, the number of open and
closed bubbles on each subgraph is the same, it is direct to get
$\sum_{A\Subset\mathcal{G};e\notin A}(\cdot)=\mathfrak{T}_{\mathcal{G}\vee
e}$. Observe that the class of $\mathcal{G}\vee e$ defines $\mathfrak{G}\vee
e$ from Lemma 5 and we know that all exponents in
$\mathfrak{T}_{\tilde{\mathcal{G}}}$ do not depend on the representative
$\tilde{\mathcal{G}}$ of the class of $\tilde{\mathfrak{G}}$. Hence
$\mathfrak{T}_{\mathcal{G}\vee e}=\mathfrak{T}_{\mathfrak{G}\vee e}$ does not
depend on the representative.

Let us focus on the second term of (132). Consider $e$, its end vertices
$v_{1}$ and $v_{2}$ and its 3 strands with color pairs $(aa_{1})$, $(aa_{2})$
and $(aa_{3})$ and consider the set bubbles in $\mathfrak{G}$. Some bubbles do
not use any strand of $e$ and three bubbles can be formed using these strands
(these are of colors $(aa_{i}a_{j})$, $i<j$). Contracting now $e$, the vertex
obtained is connected by Lemma 4. The former three strands $e$ are clearly
preserved after the contraction. The bubbles not passing through $e$ are not
affected at all by the procedure. The three bubbles passing through $e$ are
also preserved since the contraction do not delete faces or strands. The faces
passing through $e$ getting only shortened, the net result from the point of
view of the bubbles passing through $e$ is simply an ordinary ribbon edge
contraction in the sense of ribbon graphs. From this results
$\sum_{A\Subset\mathcal{G};e\in A}(\cdot)=\mathfrak{T}_{\mathcal{G}/e}$. By
Lemma 5, we define $\mathfrak{G}/e$ and conclude that the polynomial
$\mathfrak{T}_{\mathfrak{G}/e}$ does not depend on the representative
$\mathcal{G}/e$ of $\mathfrak{G}/e$.

Let us focus now on the bridge case and (133). Cutting a bridge yields, as in
the ordinary case, from the sum $\sum_{A\Subset\mathcal{G};e\notin A}$ the
product $(x-1)\mathfrak{T}_{\mathcal{G}\vee e}$. The second sum remains as it
is using the mapping between $\\{A\Subset\mathcal{G};e\in A\\}$ and
$\\{A\Subset\mathcal{G}/e\\}$ and provided the fact that there is no change in
the vertex contracted which is ensured by Lemma 4. The last stage relates
$\mathfrak{T}_{\mathcal{G}\vee e}$ and $\mathfrak{T}_{\mathcal{G}/e}$. This
can be achieved by using the bijection between $A\Subset\mathcal{G}\vee e$ and
$A^{\prime}\Subset\mathcal{G}/e$ where each $A$ and $A^{\prime}$ are both
uniquely related to some $A_{0}\Subset\mathcal{G}$ as $A=A_{0}\vee e$ and
$A^{\prime}=A_{0}/e$. Using Lemma 8, the relation (133) follows. Then, Lemma 5
allows us to define ($\mathfrak{G}\vee e$ and) $\mathfrak{G}/e$ and to
conclude that the resulting polynomials ($\mathfrak{T}_{\mathcal{G}\vee e}$
and) $\mathfrak{T}_{\mathcal{G}/e}$ are independent on the representative
($\mathcal{G}\vee e$ and) $\mathcal{G}/e$.

Next, we discuss the trivial $p$–inner self-loop case and prove (134). The
property that the sum $\sum_{A\Subset\mathcal{G};e\notin
A}(\cdot)=\mathfrak{T}_{\mathcal{G}\vee e}$ should be direct. The second sum
is now studied.

The question is whether or not $e$ being a trivial $p$–inner self-loop in
$\mathcal{G}$ remains as such in $A$. The answer for that question is yes
because $A$ contains the end vertex of $e$. We may cut some edges in each
sectors $v_{i}$ (see Figure 30) for defining $A$ but the resulting sectors are
still totally separated in $A$.

Contracting a trivial $p$–inner self-loop generates $p$ discs and $3-p$ non
trivial vertices. Now, from Lemma 6, we know how evolve all numbers of
components in the graph: the nullity is again $n(A)=n(A^{\prime})+1$, and the
exponent of $z$ becomes

$\displaystyle 5k(A)-(3(V(\mathcal{G})-E(A))+2(F_{\rm int}(A)-B_{\rm
int}(A)-B_{{\rm ext}}(A))=$ (135) $\displaystyle
5(k(A^{\prime})-2)-\Big{[}3[V(\mathcal{G}/e)-2)-(E(A^{\prime})+1)]+2(F_{\rm
int}(A^{\prime})-B_{\rm int}(A^{\prime})-B_{{\rm
ext}}(A^{\prime})+\alpha_{p})\Big{]}$ (136)
$\displaystyle=5k(A^{\prime})-(3(V(\mathcal{G}/e)-E(A^{\prime}))+2(F_{\rm
int}(A^{\prime})-B_{\rm int}(A^{\prime})-B_{{\rm ext}}(A^{\prime}))-7+4p\,,$
(137)

where $\alpha_{p}=3-2p$, $A$ and $A^{\prime}$ refer to the subgraphs related
by bijection between spanning c-subgraphs of $\\{A\Subset\mathcal{G};e\in
A\\}$ and $\\{A^{\prime}\Subset\mathcal{G}/e\\}$. At the end, one gets
$(y-1)z^{4p-7}\mathfrak{T}_{\mathcal{G}/e}$. In the similar way as done
before, we are able to conclude that this result is independent of the
representative.

∎

Some comments are in order at this point.

\- The existence of a function defined on $\mathfrak{G}$ satisfying the
relations of Theorem 4 is guaranteed by the explicit formula of
$\mathfrak{T}_{\mathfrak{G}}$ given in Definition 36. The unicity of
$\mathfrak{T}_{\mathfrak{G}}$ follows from its existence and can be only
guaranteed by suitable “boundary condition” data which are given by the bridge
contraction (133) and the data of all one-vertex (rosette) w-colored graph
polynomials. These polynomials write again using the state sum (sum over
subgraphs) of such stranded rosette graphs (113). These conditions extend
boundary conditions making unique the BR polynomial [6].

\- Note that in the equality (137) the number of bubbles $B_{{\rm
int}}+B_{{\rm ext}}$ plays a crucial role in the determination of the exponent
of $z$. In addition, the contraction of a bridge for this category of graph
with flags is close to the Tutte/BR relation for bridge contraction since we
do not need any constrain in the polynomial (for instance putting $s=z^{-1}$
in the case of the BR polynomial for graphs with flags) in order to obtain a
one term relation for such a terminal form. For the last contraction/cut for a
trivial $p$–inner self-loop (134), we still cannot map
$\mathfrak{T}_{\mathcal{G}\vee e}$ on $\mathfrak{T}_{\mathcal{G}/e}$. These
features show that the polynomial $\mathfrak{T}$ is a genuine extension of the
BR polynomial.

######  Corollary 2 (Cut/contraction relations for $\mathfrak{T}^{\prime}$).

Let $\mathfrak{G}$ be a rank 3 w-colored stranded graph with flags and any of
its representative $\mathcal{G}$. Then, for a bridge $e$, we have
$\mathfrak{T}^{\prime}_{\mathcal{G}\vee
e}=z^{6}(wq)^{3}t^{2}\mathfrak{T}^{\prime}_{\mathcal{G}/e}$ and

$\mathfrak{T}^{\prime}_{\mathcal{G}}(x,y,z,w,q,t)=[(x-1)z^{6}(qw)^{3}t^{2}+1]\mathfrak{T}^{\prime}_{\mathcal{G}/e}(x,y,z,w,q,t)\,;$
(138)

for a trivial $p$–inner self-loop $e$ in $\mathcal{G}$, $0\leq p\leq 2$, we
have

$\mathfrak{T}^{\prime}_{\mathcal{G}}(x,y,z,1,q,t)=z^{4p-6}[q^{3}t^{2}+(y-1)z^{-1}]\,\mathfrak{T}^{\prime}_{\mathcal{G}/e}(x,y,z,1,q,t)\,;$
(139)

###### Proof.

Theorem 4 implies naturally (138) after the setting $s=z^{-2}$ in
$\mathfrak{T}_{\mathcal{G}}$. We now work out the contraction/cut of the
trivial self-loops.

Same arguments as in the proof of Theorem 4 should be invoked here. The
difference now is that by changing $s\to z^{-2}$, using (1) the one-to-one
mapping between $\\{A\Subset\mathcal{G}\vee e\\}$ and
$\\{A^{\prime}\Subset\mathcal{G}/e\\}$ such that to each
$A\Subset\mathcal{G}\vee e$ one associates $A^{\prime}\Subset\mathcal{G}/e$
defined by $A^{\prime}=\tilde{A}/e$, $\tilde{A}\Subset\mathcal{G}$, and
$\tilde{A}\vee e=A$ and (2) the relations (95)-(97) of Lemma 8, we can map,
for a trivial $0,1,2$–inner self-loop $e$, $\mathfrak{T}_{\mathcal{G}\vee e}$
on $\mathfrak{T}_{\mathcal{G}/e}$. We compute the variation of the exponent of
$z$ between $A$ and $A^{\prime}$ as

$\displaystyle 5k(A)-(3(V(\mathcal{G})-E(A))+2[F_{\rm
int}(A)+C_{\partial}(A)-B_{\rm int}(A)-B_{{\rm ext}}(A)])=$ (140)
$\displaystyle
5(k(A^{\prime})-2)-(3(V(\mathcal{G}^{\prime})-2-E(A^{\prime}))+2[F_{\rm
int}(A^{\prime})+C_{\partial}(A^{\prime})-2-B_{\rm int}(A^{\prime})-B_{\rm
ext}(A^{\prime})+\alpha_{p}])$ (141)
$\displaystyle=5k(A^{\prime})-(3(V(\mathcal{G}/e)-E(A^{\prime}))+2[F_{\rm
int}(A^{\prime})-B_{\rm int}(A^{\prime})-B_{\rm
ext}(A^{\prime})])-2\alpha_{p}\,.$ (142)

The rest of the variations can be easily identified using the same lemma. It
can be proved that the result does not depend on the representative.

∎

As a comment, for $w\neq 1$, $\mathfrak{T}^{\prime}_{\mathcal{G}\vee e}$
cannot be fully mapped onto $\mathfrak{T}^{\prime}_{\mathcal{G}/e}$ due to the
fact that the number $F_{\partial}$ of face of the boundary of each subgraph
cannot be determined in that general situation. Nevertheless, we have two
other noteworthy reductions which solve this issue without putting $w=1$.

######  Corollary 3 (Cut/contraction rules for $\mathfrak{T}^{\prime\prime}$
(and $\mathfrak{T}^{\prime\prime\prime}$)).

Let $\mathfrak{G}$ be a rank 3 w-colored stranded graph with flags and any of
its representative $\mathcal{G}$.

Then, for a bridge $e$, we have $\mathfrak{T}^{\prime\prime}_{\mathcal{G}\vee
e}=z^{6}\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}$ and

$\mathfrak{T}^{\prime\prime}_{\mathcal{G}}=[(x-1)z^{6}+1]\,\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}\,;$
(143)

for a trivial $p$–inner self-loop $e$ in $\mathcal{G}$, $0\leq p\leq 2$, we
have $\mathfrak{T}^{\prime\prime}_{\mathcal{G}\vee
e}=z^{4p-6}\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}$ and

$\mathfrak{T}^{\prime\prime}_{\mathcal{G}}=z^{4p-6}[1+(y-1)z^{-1}]\,\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}\,.$
(144)

The same contraction and cut rules applies for
$\mathfrak{T}^{\prime\prime\prime}_{\mathcal{G}}$.

###### Proof.

The new ingredient to achieve the proof of this statement is (98) of Lemma 8.

∎

More remarks can be made at this stage.

\- $\mathfrak{T}^{\prime\prime}$ and $\mathfrak{T}^{\prime\prime\prime}$
satisfy the same rules. Hence $\mathfrak{T}^{\prime\prime}$ which is more
general becomes more relevant.

\- We did not use the notion of product of graphs for mapping the polynomials
for $\mathcal{G}\vee e$ onto those of $\mathcal{G}/e$. In fact, even for
ribbon graphs our approach applies. The dot product of graphs is however an
elegant way to encode such a map. Certainly, this notion deserves to be
defined in the framework of stranded graphs in order to achieve more
calculations. For instance, how the calculation of more involved graphs with
many trivial self-loops can be inferred from the product of contributions of
much simpler trivial self-loops.

\- The exponents of $z^{4p-7}$ or of $z^{4p-6}$ can be negative in some
expressions. This simply implies that in the polynomial
$\mathfrak{T}^{\prime}_{\mathcal{G}/e}$ or
$\mathfrak{T}^{\prime\prime}_{\mathcal{G}/e}$ all monomials should contain an
enough large power of $z$ to render the overall expression a well defined
polynomial.

If the notion of point graph multiplication of two disjoint graphs is not
defined at this stage, in contrast, the disjoint union operation on graph
extends naturally in the present formulation.

######  Lemma 9 (Disjoint union).

Let $\mathfrak{G}_{1}$ and $\mathfrak{G}_{2}$ two disjoint rank 3 $w$-colored
graphs, then

$\mathfrak{T}_{\mathfrak{G}_{1}\sqcup\mathfrak{G}_{2}}=\mathfrak{T}_{\mathfrak{G}_{1}}\,\mathfrak{T}_{\mathfrak{G}_{2}}\,.$
(145)

###### Proof.

This is totally standard as in the ordinary procedure using additive
properties of exponents in $\mathfrak{T}_{\mathcal{G}}$.

∎

######  Corollary 4 (3–inner self-loop contraction).

Given a rank 3 w-colored graph $\mathfrak{G}$ and $\mathcal{G}$ one of its
representative containing a 3–inner self-loop $e$ then

$\mathfrak{T}_{\mathcal{G}}=z^{5}(z^{3}s(wq)^{3}t^{2}+y-1)\mathfrak{T}_{\mathcal{G}/e}\,.$
(146)

###### Proof.

We use the fact that $e$ is a 3–inner then it forms a separate subgraph $g$.
In order to compute the polynomial of $\mathcal{G}$, Lemma 9 can be applied
and a direct evaluation of the graph $g$ yields the desired factor.

∎

######  Definition 37 (Multivariate form).

The multivariate form associated with (113) is defined by:

$\displaystyle\widetilde{\mathfrak{T}}_{\mathfrak{G}}(x,\\{\beta_{e}\\},\\{z_{i}\\}_{i=1,2,3},s,w,q,t)=\widetilde{\mathfrak{T}}_{\mathcal{G}}(x,\\{\beta_{e}\\},\\{z_{i}\\}_{i=1,2,3},s,w,q,t)$
(147) $\displaystyle=\sum_{A\Subset\mathcal{G}}x^{{\rm r}\,(A)}(\prod_{e\in
A}\beta_{e})z_{1}^{F_{{\rm int}}(A)}z_{2}^{B_{{\rm int}}(A)}z_{3}^{B_{{\rm
ext}}(A)}\,s^{C_{\partial}(A)}\,w^{F_{\partial}(A)}q^{E_{\partial}(A)}t^{f(A)}\,,$

for $\\{\beta_{e}\\}_{e\in\mathcal{E}}$ labeling the edges of the graph
$\mathcal{G}$.

It is direct to prove the following statement by ordinary techniques.

######  Proposition 15.

For all regular edge $e$,

$\displaystyle\widetilde{\mathfrak{T}}_{\mathfrak{G}}=\widetilde{\mathfrak{T}}_{\mathfrak{G}\vee
e}+x\beta_{e}\,\widetilde{\mathfrak{T}}_{\mathfrak{G}/e}\,.$ (148)

######  Remark 2.

We can now compare $\widetilde{\mathfrak{T}}$ with Gurau’s polynomial denoted
in the following by $G$ [12]. We shall use however not the full form of
$G_{\mathcal{G}}$ denoted $P_{\mathcal{G}}$ in [12] but we will use instead
two improvements: (1) the normalized form
$P_{\mathcal{G}}(\\{\beta_{e}x_{1}\\},\dots)$ of $P_{\mathcal{G}}$, where
$x_{1}$ is the variable associated with the number of edges which brings an
inessential overall factor of $x_{1}^{E(A)}$ consistently absorbed by the
$\beta_{e}$ and (2) a rank formulation of
$G_{\mathcal{G}}(\\{\beta_{e}\\},\dots)=P_{\mathcal{G}}(\\{\beta_{e}x_{1}\\},\dots)$,
i.e. rather then using two variables $x_{0}$ for the vertices and $x_{4}$ for
the number of connected components of the rank 3 colored tensor graph, we will
simply use $x^{{\rm r}\,(A)}$, $A\Subset\mathcal{G}$. This notation is more
compact and ${\rm r}\,(A)$ is the quantity depending only on the class of $A$.

For a rank 3 colored tensor graph $\mathcal{G}$, the polynomials
$\widetilde{\mathfrak{T}}$ and $G$ are related by

$\widetilde{\mathfrak{T}}_{\mathcal{G}}(x,\\{\beta_{e}\\},z_{1},z_{2},z_{3}=1,s,w,q,t)=G_{\mathcal{G}}(x,\\{\beta_{e}\\},z_{1},z_{2},s,q,w,t)\,,$
(149)

with, according to the convention in [12], we have

$\displaystyle C_{\partial}=|\mathfrak{B}_{\partial}^{3}|,\quad
F_{\partial}=|\mathfrak{B}_{\partial}^{2}|,\quad
E_{\partial}=|\mathfrak{B}_{\partial}^{1}|,\quad
f=|\mathfrak{B}_{\partial}^{0}|,\quad B_{{\rm int}}=|\mathfrak{B}^{3}|,\quad
F_{{\rm int}}=|\mathfrak{B}^{2}|.$ (150)

As expected, for this category of graphs the polynomial
$\widetilde{\mathfrak{T}}$ is more general because it contains one additional
variable (denoted above by $z_{3}$) which is associated with the number of
external bubbles of the graph. This variable can be introduced by hand in $G$
making it a little more general. This additional variable does not lead to any
new features for the multivariate form however, as seen in our developments,
it plays an important role in the non-multivariate form.

We finally wrap up this section with additional insights:

$\bullet$ We have successfully defined the classes of rank 3 w-colored
stranded graphs for which a natural contraction/cut rule makes a sense and the
associated polynomial invariants $\mathfrak{T},\mathfrak{T}^{\prime}$ and
$\mathfrak{T}^{\prime\prime}$. Our results extend both BR polynomial to
arbitrary contractions of colored tensor graphs of rank 3. The achievement of
this requires to relax and to modify several initial concepts introduced by
Gurau (to mention a few, the meaning of 3-bubbles with now no fixed valence
and the very notion of contraction itself, for instance). The form of our
polynomial is not primarily given in the multivariate form (as the one by
Gurau). Thus, it has required a full-fledge proof in order to establish that
this polynomial is a genuine invariant for the present graphs. We have also
successfully identified terminal forms of the contraction/cut sequence for
this type of graphs. Such a study of terminal forms has been never addressed
in former works on stranded graphs [12][18].

$\bullet$ We stress that after the contraction of some $p$–inner self-loops,
other remaining trivial $p^{\prime}$–inner self-loops may change their number
of $p^{\prime}$ inner faces. This is the reason why given a certain number of
$m_{p}$ trivial $p$–inner self-loops, we cannot easily infer the number of
trivial $0,1,2$– or $3$–inner self-loops which shall really contribute to the
final evaluation of the terminal form. In fact, we can show that contracting a
trivial $p$–inner self-loop at some point of the sequence may only be neutral
for or increase by 1 the number of inner faces of other $p$–inner self-loops.
Hence, roughly speaking, after contraction there may or may not have a
“migration” of the elements of the set of $p$–inner self-loops to set of the
$(p+1)$–inner self-loops, $p+1\leq 3$.

In any case, the polynomial $\mathfrak{T}^{\prime\prime}$ of some terminal
form $\mathcal{G}$ representative of a w-colored graph made only with bridges
and trivial $p$–inner self-loops can be computed as

$x^{m}\,\left[\prod_{p=0}^{3}\,\\{z^{4p-6}[1+(y-1)z^{-1}]\\}^{n_{p}}\right]\times\mathfrak{T}^{\prime\prime}(\mathcal{G}^{0})\,,$
(151)

for some integer $m,n_{0,1,2,3}$, and where
$\mathfrak{T}^{\prime\prime}(\mathcal{G}^{0})$ is the polynomial of
$\mathcal{G}^{0}$ graph having only vertices and hence is given by (128).
After changing $s\to z^{-2}s^{2}$, one gets

$\mathfrak{T}^{\prime\prime}(\mathcal{G}^{0})=z^{2B_{{\rm
ext}}(\mathcal{G}^{0})}s^{2C_{\partial}(\mathcal{G}^{0})-(f(\mathcal{G}^{0})-E_{\partial}(\mathcal{G}^{0})+F_{\partial}(\mathcal{G}^{0}))}\,.$
(152)

Note that the contribution (151) only makes a sense as a polynomial if
exponents $n_{p}$ are not arbitrary. One must have indeed

$5n_{3}+n_{2}-3n_{1}-7n_{0}+2B_{\rm ext}(\mathcal{G}^{0})\geq 0\,,$ (153)

which should follow from Proposition 12. However it is interesting to show
that this is indeed true in a different (and quick) manner. The issue here is
to show that the total number of trivial 1–inner and 0–inner contributions
cannot exceed the number of 3–inner, 2–inner contributions and external
bubbles after the overall contraction. In fact, whenever we contract a trivial
1–inner or 0–inner self-loop, we split the one-vertex graph in $m_{1}=2$ or
$m_{0}=3$ one-vertex (non discs) graphs and get either a power of $z^{-3}$ or
$z^{-7}$, respectively. On each of these resulting graphs, we can pursue the
contraction. But, at the end, either we are led to a connected component of
the form of a 3–inner self-loop and we gain a power of $z^{5}$ per connected
component obtained, or to a final vertex with only flags (see Lemma 1). The
simplest non trivial vertex-graph with the minimal number of external bubbles
is given $\mathcal{G}_{1}$ in Figure 29 and yields $B_{{\rm
ext}}(\mathcal{G}_{1})=3$. Call $\beta_{3}$ the number of components yielding
a 3–inner self-loop and $\beta_{x}$ the number of components yielding an empty
vertex graph. Then

$\beta_{3}+\beta_{x}=2n_{1}+3n_{0}\,,\qquad\beta_{3}\leq n_{3}\,,\qquad
3\beta_{x}\leq B_{{\rm ext}}(\mathcal{G}^{0})\,,$ (154)

and hence

$\displaystyle-3n_{1}-7n_{0}+5n_{3}+2B_{{\rm ext}}(\mathcal{G}^{0})$
$\displaystyle\geq$ $\displaystyle-3n_{1}-7n_{0}+5\beta_{3}+2B_{{\rm
ext}}(\mathcal{G}^{0})$ (155) $\displaystyle\geq$ $\displaystyle
7n_{1}+8n_{0}+2(B_{{\rm ext}}(\mathcal{G}^{0})-5\beta_{x}/2)\geq 0\,.$ (156)

In all situation, we can always beat the negative power introduced by the
trivial $0-$ or $1$–inner edge.

$\bullet$ Studying the limit cases, we have the following reduced polynomials:
for a w-colored graph $\mathfrak{G}$ and any representative $\mathcal{G}$ of
$\mathfrak{G}$,

$\mathfrak{T}_{\mathcal{G}}(x,y,z=1,s=1,w=1,q=1,t)=\mathcal{T}_{\mathcal{G}}(x,y,t)\,,$
(157)

where, computing $\mathcal{T}$ for $\mathcal{G}$, we consider in its simple
graph (collapsed) form.

There is a priori no direct mapping between $\mathfrak{T}$ and BR polynomials
(with and without flags) because these two polynomials have a different
“dimensionality” or rank with respect to stranded structures. One might think
about a combinatorial manipulation by collapsing the tensor structure to a
ribbon graph by removing a strand. This deserves to be fully understood.
However, in order to find such a mapping with BR polynomials, a natural way
would be to define the classes of “rank 2” w-colored graphs and their
associated polynomials. These latter should coincide with $\mathcal{R}$ and
$\mathcal{R}^{\prime}$ after identifications and reduction of some variables.
There will be nothing more general than $\mathcal{R}$ and
$\mathcal{R}^{\prime}$ in such context. The point raised in rank $D\geq 3$ is
the presence of $p\geq 3$-bubbles.

Few important properties have to be addressed:

$\bullet$ The universality property of $\mathfrak{T}$, $\mathfrak{T}^{\prime}$
or $\mathfrak{T}^{\prime\prime}$ which is the fact that any other function
satisfying (132) and (133) can be expressed in terms of one of these
polynomial needs to be understood. This task is not performed in this work
which has mainly focused on the definition of the relevant graph objects for
which a polynomial invariant can be identified. The universality will be
addressed in a forthcoming work and will require a complete understanding of
the one-vertex w-colored graphs as they appear in the formalism. We think that
the introduction of colors will help in reducing the type of graph structures
that we need to classify. Hence, we aim at finding suitable classes of graphs
for which one of the polynomials $\mathfrak{T}$, $\mathfrak{T}^{\prime}$ or
$\mathfrak{T}^{\prime\prime}$ will prove to be universal.

$\bullet$ In fact, before addressing the universality issue on stranded
structures, one first needs to understand to what extent the universality
property can be recovered for the extension $\mathcal{R}$ of the BR polynomial
for ribbon graphs with flags (without pinching). This will amount to
generalize the notion of chord diagrams $\mathcal{D}$, their equivalence
relation under rotations about chords and finally their associated basic
canonical diagrams $\mathcal{D}_{ijk}$ (counting the nullity $i$, genus $j$ of
the diagram, $k$ is associated with the orientation of the surface associated
with diagram) [6, 5] to chord diagrams with flags. Such a preliminary task
will certainly imply the existence of a new canonical diagram
$\mathcal{D}_{ijkl}$, where $i$, $j$, $k$ keep their meaning and $l$ defines
the number of connected component associated with the boundary of the open
ribbon graph.

$\bullet$ There exists another formulation of the Tutte and BR polynomials in
terms of spanning tree expansions. Such a notion deserves to be investigated
as well in the context of w-colored stranded graphs.

Acknowledgements

Discussions with Vincent Rivasseau at early stage of this work are gratefully
acknowledged. RCA thanks the Perimeter Institute for its hospitality. Research
at Perimeter Institute is supported by the Government of Canada through
Industry Canada and by the Province of Ontario through the Ministry of
Research and Innovation. This work is partially supported by the Abdus Salam
International Centre for Theoretical Physics (ICTP, Trieste, Italy) through
the Office of External Activities (OEA)-Prj-15. The ICMPA is also in
partnership with the Daniel Iagolnitzer Foundation (DIF), France.

## Appendix: Examples

We carry out explicitly two examples in order illustrate our results in the
present appendix. We will not focus on the equivalence class $\mathfrak{G}$
but on a particular representative always denoted by $\mathcal{G}$.

Example 1: A simple colored graph. Consider the graph $\mathcal{G}$ given by
Figure 32. Computing the multivariate form of the polynomial, we obtain by the
spanning c-subgraph summation in (147) with $\beta_{i}$ is associated with the
edge of color $i$,

$\displaystyle\mathfrak{T}_{\mathcal{G}}(x,\\{\beta_{e}\\},z,s,q,t)$
$\displaystyle=$
$\displaystyle\beta_{0}\beta_{1}\beta_{2}\beta_{3}\,xz_{1}^{6}z_{2}^{4}$ (A.1)
$\displaystyle+$
$\displaystyle(\beta_{0}\beta_{1}\beta_{2}+\beta_{0}\beta_{2}\beta_{3}+\beta_{1}\beta_{2}\beta_{3}+\beta_{0}\beta_{1}\beta_{3})\,xz_{1}^{3}z_{2}z_{3}^{3}sw^{3}q^{3}t^{2}$
(A.2) $\displaystyle+$
$\displaystyle(\beta_{0}\beta_{1}+\beta_{0}\beta_{2}+\beta_{0}\beta_{3}+\beta_{1}\beta_{2}+\beta_{1}\beta_{3}+\beta_{2}\beta_{3})\,xz_{1}z_{3}^{4}sw^{4}q^{6}t^{4}$
(A.3) $\displaystyle+$
$\displaystyle(\beta_{0}+\beta_{1}+\beta_{2}+\beta_{3})\,xz_{3}^{5}sw^{5}q^{9}t^{6}+z_{3}^{8}s^{2}w^{8}q^{12}t^{8}\,.$
(A.4)

Then this polynomial coincides exactly with the normalized Gurau’s polynomial
$G_{\mathcal{G}}$ after setting the additional variable $z_{3}$ associated
with external bubbles to 1. Note that reversely, introducing a new variable
associated with the same component in $C_{\mathcal{G}}$ both polynomials
match, for this example. Note also that, in this specific example the exponent
of $z_{3}$ and of $w$ always coincide. However it is not always true that each
open bubble would have a unique boundary component.

Cutting one edge, say the one of color 0, yields $\mathcal{G}\vee e$, so that
we evaluate

$\displaystyle\mathfrak{T}_{\mathcal{G}\vee e}(x,\\{\beta_{e}\\},z,s,q,t)$
$\displaystyle=$
$\displaystyle\beta_{1}\beta_{2}\beta_{3}\,xz_{1}^{3}z_{2}z_{3}^{3}sw^{3}q^{3}t^{2}$
(A.5) $\displaystyle+$
$\displaystyle(\beta_{1}\beta_{2}+\beta_{1}\beta_{3}+\beta_{2}\beta_{3})xz_{1}z_{3}^{4}sw^{4}q^{6}t^{4}$
(A.6) $\displaystyle+$
$\displaystyle(\beta_{1}+\beta_{2}+\beta_{3})xz_{3}^{5}sw^{5}q^{9}t^{6}+z_{3}^{8}s^{2}w^{8}q^{12}t^{8}$
(A.7)

and contracting the same edge gives

$\displaystyle\mathfrak{T}_{\mathcal{G}/e}(x,\\{\beta_{e}\\},z,s,q,t)$
$\displaystyle=$
$\displaystyle\beta_{1}\beta_{2}\beta_{3}\,z_{1}^{6}z_{2}^{4}$ (A.8)
$\displaystyle+$
$\displaystyle(\beta_{1}\beta_{2}+\beta_{2}\beta_{3}+\beta_{1}\beta_{3})z_{1}^{3}z_{2}z_{3}^{3}sw^{3}q^{3}t^{2}$
(A.9) $\displaystyle+$
$\displaystyle(\beta_{1}+\beta_{2}+\beta_{3})z_{1}z_{3}^{4}sw^{4}q^{6}t^{4}+z_{3}^{5}sw^{5}q^{9}t^{6}\,.$
(A.10)

Thus, we get

$\displaystyle\mathfrak{T}_{\mathcal{G}}=\mathfrak{T}_{\mathcal{G}\vee
e}+x\beta_{0}\mathfrak{T}_{\mathcal{G}/e}\,.$ (A.11)

It should be also emphasized that the contraction of edge is performed in the
sense of Definition 27. This definition allows us to improve the
active/passive contraction scheme as used in [12].

Figure 32.  A rank 3 colored tensor graph $\mathcal{G}$ and its cut
$\mathcal{G}\vee e$ and contraction $\mathcal{G}/e$ for a regular edge $e$ of
color 2.

$\mathcal{G}\vee e$$\mathcal{G}/e$$e$$3$$0$$1$$\mathcal{G}$

Example 2: A planar w-colored graph. We can compute $\mathfrak{T}$ for other
types of graphs which are not colored tensor graphs. In a specific instance,
consider the graph $\mathcal{G}$ of Figure 33. It combines both one colored
vertex and another type vertex. For simplicity, we change the variables
$(x-1)\to x$ and $(y-1)\to y$.

Figure 33.  A representative graph $\mathcal{G}$ of a w-colored graph
(displaying conjugate colors at all flags), the cut graph $\mathcal{G}\vee
e_{2}$ and the contracted graph $\mathcal{G}/e_{2}$ with respect to $e_{2}$.

$e_{2}$$e_{1}$$e_{0}$$e_{0}$$e_{0}$$\mathcal{G}\vee
e_{2}$$\mathcal{G}/e_{2}$$e_{1}$$e_{1}$$\mathcal{G}$

By the spanning c-subgraph summation, we get

$\displaystyle\mathfrak{T}_{\mathcal{G}}(x,y,z,s,q,t)=[xz^{10}sw^{5}q^{9}t^{6}+xyz^{9}sw^{4}q^{6}t^{4}+2z^{2}w^{2}q^{6}t^{4}+3yzwq^{3}t^{2}+y^{2}]z^{14}sw^{5}q^{9}t^{6}\,.$
(A.12)

We want to compare (A.12) with the result obtained by contraction and cut
procedure using a much as possible results on terminal forms. Using the
notations of Figure 33, we must check that

$\mathfrak{T}_{\mathcal{G}}=\mathfrak{T}_{\mathcal{G}\vee
e_{2}}+\mathfrak{T}_{\mathcal{G}/e_{2}}\,.$ (A.13)

Evaluating $\mathfrak{T}_{\mathcal{G}\vee e_{2}}$ and
$\mathfrak{T}_{\mathcal{G}/e_{2}}$, one finds

$\displaystyle\mathfrak{T}_{\mathcal{G}\vee
e_{2}}=[xz^{8}s(wq)^{3}t^{2}+1]\mathfrak{T}_{(\mathcal{G}\vee
e_{2})/e_{0}}\,,\quad\mathfrak{T}_{(\mathcal{G}\vee
e_{2})/e_{0}}=\mathfrak{T}_{((\mathcal{G}\vee e_{2})/e_{0})\vee
e_{1}}+yz\,\mathfrak{T}_{((\mathcal{G}\vee e_{2})/e_{0})/e_{1}}\,,$ (A.14)
(A.15)
$\displaystyle\mathfrak{T}_{\mathcal{G}/e_{2}}=\mathfrak{T}_{(\mathcal{G}/e_{2})\vee
e_{1}}+yz\,\mathfrak{T}_{(\mathcal{G}/e_{2})/e_{1}}\,.$ (A.16)

We used in (A.15) the fact that $e_{0}$ is a bridge in $\mathcal{G}\vee e_{2}$
and that $e_{1}$ is a 2–inner self-loop in $(\mathcal{G}\vee e_{2})/e_{0}$.
Meanwhile, in (A.16), we used the fact that $e_{1}$ is a 2–inner self-loop in
$\mathcal{G}/e_{2}$.

A straightforward calculation yields

$\displaystyle\mathfrak{T}_{((\mathcal{G}\vee e_{2})/e_{0})\vee
e_{1}}=z^{16}sw^{7}q^{15}t^{10}\,,\qquad\mathfrak{T}_{((\mathcal{G}\vee
e_{2})/e_{0})/e_{1}}=z^{14}sw^{6}q^{12}t^{8}\,,$ (A.17)
$\displaystyle\mathfrak{T}_{(\mathcal{G}/e_{2})\vee
e_{1}}=z^{16}sw^{7}q^{15}t^{10}+yz^{15}sw^{6}q^{12}t^{8}\,,\qquad\mathfrak{T}_{(\mathcal{G}/e_{2})/e_{1}}=z^{14}sw^{6}q^{12}t^{8}+yz^{13}sw^{5}q^{9}t^{6}\,.$
(A.18)

Plugging these results on (A.15) and (A.16) and summing these contributions in
(A.13), we recover (A.12).

## References

  * [1] R. C. Avohou, J. Ben Geloun and E. R. Livine, “On terminal forms for topological polynomials for ribbon graphs: The $N$-petal flower,” arXiv: 1212.5961[math.CO].
  * [2] J. Ben Geloun, T. Krajewski, J. Magnen and V. Rivasseau, “Linearized Group Field Theory and Power Counting Theorems,” Class. Quant. Grav. 27, 155012 (2010) [arXiv:1002.3592 [hep-th]].
  * [3] J. Ben Geloun, J. Magnen and V. Rivasseau, “Bosonic Colored Group Field Theory,” Eur. Phys. J. C 70, 1119 (2010) [arXiv:0911.1719 [hep-th]].
  * [4] J. Ben Geloun and V. Rivasseau, “A Renormalizable 4-Dimensional Tensor Field Theory,” arXiv:1111.4997 [hep-th].
  * [5] B. Bollobas, “Modern Graph Theory” (Springer, NY, 1998).
  * [6] B. Bollobas and O. Riordan, “Polynomial of graphs on surfaces,” Math. Ann. 323, 81–96 (2002).
  * [7] B. Bollobas and O. Riordan, “A polynomial invariant of graphs on orientable surfaces,” Proc. London Math. Soc. 83, 513–531 (2001).
  * [8] F. Caravelli, “A Simple Proof of Orientability in Colored Group Field Theory,” SpringerPlus 1, 6 (2012) [arXiv:1012.4087 [math-ph]].
  * [9] S. Chmutov, “Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial,” to appear in J. Combin. Theory Ser. B (2009), arXiv:0711.3490v3 [math.CO].
  * [10] R. Gurau, “Colored Group Field Theory,” Commun. Math. Phys. 304, 69 (2011) [arXiv:0907.2582 [hep-th]].
  * [11] R. Gurau, “Lost in Translation: Topological Singularities in Group Field Theory,” Class. Quant. Grav. 27, 235023 (2010) [arXiv:1006.0714 [hep-th]].
  * [12] R. Gurau, “Topological Graph Polynomials in Colored Group Field Theory,” Annales Henri Poincare 11, 565 (2010) [arXiv:0911.1945 [hep-th]].
  * [13] R. Gurau and J. P. Ryan, “Colored Tensor Models - a review,” SIGMA 8, 020 (2012) [arXiv:1109.4812 [hep-th]].
  * [14] R. Gurau, “The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders,” Nucl. Phys. B 865, 133 (2012) [ar(x-1)iv:1203.4965 [hep-th]].
  * [15] T. Krajewski, V. Rivasseau and F. Vignes-Tourneret, “Topological graph polynomials and quantum field theory. Part II. Mehler kernel theories,” Annales Henri Poincare 12, 483 (2011) [arXiv:0912.5438 [math-ph]].
  * [16] T. Krajewski, V. Rivasseau, A. Tanasa and Zhituo Wang, “Topological Graph Polynomials and Quantum Field Theory. Part I. Heat Kernel Theories,” annales Henri Poincare ${\bf 12}$, 483 (2011) [arXiv:0811.0186v1 [math-ph]].
  * [17] A. Sokal, “The multivariate Tutte polynomial (alias Potts model) for graphs and matroids,” Surveys in combinatorics 2005, 173-226, London Math. Soc. Lecture Note Ser., 327, (Cambridge Univ. Pr., Cambridge ,2005) arXiv:math/0503607; J. Ellis-Monaghan and C. Merino, “Graph polynomials and their applications I: The Tutte polynomial,” arXiv:0803.3079 [math.CO].
  * [18] A. Tanasa, “Generalization of the Bollobas-Riordan polynomial for tensor graphs,” J. Math. Phys. 52, 073514 (2011) [arXiv:1012.1798 [math.CO]].
  * [19] W. T. Tutte, “Graph theory”, vol. 21 of Encyclopedia of Mathematics and its Applications (Addison-Wesley, Massachusetts, 1984).