# Fundamental Heaps for Surface Ribbons and Cocycle Invariants

Masahico Saito  Department of Mathematics, University of South Florida, Tampa,
FL 33620, U.S.A<EMAIL_ADDRESS>and  Emanuele Zappala  Institute of
Mathematics and Statistics, University of Tartu
Narva mnt 18, 51009 Tartu, Estonia<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of fundamental heap for compact orientable surfaces
with boundary embedded in $3$-space, which is an isotopy invariant of the
embedding. It is a group, endowed with a ternary heap operation, defined using
diagrams of surfaces in a form of thickened trivalent graphs called surface
ribbons. We prove that the fundamental heap has a free part whose rank is
given by the number of connected components of the surface. We study the
behavior of the invariant under boundary connected sum, as well as
addition/deletion of twisted bands, and provide formulas relating the number
of generators of the fundamental heap to the Euler characteristics. We
describe in detail the effect of stabilization on the fundamental heap, and
determine that for each given finitely presented group there exists a surface
ribbon whose fundamental heap is isomorphic to it, up to extra free factors. A
relation between the fundamental heap and the Wirtinger presentation is also
described. Moreover, we introduce cocycle invariants for surface ribbons using
the notion of mutually distributive cohomology and heap colorings. Explicit
computations of fundamental heap and cocycle invariants are presented.

###### Contents

  1. 1 Introduction
  2. 2  Preliminaries
    1. 2.1 Diagrams of surface ribbons and their moves
    2. 2.2 Heaps
    3. 2.3 Ternary self-distributive homology
  3. 3 The fundamental heap of surface ribbons
    1. 3.1 Definitions and examples
    2. 3.2 Adding a twisted band and realizations of the fundamental heap
    3. 3.3 A relation to the Wirtinger presentation
    4. 3.4 Effect under stabilization
    5. 3.5 Realization problem for the fundamental heap of surface ribbons
  4. 4 Reversibility and additivity conditions
    1. 4.1 Reversibility and additivity for TSD operations
    2. 4.2 Reversibility and additivity for TSD 2-cocycles
    3. 4.3 Mutually distributive RA cocycles
  5. 5 Colorings and cocycle invariants of ribbon graphs
    1. 5.1 Colorings
    2. 5.2 Cocycle invariants with respect to the boundary curves

##  1\. Introduction

The purpose of this article is to introduce, and investigate, invariants of
compact orientable surfaces with boundary embedded in $3$-space up to isotopy,
using ternary self-distributive (TSD) operations and their cohomology theory.
More specifically, we focus on heaps, ternary structures that are epitomized
by the operation $(x,y,z)\mapsto xy^{-1}z$ in a group $G$, and the notion of
mutually distributive cocycles of TSD operations, applied to heaps. Any
compact orientable surface embedded in 3-space can be represented by a thin
ribbon neighborhood of a trivalent graph that is its spine, which we call a
surface ribbon. Their diagrams and Reidemeister type moves were studied in
[Matsu]. We utilize this diagrammatic.

Self-distributive (binary) operations have been used since the 1980’s to
construct invariants of knots and links, following the articles [Joyce, Mat],
where the notion of fundamental quandle was introduced, defined topologically
and diagrammatically. Homology and cohomology theories of quandles were then
introduced, and used to construct invariants of links in $3$-space, as well as
knotted surfaces in $4$-space [CJKLS]. These invariants are defined via
certain partition functions, roughly described in the case of links in
$3$-space as follows. The initial data of the construction is a quandle $X$,
along with a $2$-cocycle of $X$ with coefficients in an abelian group $A$.
First, one defines the set of $X$-colorings of a fixed diagram $D$ of a link
$L$ as the set of homomorphisms from the fundamental quandle of $L$ (obtained
through $D$) to $X$. A coloring is also regarded as an assignment of elements
of $X$ to arcs of $D$, and assigned elements are called colors. For each
coloring, then, one takes the Boltzmann weights of each crossing of $D$, where
the $2$-cocycle is evaluated at the pair of colors of the underpassing and
overpassing arcs, then for each coloring, all these weights are multiplied
together over all crossings. Upon summing over all $X$-colorings this quantity
results to be invariant with respect to Reidemeister moves and, therefore, is
independent of the choice of diagram of $D$.

$x$$x*y$$y$$x$$y$$x*y$ Figure 1. Positive (left) and negative (right)
crossings and their colorings for binary quandles.

The intuitive and diagrammatic interpretation that underlies the above
described paradigm relies on the scheme depicted in Figure 1, where the
orientation is used to determine the sign of the crossing and the consequent
sign of the Boltzmann weight. In the figure, a coloring rule is given at a
crossing, where two colors $x,y$ determine the third color to be $x*y$, where
$*$ denotes the quandle operation. Using TSD operations, a similar
diagrammatic interpretation is introduced, where the arcs are doubled and the
colors change at given crossings by means of the TSD operation, following the
rules depicted in Figure 2 (A), where $z=xu^{-1}v$ and $w=yu^{-1}v$. The
colors of the underpassing arcs change from $x$, resp. $y$, to $z=T(x,u,v)$,
resp. $w=T(y,u,v)$, where $T$ is a given TSD operation. Among known examples
of TSD operations we have compositions of binary self-distributive operations
of mutually (binary) self-distributive operations [ESZcascade], as well as
heaps, which are not compositions of lower arity operations. The ribbon
cocycle invariant is a framed link invariant constructed in [SZframedlinks],
using the diagrammatic interpretation of heaps given above. Concisely, in
order to define the ribbon cocycle invariant the notion of fundamental heap is
introduced, and consequently, that of heap coloring of a framed diagram of a
framed link as well. For each heap coloring with a given heap $X$ the
Boltzmann weight is assigned at each crossing from two terms associated at
each instance of $T$ as described above, and the weights derived from
evaluating a fixed TSD $2$-cocycle are then multiplied over all crossings.
Summing over all colorings by $X$ of the given framed link diagram one obtains
an object that is invariant under framed Reidemeister moves. Interestingly,
the fact that the Boltzmann weights are defined independently on each of the
two underpassing arcs at a crossing, by means of the ternary operation,
induces an invariant that is an element of the tensor product of algebras
$\mathbb{Z}[A]\otimes\mathbb{Z}[A]$, where $A$ is the abelian coefficient ring
used for cohomology. This is a fundamental difference between the binary and
the ternary approaches.

Figure 2. Building blocks

In the present article, we employ the same principles used to obtain the
fundamental heap and the ribbon cocycle invariant for framed links
[SZframedlinks] to define those for surface ribbons. Differences from earlier
work [SZframedlinks] are described as follows. These objects are proved to be
isotopy invariants using local moves that consist of framed Reidemeister
moves, as well as moves that involve trivalent fat vertices of surface ribbons
[Matsu], which present the main difference with respect to the case of framed
links. In addition, since the boundary components of surface ribbons need to
be oriented in antiparallel fashion, a further difference appears between the
cocycle invariants of framed links and those of surface ribbons. Lastly,
whilst the ribbon cocycle invariant for framed links employs by definition a
single heap $2$-cocycle with coefficients in some abelian group $A$, the
initial data to construct cocycle invariants of surface ribbons is a family of
mutually distributive $2$-cocycles, in the sense of [ESZcascade], assigned to
each connected component of a surface ribbon. Then, as in the case of framed
links, the invariant takes values in a tensor product of copies of the group
algebra of $A$, where tensor factors correspond to boundary connected
components. Thus the invariant is stratified to connected components of both
surface ribbons and their boundary curves.

Related topics can be found, for example, in the following papers. Spatial
graphs with a move that corresponds to handle slides have been studied also
for handlebody-links [Ishii]. Corresponding algebraic structures that have
multiplication and braiding at the same time, with compatibility conditions,
have also been studied [CIST, Lebed]. Invariants for compact surfaces with
boundary represented by ribbon graphs using the moves provided in [Matsu] were
defined and studied in [IMM, SZbraidedFrob]. Knot invariants using ternary
operations have been studied, for example, in [NOO, Nie1, Nie2], in which
colorings are assigned to complementary regions, while in this paper,
colorings are assigned on doubled arcs of surface ribbons.

Figure 3. Heap and Reidemeister type III move

We now present more details regarding the constructions of invariants. The
fundamental heap of surface ribbons is defined from a given diagram $D$ of $S$
as follows. We first introduce a generator for each arc appearing in $D$.
Next, using the coloring condition as explained above, one introduces
relations. The fundamental heap is the group defined by a presentation with
these generators and relations. Its isomorphism class is invariant under the
moves of Figure 4 given by [Matsu], and it is therefore independent of the
choice of $D$, and is an isotopy invariant. We present several properties of
the fundamental heap. For instance, we prove that it is not changed by
addition/deletion of twisted bands, with the application of implying certain
inequalities between the genus of the ribbon surface and the number of
generators of the fundamental heap. Moreover, we prove that any surface ribbon
can be turned, by means of stabilizations, into a new surface ribbon whose
fundamental heap is free. We also find a solution to the realization problem
for heaps as fundamental heaps of surface ribbons. Specifically, given a
finitely presented group $X$, there exists a surface ribbon whose fundamental
heap is isomorphic to the group heap of $X$, up to some free factor. The rank
of the free factor is related to the Euler characteristic and the number of
connected components.

The framed Reidemeister move III with the antiparallel convention for boundary
components, and heap colorings, is given in Figure 3. The figure also
indicates that different ribbons (belonging to different connected components)
are decorated with possibly different $2$-cocycles, with the fundamental
assumptions that all pairs of cocycles are mutually distributive. Then, the
coloring condition at a framed Reidemeister move III is guaranteed by the
self-distributivity of heap operations, while the invariance of weights is
equivalent to mutual distributivity of the cocycles. Moreover, the presence of
trivalent fat vertices also requires extra conditions on the labeled
cohomology of [ESZcascade]. These conditions, hereby called reversibility and
additivity, ensure that ribbons can be slid above and below fat vertices,
which is one of the moves that determine the isotopy class of the embedding
[Matsu].

The article is organized as follows. In Section 2 we recall some algebraic and
topological preliminaries. More specifically, we provide the definition of
surface ribbons and their diagrams, and we recall the diagrammatic moves
[Matsu] that determine their isotopy class. Then we introduce heaps, and
recall ternary self-distributive (co)homology and labeled cohomology in the
specific case of mutually distributive $2$-cocycles. In Section 3 we introduce
the fundamental heap of surface ribbons. Then we study the properties of
fundamental heaps under stabilization, twisted band addition/deletion, and
boundary connected sum. We investigate its relation to Euler characteristic
and genus of surface ribbons, and provide a positive answer to the realization
problem of heaps as fundamental heaps of surface ribbons. A connection with
the Wirtinger presentation is also provided, as well as some classes of
examples. Section 4 is devoted to the definition of a subgroup of the mutually
distributive second cohomology group of heaps, determined by two additional
conditions. Families of examples of cocycles satisfying the extra conditions
are also provided. In Section 5 we introduce colorings of surface ribbons by
heaps, and use this notion along with the cohomology of Section 4 to construct
cocycle invariants of surface ribbons. We provide nontrivial examples and
discuss a formula for the cocycle invariants of boundary connected sums.

##  2\.  Preliminaries

In this section we review materials used in this paper.

###  2.1. Diagrams of surface ribbons and their moves

In this section we review diagrams representing compact orientable surfaces
with boundary embedded in 3-space (spatial surfaces with boundary). Our
discussion is based on [Matsu]. By compact surfaces with boundary, we mean
surfaces that are compact and such that each component has a non-empy
boundary. Compact surfaces with boundaries embedded in $3$-space are
determined by their spines. Recall that a spine for a surface $S$ is a
trivalent graph $G$ such that a normal neighborhood of $G$ in $S$ is
homeomorphic to $S$ with a normal neighborhood of $\partial S$ removed. We
therefore represent compact surfaces with boundary, diagrammatically, as
fattened trivalent graphs where each edge is given by a pair of parallel arcs,
while vertices are represented by triples of arcs as in Figure 2 (C). We call
such representations surface ribbons throughout the paper. Thus a surface
ribbon is a compact orientable surface with boundary in the form of a
thickened flat trivalent graph. The fundamental diagrammatic units are given
in Figure 2, where in (A) a fattened crossing is represented. For simplicity
we also represent surface ribbons by trivalent graphs as in (B) and (D) in the
figure.

Figure 4. Moves

In [Matsu], it was shown that the isotopy class of a compact orientable
surface with boundary in a surface ribbon form is determined diagrammatically
by the moves given in Figure 4. Moves RII, RIII and CL are the framed
Reidemeister moves for framed links. Moves IY, YI and IH appear also in the
study of handlebody knots in $3$-space, see for instance [Ishii]. In
particular, we mention that the IH move is important in the well posedness of
the diagrammatic interpretation in terms of trivalent graphs (spines), since
it allows to arbitrarily desingularize higher order vertices. Matsuzaki has
determined the moves for non-oriented surfaces as well [Matsu], although we do
not consider this case here. The main difference with the present case is that
a half-twist is specified in trivalent graphs, and further moves involving
half-twists are considered as well. In the context of orientable surfaces no
half-twist needs to be taken into account, as half twists appear in even
numbers for orientable surfaces, and two half-twists are represented by a
small loop as in Figure 5.

Figure 5. A loop corresponds to a full twist

###  2.2. Heaps

In this section we recall the definition and basic properties of heaps. Given
a set $X$ with a ternary operation $[-]$, the set of equalities

$[[x_{1},x_{2},x_{3}],x_{4},x_{5}]=[x_{1},[x_{4},x_{3},x_{2}],x_{5}]=[x_{1},x_{2},[x_{3},x_{4},x_{5}]]$

is called para-associativity. The equations $[x,x,y]=y$ and $[x,y,y]=x$ are
called the degeneracy conditions. A heap is a non-empty set with a ternary
operation satisfying the para-associativity and the degeneracy conditions
[ESZheap].

A typical example of a heap is a group $G$ where the ternary operation is
given by $[x,y,z]=xy^{-1}z$, which we call a group heap. If $G$ is abelian, we
call it an abelian (group) heap. Conversely, given a heap $X$ with a fixed
element $e$, one defines a binary operation on $X$ by $x*y=[x,e,y]$ which
makes $(X,*)$ into a group with $e$ as the identity, and the inverse of $x$ is
$[e,x,e]$ for any $x\in X$. Moreover, the associated group heap coincides with
the initial heap structure. Focusing on group heaps is therefore not a strong
restriction, as it is always possible to construct a group whose group heap
coincides with an arbitrary heap.

Let $X$ be a set with a ternary operation $(x,y,z)\mapsto T(x,y,z)$. The
condition $T((x,y,z),u,v)=T(T(x,u,v),T(y,u,v)T(z,u,v))$ for all $x,y,z,u,v\in
X$, is called ternary self-distributivity, TSD for short. It is known and
easily checked that the heap operation $(x,y,z)\mapsto[x,y,z]=T(x,y,z)$ is
ternary self-distributive. In this paper we focus on the TSD structures of
group heaps.

###  2.3. Ternary self-distributive homology

The ternary self-distributive (co)homology, which we review, was studied in
[EGM, Green, ESZcascade]. Let $X$ be a ternary self-distributive set. The
$n$-dimensional chain group $C_{n}^{\rm SD}(X)$ is the free abelian group
generated by $(2n-1)$-tuples $(x_{1},x_{2},\ldots,x_{2n-1})$. The boundary
operator $d_{n}:C_{n}^{\rm SD}(X)\rightarrow C_{n-1}^{\rm SD}(X)$ is defined
by

$\displaystyle d_{n}(x_{1},x_{2},\ldots,x_{2n-1})=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i}[\
(x_{1},\ldots,\widehat{x_{2i},x_{2i+1}},\ldots,x_{2n-1})$ $\displaystyle\hskip
42.67912pt-(x_{1}x_{2i}^{-1}x_{2i+1},\ldots,x_{2i-1}x_{2i}^{-1}x_{2i+1},\widehat{x_{2i},x_{2i+1}},x_{2i+2},\ldots,x_{2n-1})\
].$

Cycle, boundary, homology groups are as usual denoted by $Z_{n}^{\rm SD}(X)$,
$B_{n}^{\rm SD}(X)$, and $H_{n}^{\rm SD}(X)$, respectively. For an abelian
group $A$, one defines the cochain, cocycle, coboundary and cohomology groups
by dualizing their homological counterparts, as usual. A similar notation,
with upper indices, is used to indicate these groups. We adopt the convention
that the cohomology differentials, written as $\delta^{n}:C^{n}_{\rm
SD}(X,A)\longrightarrow C^{n+1}_{\rm SD}(X,A)$, are dual to the homological
differentials $d_{n+1}:C_{n+1}^{\rm SD}(X)\longrightarrow C_{n}^{\rm SD}(X)$.

The 2-cocycle condition in this cohomology is formulated as

$(*)\quad\delta^{2}\psi(x,y,z,u,v)=\psi(x,y,z)-\psi(xu^{-1}v,yu^{-1}v,zu^{-1}v)-\psi(x,u,v)+\psi(xy^{-1}z,u,v)=0,$

where $x,y,z,u,v\in X$.

In addition, we will need the notion of mutually distributive cocycles
[ESZcascade]. Although its definition was given in [ESZcascade] for more
general settings, here we provide the definition for the special case that we
apply in this paper. Let $(X,T)$ denote a TSD structure and let $\psi_{1}$ and
$\psi_{2}$ be TSD $2$-cocycles (as given above) with an abelian coefficient
group $A$. Then, we say that the pair $(\psi_{1},\psi_{2})$ is mutually
distributive if the following two conditions hold

$\displaystyle\psi_{1}(x,y,z)+\psi_{2}(T(x,y,z),u,v)$ $\displaystyle=$
$\displaystyle\psi_{2}(x,u,v)+\psi_{1}(T(x,u,v),T(y,u,v),T(z,u,v)),$
$\displaystyle\psi_{2}(x,y,z)+\psi_{1}(T(x,y,z),u,v)$ $\displaystyle=$
$\displaystyle\psi_{1}(x,u,v)+\psi_{2}(T(x,u,v),T(y,u,v),T(z,u,v)).$

In this situation we also say that $\psi_{1}$ and $\psi_{2}$ are mutually
distributive.

A pair of mutually distributive 2-cocycles $(\psi_{1},\psi_{2})$ is called
coboundary if there exists $f\in C^{1}_{\rm SD}(X,A)$ such that
$\psi_{i}=\delta f$ for $i=1,2$.

##  3\. The fundamental heap of surface ribbons

In this section we define the fundamental heap, we show that it is an
invariant of surface ribbons, and present examples and properties.

###  3.1. Definitions and examples

######  Definition 3.1.

The fundamental heap $h(S)$ of a surface ribbon $S$ is defined as follows. Let
$D$ be a diagram of $S$ with double arcs of ribbons with building blocks as in
Figure 2 (A) at crossings and (C) at trivalent vertices. We define $h(D)$ by a
presentation using $D$ and show that it is well defined, i.e. independent of
choice of $D$. Let ${\cal A}$ be the set of arcs. Two arcs of a ribbon segment
(doubled arcs) are listed as separate (distinct) elements of ${\cal A}$. Each
arc is assigned a generator. In Figure 2, generators are represented by
letters (labels) $x,y,u,v,z,w$. Letters assigned to arcs are identified with
(the names of) the arcs themselves, and regarded as elements of ${\cal A}$.
Then the set of generators of $h(D)$ is ${\cal A}$.

For each crossing as depicted in Figure 2 (A), the relations are given by
$\\{z=xu^{-1}v,w=yu^{-1}v\\}$. Specifically, when the arc $x$ goes under the
arcs $(u,v)$, in this order, to the arc $z$, then the relation is defined as
$z=xu^{-1}v$, and similar from $y$ to $w$. The set of union of the two
relations over all crossings is denoted by ${\cal T}$ and constitutes the set
of relations of $h(D)$. For each trivalent vertex as in Figure 2 (C), each
connected arc receives the same letter, and no relation is imposed.

The fundamental heap $h(D)$ is the group heap of the group whose presentation
is given by a set of generators corresponding to double arcs, and the set of
relations assigned to all crossings: $\langle\,{\cal A}\ |\ {\cal
T}\,\rangle$. In the next lemma, it is proved that $h(D)$ does not depend on
the choice of $D$ and, therefore that it is well defined for $S$, and it is
denoted by $h(S)$. For a connected disk $B^{2}$, it is defined as
$h(B^{2})={\mathbb{Z}}$.

For diagrams of spine consisting of single arcs as in Figure 2 (B) and (D),
the letters (generators) assigned are placed at the two sides of each arc.

Figure 6. Labeled IH move Figure 7. YI and IY moves

######  Lemma 3.2.

The fundamental heap $h(S)$ is well defined, that is, the isomorphism class of
the group heap $h(D)$ is independent on the choice of $D$.

###### Proof.

Applying the moves for the spines of surface ribbons in [Matsu], we check the
invariance under the moves listed in Figure 4. Of these, Reidemeister moves
RII and RIII, as well as the CL move, are checked in [SZframedlinks]. A
diagram for checking RIII is depicted in Figure 3. In this figure, for
example, using relations at crossings, the generator assigned on the bottom
right string, is expressed in terms of $x,y,z,u,v$ as $xy^{-1}zu^{-1}v$ and
$(xu^{-1}v)(yu^{-1}v)^{-1}(zu^{-1}v)$, in the left and right figures,
respectively, and they coincide.

The IH move does not involve any change of relations, and the presentation
does not change. The way the connected components of arcs receive consistently
the same letters is depicted in Figure 6. The remaining moves, YI and IY, are
also checked diagrammatically as depicted in Figure 7, as desired. We note
that if we write $T(x,y,z)=xy^{-1}z$ as a ternary operation, then the YI move
requires $T(T(u,x,y),y,z)=T(u,x,z)$ (left of Figure 7), and the IY move
requires $T(y,u,v),v,u)=y^{\prime}=y$ (right of Figure 7), both of which hold
for the group heap operation. These properties of heaps are also used in the
proof of Lemma 5.2, accordingly. ∎

Recall that a group $G$ is said to be finitely presented if there exists a
presentation of $G$ with a finite number of generators and a finite number of
relators. From the definition, we have that the fundamental heap of a surface
ribbon is the group heap of a finitely presented group. From the definition we
have the following as well.

######  Lemma 3.3.

Let $S_{1},S_{2},\ldots,S_{\nu}$ be surface ribbons with fundamental heaps
$h(S_{i})$, $i=1,\ldots,n$. Then the surface ribbon $S=\sqcup_{i}S_{i}$, a
split sum (disjoint union), has fundamental heap $h(S)\cong
h(S_{1})*h(S_{2})*\cdots*h(S_{\nu})$.

The following is a generalization of the corresponding result in
[SZframedlinks], which was proved for framed links.

######  Theorem 3.4.

Let $S=S_{1}\cup\cdots\cup S_{\nu}$ be a surface ribbon written as the union
of connected components. Then $h(S)\cong F_{\nu}*\hat{h}(S)$ for some group
$\hat{h}(S)$, where $F_{\nu}$ denotes the free group of rank $\nu$.

###### Proof.

Let $x,y$ be a pair of generators assigned to a single ribbon $R$. We call the
elements $x^{-1}y$ and $y^{-1}x$ ribbon terms, and a word in ribbon terms a
ribbon word.

Figure 8. Base of surface ribbons

First assume that $S$ is connected. Any connected surface ribbon can be put in
a “standard form” as depicted in Figure 8, where dotted arcs are ribbons
connecting them, and they may be knotted and linked with each other. In Figure
8, the left portion indicates a pair of crossed handles (ribbons) that
contribute to the genus by one, and the right portion indicates a single
trivial handle that contributes to the number of connected components by one.
Although in general there are multiple numbers of these, we make computations
for the surface having one for each type, as the general case is similar.
Assigned letters are indicated in the figure, where greek letters indicate
ribbon words.

The presentation of $h(S)$ with arcs as generators and relations at crossings
as defined in Definition 3.1 is modified as follows, that represent an
isomorphic group. For this proof, we choose a set of generators to be the
(letters assigned to) arcs ${\mathcal{A}}_{0}$ of ribbons connected to the
base separately, in addition to arcs away from the base, and a set of relators
${\mathcal{T}}_{0}$ to be equalities when two feet of ribbons are adjacent at
the base, in addition to relations derived from crossings as in Definition
3.1. For example, the arc with letter $y$ at the left most ribbon foot in
Figure 8 is adjacent to the arc with letter $u$ at the second foot, so that a
relation $y=u$ is part of ${\mathcal{T}}_{0}$. From Figure 8, one obtains a
set of relations: (1) $y=u$, (2) $v=y\Theta u^{-1}v$, (3) $x\Theta
u^{-1}v=v\Omega$, (4) $u\Omega=s$, (5) $t=t\Lambda$, and (6) $s\Lambda=x$ from
connected segments. Thus these relations are contained in ${\mathcal{T}}_{0}$.
This is a starting presentation
$\langle\,{\mathcal{A}}_{0}\mid{\mathcal{T}}_{0}\,\rangle$.

If a pair of letters $x,y$ are assigned to the boundary arcs of a ribbon, then
we add a new generator $\alpha$ that is a ribbon term, and add a relation
$\alpha=x^{-1}y$ that is a defining relation of the ribbon term. Perform this
process to obtain a new presentation
$\langle\,{\mathcal{A}}_{1}\mid{\mathcal{T}}_{1}\,\rangle$.

When one traces the boundary curve labeled $x$ along the dotted line and
encounter the first crossing as in Figure 2 (A), together with the generator
$z$ assigned on the arc on the other side of a pair of arcs labeled $u$ and
$v$, we have a relation $z=xu^{-1}v=x\beta$, where $\beta=u^{-1}v$ is a ribbon
term. By this relation, $z$ is eliminated from the set of generators and
replaced by $x\beta$, and in all relations having $z$ in them are replaced by
those with $z$ substituted by $x\beta$. By this process, the generator $z$ is
replaces by $x\beta$. Similarly, $w$ is replaced by $y\beta$.

Continuing this process, when the arc labeled $x$ reaches to the base of the
surface as in Figure 8 at the arc third from the left, it is labeled by
$x\Theta u^{-1}v$, where $\Theta$ is a ribbon word and $u^{-1}v$ is a ribbon
term $\beta$. The other arcs are similarly labeled in the figure. The other
relations at the base are changed as follows: (1) $u=x\alpha$, (2) $v=y\Theta
u^{-1}v=x\alpha\Theta\beta$, (3)
$(v=)\Theta\beta\Omega^{-1}=\alpha\Theta\beta$, (4) replaces
$s=x\alpha\Omega$, (5) $\Lambda=1$ for ribbon word $\Lambda$, and $t$ is
replaced using a ribbon term $\gamma=s^{-1}t$ to be
$t=s\gamma=x\alpha\Omega\gamma$, and (6) $s=x\Lambda^{-1}$. Thus we obtain a
group presentation with all the generators original assigned to arcs are
expressed by $x\Phi$ for some ribbon word $\Phi$.

In summary the original presentation of $h(S)$, $\langle\,{\cal A}\ |\ {\cal
T}\,\rangle$ as in Definition 3.1 is replaced by a new resulting presentation
$\langle\,x,{\mathcal{B}}\mid{\mathcal{R}}\,\rangle$, where $x$ is a free
generator, ${\mathcal{B}}$ is the set of generating ribbon terms corresponding
to ribbon segments, and ${\mathcal{R}}$ is a set of relations among ribbon
words. Hence $h(S)$ is written as $\langle\,x\ \rangle*\langle\
{\mathcal{B}}\mid{\mathcal{R}}\ \rangle={\mathbb{Z}}*\hat{h}(S)$ where
$\hat{h}(S)=\langle\ {\mathcal{B}}\mid{\mathcal{R}}\,\rangle$. The argument is
repeated to higher genus and with more than one trivial bands. The argument is
also repeated, with a single free generator for each connected component
$S_{i}$, $i=1,\ldots,\nu$, with one free generator for each component. ∎

######  Definition 3.5.

By uniqueness of free product of groups [Scott-Wall], the group $\hat{h}(S)$
in Theorem 3.4 is well defined up to isomorphism. We call the group heap of
$\hat{h}(S)$ the reduced fundamental heap of $S$.

######  Definition 3.6.

For a surface ribbon $S$, the maximum rank $s$ of the free group factor
$h(S)\cong F_{s}*G$ of the fundamental heap is called the rank of $h(S)$, or
simply, of $S$, and denoted by ${\rm rank}(S)$.

######  Remark 3.7.

We observe that the rank of a surface ribbon is well defined, as a consequence
of the uniqueness of free product factorization of groups [Scott-Wall].

######  Definition 3.8.

Denote by $\mu(G)$ the minimum number of generators of a finitely generated
group $G$. For a surface ribbon $S$ let $\mu(S)$ denote $\mu(h(S))$.

From Theorem 3.4, we have ${\rm rank}(S)\geq\nu$, where $\nu$ denotes the
number of connected components of $S$. In general the inequality is strict, as
we see below.

Figure 9. Adding trivial bands

######  Example 3.9.

A trivial single band $B_{1}$ and a trivial crossed band pair $B_{2}$ are
depicted in (A) and (B) in Figure 9, respectively. If the two end points are
closed by trivial arcs, then both result in the ribbon surface with the
fundamental heap isomorphic to $F_{2}=\langle\,x,y\,\rangle$, the free group
of rank two, as seen from the figure. Let $S=(B_{1})^{m}(B_{2})^{n}$ denote
the closure of concatenation of $m$ trivial bands and $n$ pairs of crossed
band pairs. Then we have $h(S)\cong F_{m+n+1}$, the free group of rank
$m+n+1$.

Figure 10. Ribbon with loops

######  Example 3.10.

A band with $m$ loops is depicted in Figure 10. At the end points, $x$ and $y$
are assigned as generators. We set the ribbon element to be $\alpha=x^{-1}y$.
Inductively at the right end of the arcs receive the labels $x\alpha^{m}$ and
$y\alpha^{m}$ as depicted. This computation was done in [SZframedlinks]. As
depicted, we obtain a relation $\alpha^{m}=1$, and the bottom end receives the
label $y$, which coincides with the top label. If we close the end points, we
obtain an annulus with the fundamental heap isomorphic to
$\langle\,x,\alpha\mid\alpha^{m}\,\rangle\cong F_{1}*{\mathbb{Z}}_{m}$. If we
concatenate copies of the band with $m_{i}$ loops vertically, $i=1,\ldots,n$,
and close the end points, then we obtain an $n$ punctured disk, denoted
$D(m_{1},\ldots,m_{n})$, such that $h(D(m_{1},\ldots,m_{n}))\cong
F_{1}*{\mathbb{Z}}_{m_{1}}*\cdots*{\mathbb{Z}}_{m_{n}}$. Furthermore, if we
concatenate $k-1$ copies of the trivial band in Figure 9 (A), then we obtain
an $n+k$ punctured disk $D(m_{1},\ldots,m_{n};k)$ such that
$h(D(m_{1},\ldots,m_{n};k))\cong
F_{k+1}*{\mathbb{Z}}_{m_{1}}*\cdots*{\mathbb{Z}}_{m_{n}}$. Variations of this
construction are found below in Example 3.21.

Let ${\rm Ab}[G]$ denote the abelianization of a group $G$. By applying Lemma
3.3 to the abelianization of the fundamental heaps in Example 3.9 and Example
3.10, we obtain the following.

######  Proposition 3.11.

For any finitely generated abelian group $A$, there exists a connected surface
ribbon $S$ such that ${\rm Ab}[h(S)]\cong A$.

Figure 11. Punctured torus

######  Example 3.12.

Let $T_{1}(k)$ denote the surface ribbon obtained by braiding two ribbons
$(2k+1)$-times and closing them to give braided handles of a surface. This
surface is shown in Figure 11, where top and bottom arcs are joined to give a
punctured torus. For the labels $x,y,u,v$ assigned to the arcs at the right of
the figure, the middle two bands at the left of the figure have arcs with the
label

$(u\beta^{-k}(\alpha\beta)^{k},v\beta^{-k}(\alpha\beta)^{k})\times(x\alpha^{-(k+1)}(\alpha\beta)^{k+1},y\alpha^{-(k+1)}(\alpha\beta)^{k+1})$

as indicated in the figure. The fundamental heap with the open end points
labeled $y$ and $u$ is generated by $x,y,u,v$ with relations (1)
$x=u\beta^{-k}(\alpha\beta)^{k}$, (2)
$v\beta^{-k}(\alpha\beta)^{k}=x\alpha^{-(k+1)}(\alpha\beta)^{k+1}$, and (3)
$y\alpha^{-(k+1)}(\alpha\beta)^{k+1}=v$, as read from the figure. If we
connect the arcs labeled by $y$ and $u$, we obtain a punctured torus, and an
additional relation (4) $y=u$ holds.

Substituting (3) into (2) we obtain $(\alpha\beta)^{k+1}=\beta^{k+1}$. Then
(1) implies

$x=u\beta^{-k}(\alpha\beta)^{k}=u\beta^{-k}\cdot\beta^{k+1}(\alpha\beta)^{-1}=u\alpha^{-1}=u(x^{-1}y)^{-1}=uy^{-1}x,$

and we obtain the relation (4) $y=u$. Also (3) implies
$v=y\alpha^{-(k+1)}\cdot\beta^{k+1}=u\alpha^{-(k+1)}\beta^{k+1}$ which implies
$\beta=u^{-1}v=\alpha^{-(k+1)}\beta^{k+1}$, hence $\alpha^{k+1}=\beta^{k}$.
Therefore the punctured torus $T_{1}(k)$ has the fundamental heap
$h(T_{1}(k))\cong\langle\,y,\alpha,\beta\mid\alpha^{k+1}=\beta^{k},(\alpha\beta)^{k+1}=\beta^{k+1}\,\rangle$.

Figure 12. Boundary connected sum

The construction of concatenations in the above examples can be generalized to
boundary connected sum of surfaces, that are commonly used, as follows. Let
$S_{1}$, $S_{2}$ be surface ribbons. Regard them as embedded in disjoint balls
as indicated in Figure 12 (A). Specify a boundary component $b_{i}$ in
$S_{i}$, $i=1,2$, and isotope a small portion of $b_{i}$ out of the boundary
of the ball as indicated in (A). Then connect the small portions by a short
straight band as in (B). The resulting surface $S$ is denoted by
$S_{1}\natural_{(b_{1},b_{2})}S_{2}$, or $S_{1}\natural S_{2}$ for simplicity.
The ambient isotopy type of $S$ depends only on the choice of the boundary
components $b_{i}$.

######  Proposition 3.13.

Let $S_{1}$ and $S_{2}$ be surface ribbons. Then there is an epimorphism

$\Gamma:\hat{h}(S_{1}\natural
S_{2})\longrightarrow\hat{h}(S_{1})*\hat{h}(S_{2}).$

Moreover, $\Gamma$ can be extended to an epimorphism
$\tilde{\Gamma}:h(S_{1}\natural S_{2})\longrightarrow
F_{1}*\hat{h}(S_{1})*\hat{h}(S_{2})$.

###### Proof.

We proceed to give an epimorphism for presentations of the fundamental heaps
of $S_{1}$, $S_{2}$ and $S_{1}\natural S_{2}$, where the generators $x_{1}$
and $x_{2}$ corresponding to arcs of the boundaries $b_{1}$ and $b_{2}$ of
Figure 12 (A), do not appear in the relators. This is possible, as in the
proof of Theorem 3.4, since along each boundary we have a relation of the form
$x_{i}\Phi_{i}=x_{i}$ for some ribbon word $\Phi_{i}$. Let $a_{1}$ and $a_{2}$
be the arcs on $b_{1}$ and $b_{2}$ that are protruding out from $S_{1}$ and
$S_{2}$ as in (A). The presentation of $\hat{h}(S_{1})$ is obtained by moving
counterclockwise along the boundary $b_{1}$, starting from the arc $a_{1}$ and
imposing relators corresponding to the labeling conditions encountered along
$b_{1}$. Finally, one obtains a relator of type $x_{1}\Phi_{1}x_{1}^{-1}$,
corresponding to the equality $x_{1}\Phi_{1}=x_{1}$ at $a_{1}$. Similarly, we
have another relator $x_{2}\Phi_{2}x_{2}^{-1}$ for $\hat{h}(S_{2})$. Clearly
these relators are equivalent to $\Phi_{1}$ and $\Phi_{2}$, respectively.
After performing the boundary connected sum $S_{1}\natural S_{2}$, as in
Figure 12 (B), let $a_{1}^{\prime}$ and $a_{2}^{\prime}$ be upper and lower
boundary arcs in the bands connecting $S_{1}$ and $S_{2}$, respectively, and
let $x_{1}^{\prime}$ and $x_{2}^{\prime}$ be generators assigned to them. We
proceed counterclockwise along the boundary starting from $a_{1}^{\prime}$ and
read off relations at crossings. We meet all the crossings that gave the
presentation of $\hat{h}(S_{1})$ obtaining all the same relators until we
reach the lower portion $a_{2}^{\prime}$ of the band, and obtain
$x_{1}^{\prime}\Phi_{1}=x_{2}^{\prime}$. We proceed counterclockwise along the
boundary $b_{2}$ of $S_{2}$. The label of the arc outcoming from $S_{2}$ on
top of the band connecting $S_{1}$ and $S_{2}$ now is given by
$x_{2}^{\prime}\Phi_{2}=x_{1}^{\prime}$, hence
$x_{1}^{\prime}\Phi_{1}\Phi_{2}=x_{1}^{\prime}$. As a consequence we have a
presentation of $\hat{h}(S_{1}\natural S_{2})$ where the generators are the
union of the generators of $\hat{h}(S_{1})$ and $\hat{h}(S_{2})$, while all
the relators are obtained by the union of all relators of $\hat{h}(S_{1})$ and
$\hat{h}(S_{2})$ but $\Phi_{1}$ and $\Phi_{2}$, which are now combined into a
new relator $\Phi_{1}\Phi_{2}$. Let us denote $y_{i}$, $i=1,\ldots,n$ the
generators of $\hat{h}(S_{1})$, and $z_{j}$, $j=1,\ldots,m$ the generators of
$\hat{h}(S_{2})$. We distinguish the generators and relators of
$\hat{h}(S_{1}\natural S_{2})$ from those of $\hat{h}(S_{1})$ and
$\hat{h}(S_{2})$ by introducing a “tilde” symbol on top. The respective
relators are named $R_{k}$, $\Phi_{1}$ and $Q_{t}$, $\Phi_{2}$. We define the
map $\hat{h}(S_{1}\natural S_{2})\cong\langle\,\tilde{y}_{i},\tilde{z}_{j}\ |\
\tilde{R}_{k},\tilde{Q}_{t},\tilde{\Phi}_{1}\tilde{\Phi}_{2}\,\rangle$ into
$\hat{h}(S_{1})*\hat{h}(S_{2})\cong\langle\,y_{i},z_{j}\ |\
R_{k},Q_{t},\Phi_{1},\Phi_{2}\,\rangle$ by sending $\tilde{y}_{i}$ to $y_{i}$
and $\tilde{z}_{j}$ to $z_{j}$ for all $i$ and $j$. The map is well defined
because all the relators $\tilde{R}_{i}$ and $\tilde{Q}_{t}$ are mapped to
relators without tilde, hence vanish, while $\tilde{\Phi}_{1}\tilde{\Phi}_{2}$
maps to the product $\Phi_{1}\Phi_{2}$ which vanishes as well, since
$\Phi_{1}$ and $\Phi_{2}$ are relators in the free product separately.

The second statement is obtained by mapping the free generator of
$h(S_{1}\natural S_{2})$, obtained from Theorem 3.4, onto the free factor
$F_{1}$. ∎

######  Remark 3.14.

We observe that if $S_{1}$ and $S_{2}$ are surface ribbons such that the
presentations of $\hat{h}(S_{1})$ and $\hat{h}(S_{2})$ admit no nontrivial
relator of type $\Phi_{i}$ in the notation of Proposition 3.13, then the proof
of the proposition gives that $\Gamma$ is an isomorphism of heaps.

######  Example 3.15.

Proposition 3.13 and Remark 3.14 apply directly to the surfaces of Example
3.9, since the surfaces considered can be constructed as boundary connected
sums of surface ribbons whose (reduced) fundamental heaps are free.

######  Example 3.16.

Let $D_{i}:=D(m_{i})$, for $i=1,\ldots,n$ denote a family of $n$ surface
ribbons as in Example 3.10, obtained by closing the ends of the diagram in
Figure 10. Then the presentation of each $\hat{h}(D_{i})$ does not contain
relators of type $\Phi_{i}$ in the notation of Proposition 3.13, as the
computation in Example 3.10 shows. In fact, observe that the labels of the
arcs at the top and bottom of Figure 10 coincide, once the relator
$\alpha^{m_{i}}=1$ is imposed. Then Proposition 3.13 and Remark 3.14, together
with a simple inductive argument, imply that $h(D(m_{1},\ldots,m_{n}))\cong
F_{1}*\hat{h}(D_{1})*\cdots*\hat{h}(D_{n})$. The computations in Example 3.10
show directly that $\hat{h}(D_{i})={\mathbb{Z}}_{m_{i}}$. This gives the
fundamental heap of $h(D(m_{1},\ldots,m_{n}))$ as expected.

######  Example 3.17.

We let $T_{g}(k_{1},\ldots,k_{g})$ denote the boundary connected sum of $g$
surfaces $T_{1}(k_{i})$ in Example 3.12, where the boundaries used for the
connected sum are chosen to be the base of each copy $T_{1}(k_{i})$ (in
standard form, the left most arcs in Figure 11). Then, the fundamental heap of
$T_{g}(k_{1},\ldots,k_{g})$ is
$h(T_{g}(k_{1},\ldots,k_{g}))=*_{i=1}^{g}h(T_{1}(k_{i}))$, obtained from the
computation for the torus in Example 3.12 by applying Proposition 3.13 and
Remark 3.14, since the relator coming out of connecting arcs at the left of
the figure for each $T_{1}(k_{i})$ is trivial.

In the following example we see that Proposition 3.13 can be used to determine
the minimum number of generators of the fundamental heap.

######  Example 3.18.

Grusko’s Theorem implies that $\mu(G_{1}*G_{2})=\mu(G_{1})+\mu(G_{2})$, see
[Scott-Wall]. From the epimorphism of Proposition 3.13 we see that
$\mu(S_{1}\natural S_{2})+1\geq\mu(S_{1})+\mu(S_{2})$.

In particular, for those surfaces for which $\tilde{\Gamma}$ is an
isomorphism, we have an equality $\mu(S_{1}\natural
S_{2})+1=\mu(S_{1})+\mu(S_{2})$. This formula can be applied successively for
boundary connected sums of more than two surface ribbons. This is in fact the
case with the surfaces of Examples 3.9, 3.10 and 3.12. We have that
$\mu((B_{1})^{m}(B_{2})^{n})=m+n+1$ and $\mu(D(m_{1},\ldots,m_{n}))=n+2$.
Pertaining to the surface $T_{g}(k_{1},\dots,k_{g})$ of Example 3.12, we
obtain that $\mu(T_{g}(k_{1},\dots,k_{g}))=g+1$ or
$\mu(T_{g}(k_{1},\dots,k_{g}))=2g+1$, depending on whether
$\mu(\langle\,y,\alpha,\beta\mid\alpha^{k+1}=\beta^{k},(\alpha\beta)^{k+1}=\beta^{k+1}\,\rangle)=2,3$.
Set
$H=\langle\,y,\alpha,\beta\mid\alpha^{k+1}=\beta^{k},(\alpha\beta)^{k+1}=\beta^{k+1}\,\rangle$
for simplicity. It is well known that the number of generators of $G/[G,G]$
provides a lower bound to the minimum number of generators of a group $G$.
From the abelianization of $H$, namely
$H/[H,H]={\mathbb{Z}}\oplus{\mathbb{Z}}_{k+1}\oplus{\mathbb{Z}}_{k}$, we see
that $H$ has at least three generators. It follows that
$\mu(T_{g}(k_{1},\dots,k_{g}))=2g+1$. We observe that this is independent of
the crossings $k_{i}$ of each torus surface component of
$T_{g}(k_{1},\dots,k_{g})$.

######  Remark 3.19.

It is not clear whether the epimorphism $\Gamma$ of Proposition 3.13 is also
always monic. This situation affects a formula for the cocycle invariant under
boundary connected sum. If $\Gamma$ is monic, then the extra term in the
cocycle invariant considered in Remark 5.13 would vanish.

###  3.2. Adding a twisted band and realizations of the fundamental heap

In Figure 13, a local operation of a surface ribbon is depicted. On the left,
a single ribbon portion of a surface ribbon $S$ is depicted. A twisted band is
attached to the ribbon as depicted in the figure to obtain a new surface
ribbon $S^{\prime}$. The symbols involving $\psi$ will be used later. We call
this operation an addition of a (positively) twisted band. An addition of a
negatively twisted band is similarly defined with the opposite crossing
information for the added band. Note that the number of connected components
of the surface does not change under this operation, and the number of
connected components of the boundary curves changes by one under this
operation; if a band is attached to distinct boundary components, then the
number reduces by one, and the opposite otherwise.

Figure 13. Adding a twisted band

######  Lemma 3.20.

Let $S^{\prime}$ be a surface ribbon obtained from $S$ by adding a twisted
band. Then we have $h(S^{\prime})\cong h(S)$ and
$\hat{h}(S^{\prime})\cong\hat{h}(S)$.

###### Proof.

As depicted in Figure 13, the generators assigned on the arcs of the added
band are uniquely expressed by $x$ and $y$, the generators of the ribbon of
$S$ where the band is attached, by means of relations at crossings. This shows
that the presentations of the diagrams of $S$ and $S^{\prime}$ define
isomorphic groups. Since the number of connected components of $S$ and
$S^{\prime}$ is the same as noted above, Theorem 3.4 and the uniqueness of
free product of groups [Scott-Wall] imply that
$\hat{h}(S^{\prime})\cong\hat{h}(S)$. ∎

######  Example 3.21.

Let $D(n,k)$ be a disk with $n$ bands with $m_{i}$ loops, $i=1,\ldots,n$, and
$k$ trivial bands, as in Example 3.9. Add $\ell(\leq n+k)$ twisted loops as in
Figure 13 onto $g(>0)$ distinct bands among $n+k$ bands attached to the disk,
to obtain a surface $S(g,n+k)$ of genus $g$ and the number of boundary
component $b:=(n+k+1)-g$. This surface $S(g,n+k)$ has the same fundamental
heap as that of $D(n,k)$ in Example 3.9, but has a non-zero genus $g$.

Proposition 3.11 is restated as follows, using Example 3.21.

######  Proposition 3.22.

Let $A={\mathbb{Z}}^{\oplus(k+1)}\oplus[\oplus_{i=1}^{n}{\mathbb{Z}}_{m_{i}}]$
be a finitely generated abelian group. Then, for any $g,b\in{\mathbb{Z}}_{\geq
0}$ such that g+b = n+k+1, there exists a connected surface ribbon $S(g,b)$ of
genus $g$ and $b$ boundary components such that ${\rm Ab}[(h(S(g,b))]\cong A$.

######  Proposition 3.23.

Let $S$ be a connected surface ribbon with the genus $g(S)$ and the number of
boundary components $b(S)$ such that $b(S)>1$. Then for any integers
$g^{\prime}\geq 0$ and $b^{\prime}>0$ such that $g^{\prime}\geq g(S)$ and
$b^{\prime}\geq b(S)-(g^{\prime}-g(S))$, there exists a surface ribbon
$S^{\prime}$ with ${h}(S^{\prime})\cong{h}(S)$ such that
$g(S^{\prime})=g^{\prime}$ and $b(S^{\prime})=b^{\prime}$.

In particular, for any $S$ with $b(S)>1$ and any $\chi^{\prime}\leq\chi(S)$
which denotes the Euler characteristic, there exists $S^{\prime}$ such that
${h}(S^{\prime})\cong{h}(S)$ and $\chi(S^{\prime})=\chi^{\prime}$.

If $b(S)=1$, the statement holds for any $g^{\prime}\geq g(S)$ and
$b^{\prime}\geq b(S)$, and for any $\chi^{\prime}<\chi(S)$.

The statements hold for $\hat{h}$ as well.

###### Proof.

We show that for any $S$ such that $b(S)-1>0$ there exists $S^{\prime}$ and
$S^{\prime\prime}$ with $h(S^{\prime}),h(S^{\prime\prime})\cong h(S)$ such
that: (i) $g(S^{\prime})=g(S)$ and $b(S^{\prime})=b(S)+1$, and (ii)
$g(S^{\prime\prime})=g(S)+1$ and $b(S^{\prime\prime})=b(S)-1$. For a given
$g^{\prime}$ and $b^{\prime}$ as stated, then, we apply Case (ii) to obtain
$S^{\prime\prime}$ such that $g(S^{\prime\prime})=g^{\prime}$ and
$b(S^{\prime\prime})=b(S)-(g^{\prime}-g(S))$, and apply Case (i) to obtain
$S^{\prime}$ with the desired $g(S^{\prime})$ and $b(S^{\prime})$.

Let $S^{\prime}$ be the surface ribbon obtained from $S$ by adding a twisted
band to the same component of the boundary curve as in Lemma 3.20. Then by the
lemma ${h}(S^{\prime})\cong{h}(S)$ and the condition (i) is satisfied. If
$S^{\prime}$ is obtained by adding a twisted loop to two distinct components,
then (ii) is satisfied.

In Case (i), we have
$\chi(S^{\prime})=2-b(S^{\prime})-2g(S^{\prime})=2-(b(S)+1)-g(S)=\chi(S)-1$,
and in Case (ii), we have
$\chi(S^{\prime})=2-b(S^{\prime})-2g(S^{\prime})=2-(b(S)-1)-2(g(S)+1)=\chi(S)-1$,
so that the statement for $\chi$ holds.

Alternatively, attaching a twisted band corresponds, homotopically, to
attaching a loop, hence it contributes to $-1$ to the Euler characteristic,
and we have $\chi(S^{\prime})=\chi(S)-1$.

If $b(S)=1$, then Case (ii) in the proof cannot be performed. If Case (i) is
performed to $S$, we obtain $S^{\prime}$ with ${h}(S^{\prime})\cong{h}(S)$
such that $g(S^{\prime})=g(S)$ and $b(S^{\prime})=b(S)+1=2$, and
$\chi(S^{\prime})=\chi(S)-1$. If we perform Case (ii) to $S^{\prime}$, we
obtain $S^{\prime\prime}$ with ${h}(S^{\prime\prime})\cong{h}(S)$ such that
$g(S^{\prime\prime})=g(S)+1$, $b(S^{\prime\prime})=b(S^{\prime})-1=1$ and
$\chi(S^{\prime})=\chi(S)-1$. Hence the statements for $b=1$ follow.

The statement for $\hat{h}$ follows from Theorem 3.4 and the uniqueness of the
free product. ∎

######  Proposition 3.24.

Let $S$ be a surface ribbon, then there exists a surface ribbon $S^{*}$ having
maximum Euler characteristic among the surface ribbons with fundamental heap
isomorphic to $h(S)$.

###### Proof.

Let us set $t:={\rm rank}(S)$, as in Definition 3.6. Suppose, for the sake of
contradiction, that such a maximum surface ribbon does not exist. Then, there
exists a sequence $S_{k}$ with the properties that $h(S_{k})\cong h(S)$ and
$\chi(S_{k})>\chi(S_{k-1})$ for $k\in{\mathbb{N}}$. For $k$ large enough, we
have that $\chi(S_{k})>t$. But then $\chi(S_{k})$ is larger than
$\chi(\sqcup_{t}\mathbb{D})$, where $\mathbb{D}$ is the disk. Since
$\sqcup_{t}\mathbb{D}$ is the surface with $t$ connected components with the
largest Euler characteristic, it follows that $S_{k}$ has more than $t$
connected components. Consequently, from Theorem 3.4 it follows that ${\rm
rank}(S_{k})>t$, which is absurd, since $h(S_{k})\cong h(S)$ by assumption,
and the rank of a ribbon surfaces is well defined by uniqueness of free
product decomposition. ∎

######  Remark 3.25.

We say that $S^{\prime}$ is obtained from $S$ by removing a twisted band if
$S$ is obtained from $S^{\prime}$ by adding a twisted band, as described
above. Let $S$ and $S^{*}$ be as in Proposition 3.24. We claim that we cannot
remove a twisted band from $S^{*}$, in the sense that there exists no surface
ribbon $S^{\prime}$ such that $S^{*}$ is obtained from $S^{\prime}$ by adding
a twisted band. In fact, if we could find such a surface ribbon $S^{\prime}$,
then it follows (from proof of Proposition 3.23) that
$\chi(S^{\prime})>\chi(S^{*})$ and $h(S^{\prime})=h(S^{*})$, contradicting
that $\chi(S^{*})$ is maximum.

Figure 14. Possible values of $(g,b)$

We analyze possible values of genus $g$ and the number of boundary components
$b$ for a given $h(S)$. We assume that $S$ is a connected surface ribbon,
since the same argument can be applied to each component for disconnected
surface ribbons. Possible values of $(g(S^{\prime}),b(S^{\prime}))$ with
$h(S^{\prime})\cong h(S)$ are schematically depicted in Figure 14 (A). The
axes represent $g$ and $b$ as indicated. The value $(g(S),b(S))$ of a given
surface ribbon $S$ is indicted by a black dot located at the left top. The
possible range of $(g,b)=(g(S^{\prime}),b(S^{\prime}))$ for some $S^{\prime}$
guaranteed by Proposition 3.23 is depicted by (red) dark shaded region. The
region is bounded from the left by the line $g=g(S)$, and below by the lines
$b-b(S)=-(g-g(S))$ and $b=1$.

In Figure 14 (A), the white circle on the line $b=(2-\chi(S^{*}))-2g$
represents $(g(S^{*}),b(S^{*}))$. The line represents the formula
$\chi=2-b-2g$. Since $\chi(S^{*})$ is maximum, the $y$-intercept
$2-\chi(S^{*})$ is minimum among all lines though
$(g(S^{\prime}),b(S^{\prime}))$ with $h(S^{\prime})\cong h(S)$. Hence all
possible values of $(g,b)$ are bounded on the left by $g=0$, and below by
$b=(2-\chi(S^{*}))-2g$ and $b=1$.

######  Example 3.26.

We consider surface ribbons $S$ with $h(S)\cong F_{k}$, the free group of rank
$k>0$. By Example 3.9, the surface ribbon obtained by boundary connected sum
$S=(B_{1})^{m}(B_{2})^{n}$ of $m$ trivial bands $B_{1}$ and $n$ trivial
crossed band pairs $B_{2}$ have $h(S)\cong F_{m+n+1}$. In Figure 14 (B), the
left top white circle represents $S=(B_{1})^{k-1}$, with $b(S)=k$ and
$g(S)=0$. Inductively, the white dots represent
$(B_{1})^{k-3}(B_{2})^{1},(B_{1})^{k-5}(B_{2})^{2},\ldots$ along the line
$b=k-2g$. The bottom point on the line is $S=(B_{2})^{k/2}$ with
$(g(S),b(S))=((k/2),1)$ if $k$ is even, and is
$S=(B_{1})^{1}(B_{2})^{(k-1)/2}$ with $(g(S),b(S))=((k-1)/2,2)$ if $k$ is odd
(in this case there is no point on the line with $b=1$). The union of the
regions bounded by $g=0$, $b-b(S)=-(g-g(S))$ and $b=1$ described above for
these points $S$ cover the integral points bounded by $g=0$, $b=k-2g$ and
$b=1$ as represented by the green shaded region in Figure 14 (B). The green
region in (A) sweeps out that of (B) over all white points on the line of
slope $-2$.

######  Example 3.27.

Next we consider $S=D(m_{1},\ldots,m_{n})$, $m_{j}>1$, in Example 3.10, where
$h(S)\cong{\mathbb{Z}}*{\mathbb{Z}}_{m_{1}}*\cdots*{\mathbb{Z}}_{m_{n}}$. The
white point in the figure represents $S$, with $g(S)=0$ and $b(S)=n+1$. The
region of integral points representing $S^{\prime}$ with $h(S^{\prime})\cong
h(S)$ by Proposition 3.23 is represented by the green region in Figure 14 (C).

We compare this region in (C) with the region in (B). The region (B) was
realized through existence of points on the line $b=(2-\chi(S^{*}))-2g$. In
this case of $S=D(m_{1},\ldots,m_{n})$, even if we assume that this $S$ is of
maximum Euler characteristic, we do not have surface ribbons that correspond
to integral points between $b=(2-\chi(S^{*}))-2g$ and $b=(2-\chi(S^{*}))-g$,
above $b=1$, represented by the region shaded in light blue in (C). For
example, for $n=2$, since $b=3$, the point $(g,b)=(1,1)$ is on the line
$b=(2-\chi(S^{*}))-2g$ but below the line $b=(2-\chi(S^{*}))-g$, and we do not
know if there exists a surface ribbon realizing this point.

For the $y$-intercept $2-\chi(S^{*})$ in Figure 14, we have the following
bound.

######  Proposition 3.28.

Let $S$ be a surface ribbon with $\nu$ connected components. Then we have

$\mu(S)\leq 2\nu-\chi(S^{*})$

where $S^{*}$ is the surface ribbon with maximum Euler characteristic
satisfying $h(S)\cong h(S^{*})$.

###### Proof.

We assume that $S$ is connected, since an iteration of this case gives the
result in general. From the proof of Theorem 3.4, after putting $S$ in
standard form, we have a presentation of $h(S)$ with a copy of the free group
on one generator, and the reduced heap, $h(S)=F_{1}*\hat{h}(S)$. The latter
has two generators corresponding to pairs of ribbon terms for each crossed
handle pair, and one generator corresponding to a ribbon term for each trivial
handle. Therefore, we have $2g+b=2-\chi(S)$ generators. Consequently we find
that $\mu(S)\leq 2-\chi(S)$. Since $S^{*}$ has $h(S^{*})\cong h(S)$ by
assumption, it follows that $\mu(S^{*})=\mu(S)$. From $\chi(S^{*})\geq\chi(S)$
and the previous inequality for $\mu(S)$ we obtain $\mu(S)\leq 2-\chi(S^{*})$,
and the proof is complete. ∎

###  3.3. A relation to the Wirtinger presentation

The fundamental group of the complement of a spatial graph can be given by a
presentation from its oriented diagram in a manner similar to the Wirtinger
presentation of knot groups, as outlined in [MSW].

Figure 15. Wirtinger relations at a crossing (left) and a vertex (right)

######  Theorem 3.29.

For a surface ribbon $S$, let $\hat{h}(S)$ be the reduced fundamental heap,
$h(S)=F_{\nu}*\hat{h}(S)$. Then there exists an epimorphism
$\lambda:\pi_{1}(S^{3}\setminus S)\rightarrow\hat{h}(S)$.

###### Proof.

Let $\Xi$ be the core trivalent graph of $S$. We specify arbitrary
orientations on edges of $\Xi$, so that there are four possibilities of the
orientations at each vertex. The case of two-in, one-out is depicted in the
right of Figure 15. In the figures, Wirtinger generators are depicted by short
arcs behind oriented edges of $\Xi$.

At a positive crossing depicted in the left of Figure 15, we set the ribbon
terms $\alpha=x^{-1}y$, $\beta=u^{-1}v$ and $\gamma=z^{-1}w$, where
$x,y,u,v,z,w$ are generators of $h(S)$ assigned on the parallel boundary arcs
of the surface ribbon $S$. The assignment that defines $\lambda$ is such that
the Wirtinger generator of the arc corresponding to the arc labeled by $(x,y)$
is assigned $\alpha$, and similar for the other arcs. This is the same
assignment defined in [SZframedlinks] for framed links, in which the Wirtinger
relations are verified using the diagram in the left of Figure 15. Indeed,
since $z=xu^{-1}v$ and $w=yu^{-1}v$, one computes

$\gamma=z^{-1}w=(xu^{-1}v)^{-1}(yu^{-1}v)=(u^{-1}v)^{-1}(x^{-1}y)(u^{-1}v)=\beta^{-1}\alpha\beta,$

which is a Wirtinger relation. Negative crossings are checked similarly. The
group $\hat{h}(S)$ is generated by ribbon terms of the form $\alpha=x^{-1}y$,
hence the image of $\lambda$ is in $\hat{h}(S)$.

It remains to show that the relation holds at trivalent vertices. For a vertex
depicted in the right of Figure 15, the relation in $\pi_{1}(S^{3}\setminus
S)$ is $\gamma=\alpha\beta$, and this holds for $\alpha=x^{-1}y$,
$\beta=y^{-1}z$ and $\gamma=x^{-1}z$ as desired. The other three types of
orientations at vertices can be similarly checked. ∎

###  3.4. Effect under stabilization

In this section we describe a stabilization of surface ribbons and provide the
effect of a stabilization on the fundamental heap. It is known (e.g. [BFK])
that two Seifert surfaces of a link are related by a sequence of
(de/)stabilizations and isotopy, where a stabilization means a 1-handle
addition.

Figure 16. Stabilization

A 1-handle addition is depicted in Figure 16 (A). In (B), a thin portion of
the boundary is pushed towards the left foot of the handle, and wraps around
the handle to obtain a thin ribbon that was a part of the handle. The boundary
is pushed further along the handle to the right foot. The pushed boundary
curve stops short of reaching the boundary near the right foot of the handle
as depicted. By straightening and flipping, we obtain the surface in (C). In
summary, an addition of a pair of a long and a short trivial ribbons as in (C)
is regarded as a stabilization of a surface ribbon.

Figure 17. Labels for a stabilization

######  Proposition 3.30.

For any surface ribbon $S$, there exists another $S^{\prime}$ obtained from
$S$ by a sequence of stabilizations such that $h(S^{\prime})$ is a free group.

###### Proof.

We may assume that a given diagram of $S$ is connected. In Figure 17,
generators and relations for a stabilization at a ribbon is depicted. Each arc
receives a generator as indicated, and 4 ribbon crossings as indicated by red
circles give rise to relations (1) through (4). From (1), (3) and (4) we
obtain that $x=u=v=z$, and from (2) we obtain these generators are equal to
$y$. When a stabilization is performed in this way to a vertical ribbon whose
boundary curves receive generators $x$ and $y$, the effect of the
stabilization is an additional relation $x=y$ and an additional free generator
$a$.

Figure 18. Stabilization at a crossing

When two other similar operations are performed at a crossing as depicted in
Figure 18, the effect is that corresponding to stabilizations labeled (1),
(2), and (3), relations $y=x$, $s=x$ and $t=x$ are introduced as depicted, and
the original relations for $u$ and $v$ imply $u=x$ and $v=x$ as well. Three
free generators are introduced, as indicated by $a$, $b$ and $c$ corresponding
to small loops in the figure at (1), (2) and (3), respectively. Hence the
effect of this process at this crossing is that all original generators
assigned to arcs are equated, and three new free generators are introduced.

By performing this procedure at every ribbon crossing of the diagram, we
obtain $S^{\prime}$ by stabilizations such that $h(S^{\prime})$ is a free
group. ∎

###  3.5. Realization problem for the fundamental heap of surface ribbons

Recall that, as observed above, fundamental heaps of surface ribbons are
finitely presented by definition. From Theorem 3.4 it follows that for any
surface ribbon $S$, $h(S)$ contains a free factor. We show below that any
finitely presented group can be realized as a fundamental heap after adding
some free factor.

Figure 19. Constructing relators over base surface

######  Theorem 3.31.

Let $G$ be a finitely presented group. Then there exists a surface ribbon $S$
such that $h(S)\cong F_{k}*G$ for some positive integer $k$.

###### Proof.

First observe that if $G$ factors as free product of subgroups
$G=G_{1}*G_{2}*\cdots*G_{q}$ then, using Lemma 3.3, we can reduce the problem
to finding ribbon surfaces whose reduced heap is isomorphic to the group heap
of each factor $G_{i}$, since their disjoint union (split sum) would realize
$G$. We may assume that $G$ is irreducible with respect to free product
factorization. Let $G=\langle\,\alpha_{1},\ldots,\alpha_{n}\mid
R_{1},\ldots,R_{m}\,\rangle$. We construct the required surface $S$ in various
steps, at each of which we consider the effect on the reduced heap. We start
by introducing a surface ribbon $S$ consisting of $m$ single trivial handles
concatenated horizontally. We realize each relator of $G$ on the handles of
this surface, which we will refer to as “base surface”. Close to the left foot
of each handle we apply the stabilization procedure as in Figure 17. As a
consequence we have two free generators, for each stabilization, indicated by
the letters $x$ and $a_{i}$ as in Figure 17, for $h(S)$. This is depicted in
Figure 19 for the first two handles. Also, the labels of the handle of $S$ are
$x$ for both edges of the handle, outside the small region where $a_{i}$ is
located, i.e. at the left foot of each handle of the base surface. Let us now
consider the first relator $R_{1}$ in the presentation of $G$. Let
$w(\alpha_{1},\ldots,\alpha_{n})=\alpha_{i_{1}}^{n_{1}}\cdots\alpha_{i_{r}}^{n_{r}}$
be the word corresponding to $R_{1}$. We may assume that
$w(\alpha_{1},\ldots,\alpha_{n})$ is reduced and, we suppose that $n_{1}$ is
positive. We introduce an annular surface ribbon component wrapping $n_{1}$
times around the handle of $S$, as shown on the upper part of the first handle
of Figure 19. For a negative $n_{1}$, we take negative twists for the wrapping
ribbon. Let us denote by $y_{1}$ and $z_{1}$ the outer and inner arcs,
respectively, of the annular ribbon that has been introduced. Then, depending
on the positive or negative crossing, respectively, $x$ changes to
$x(y_{1}^{-1}z_{1})^{n_{1}}$ or
$x(z_{1}^{-1}y_{1})^{n_{1}}=x(y_{1}^{-1}z_{1})^{-n_{1}}$. We assume that it is
the first case, without loss of generality. We observe that since the handle
of $S$ has undergone stabilization, as in the proof of Proposition 3.30, the
two arcs delimiting the handle of $S$ have the same color $x$ and do not
modify the colors $y_{1}$ and $y_{2}$ after overpassing the annular ribbon
that has been introduced, so no relator is derived on this ribbon. This is
also crucial in the fact that the color $x$ changes to
$x(y_{1}^{-1}z_{1})^{n_{1}}$, since if the base handle were not monochromatic,
its colors would have obstructed us from obtaining a multiplication by a
simple power $(y_{1}^{-1}z_{1})^{n_{1}}$. We do not perfom any chagnes on the
other handles of the base surface. Let us denote the surface just obtained as
$S_{1}$. Proceeding as in the proof of Theorem 3.4, we obtain a presentation
of $h(S_{1})$ with $m+2$ generators $x,a_{i},y_{1},\alpha_{1}$, where we have
set $\beta_{1}:=y_{1}^{-1}z_{1}$, and a single relator $\beta_{1}^{n_{1}}$
because the band after wrapping ribbon connects to the base labeled $x$. We
repeat the same construction of adding another annular ribbons that links the
handle of $S$, but does not link other parts. Let us denote by $y_{2}$ and
$z_{2}$ the outer and inner arcs, respectively, of the newly introduced
handle. We assume, as before, that the crossing with the handle of $S$
introduces a new color $\beta_{2}:=y_{2}^{-1}z_{2}$. We let this handle wrap
around the base handle $n_{2}$ times and, moreover, we use the same convention
on signs as before. We connect these annular ribbons to the base handle
labeled $x$ by stabilization, as indicated by a small blue arcs at the right
of the annular ribbon in Figure 19. This addition introduces a relation
$x=y_{i}$ and addition of a free generator corresponding to a small loop ($y$
in Figure 16 (C)). Let us denote by $S_{2}$ the surface ribbon obtained via
this procedure. The fundamental heap of $S_{2}$ has $m+3$ free generators
(labeled $a_{i}$’s) and one relator $\beta_{1}^{n_{1}}\beta_{2}^{n_{2}}$.

Figure 20. Identifying generators through stabilization

We proceed in this way $r$ times to construct a surface ribbon which we call
$S_{r}$, until we have a relator $\beta_{1}^{n_{1}}\cdots\beta_{r}^{n_{r}}$.
The word $w(\alpha_{1},\ldots,\alpha_{n})$ might have repetitions of
$\alpha_{i}$’s in the product
$\alpha_{i_{1}}^{n_{1}}\cdots\alpha_{i_{r}}^{n_{r}}$. For each repeated pair
$\alpha_{i_{t}}=\alpha_{i_{s}}$, we add bands to connect the respective
connected components of $\beta_{i_{t}}$ and $\beta_{i_{s}}$, being careful to
let the connecting band overpass any other ribbon along the way, in such a way
that $\beta_{i_{t}}=\beta_{i_{s}}$. This is diagrammatically indicated in
Figure 20. Since the overpassing ribbons are monochromatic, no new relator is
introduced, so that the presentation of the fundamental heap $h(S_{r})$ has a
single relator that is not changed except for the effect of having certain
letters $\beta_{i}$ identified, mirroring the same word of $R_{1}$.

At the end of this procedure we have found a surface $S_{r}$ whose fundamental
heap is the free product of free factors (determined in number by the
stabilizations performed) and a single relator obtained from
$w(\alpha_{1},\ldots,\alpha_{r})$ upon substituting $\beta_{1}$ to
$\alpha_{i_{1}}$, $\beta_{2}$ to $\alpha_{i_{2}}$ etc.

Next, we construct a ribbon surface on the second handle of the base surface,
in the same way as in the previous step, that introduces a relator
$\gamma_{1}^{m_{1}}\cdots\gamma_{\ell}^{m_{\ell}}$ that coincides with $R_{2}$
upon substituting the $\gamma$’s in place of appropriate $\alpha_{i}$’s in the
reduced word representing the relator $R_{2}$. To complete this step, we need
to add untwisted bands between the handles just introduced and those in the
previous step, in order to equate the generators $\beta_{i}$ and $\gamma_{j}$
that are substituted to the same generator $\alpha_{k}$ of $G$. Letting the
band that is introduced in the process overpass every ribbon met along the
way, at each step, we do not introduce new relators. This is similar to Figure
20, where instead of introducing a band between surface ribbon over the same
base handle, we connect surface ribbons over different base handles. We also
note that the other boundary curves do not create additional relations, as
they run parallel to the outer boundary curve, and have the same letter $x$
assigned, producing the same relator words $R_{i}$ from each factor of
boundary connected sum.

After performing the previous steps $m$ times, we obtain a surface ribbon
$\tilde{S}$ whose fundamental heap consists of free products of a number of
free factors, and relators $R^{\prime}_{1},\ldots,R^{\prime}_{m}$ where
$R_{i}$ and $R^{\prime}_{i}$ correspond to the same words, and differ only by
appropriate changes of variables from $\alpha_{j}$ to $\beta_{k}$, according
to some correspondence determined during the construction of $\tilde{S}$.

Finally, there is a mapping from $G$ onto
$\bar{G}:=\langle\,\beta_{1},\ldots,\beta_{n}\mid
R^{\prime}_{1},\ldots,R^{\prime}_{m}\,\rangle$ determined by the assignment
$\alpha_{i}\mapsto\beta_{i}$. This is clearly well defined since by
construction we have that $R^{\prime}_{j}$ is obtained from $R_{j}$ upon
substituting the letters $\beta_{i}$ to $\alpha_{i}$. This gives an
isomorphism between $G$ and $\bar{G}$. Since we have that
$h(\tilde{S})=F_{k}*\bar{G}$ for some $k$ given by $\\#\\{{\rm Stabilizations\
performed}\\}+1$, this completes the proof. ∎

######  Remark 3.32.

We point out that the number of free factors $F_{k}$ appearing in the
constructive proof of Theorem 3.31 depends on the particular presentation of
the group $G$ that is chosen at the beginning. Given two different
presentations of $G$, we obtain two generally different surface ribbons
$S_{1}$ and $S_{2}$ whose fundamental heap is isomorphic to the group heap of
$G$, up to a number of free factors. It is, therefore, desirable to determine
a relation between the number of free factors appearing in the previous
construction in terms of a given presentation of a group $G$.

Let $\langle\,\alpha_{1},\ldots,\alpha_{n}\ |\ R_{1},\ldots,R_{m}\,\rangle$
denote a presentation of $G$. Let $S$ denote the surface ribbon constructed in
Theorem 3.31, where $k$ is the number of free factors appearing in $h(S)$.
Then we have $k=n+m+1$. To see this, first note that in the construction of
$S$ in Theorem 3.31, each handle of the base surface is stabilized once, and
contributes a free factor corresponding to a small loop denoted by $a_{i}$ in
Figure 19. These handles correspond to relators, so that the number of these
free factors is $m$. For the first generator $\alpha_{i_{1}}^{n_{1}}$
appearing in the first relator $R_{1}$:
$w(\alpha_{1},\ldots,\alpha_{n})=\alpha_{i_{1}}^{n_{1}}\cdots\alpha_{i_{r}}^{n_{r}}$
in the proof of Theorem 3.31, the handle wrapping around the first base handle
depicted in Figure 19 has a stabilization connected to the base handle
depicted by a small blue arc in the figure, which contributes one free factor.
When the same generator appears again, the corresponding wrapping handle is
connected to the first as depicted in Figure 20, and is not stabilized, so
that it does not contribute any additional free factor. Therefore each
generator contributes one free factor. The external boundary component labeled
$x$ runs over all handles and contributes one free factor. Hence we obtain
$k=n+m+1$.

##  4\. Reversibility and additivity conditions

In this section, we consider two algebraic conditions for TSD operations and
corresponding 2-cocycle conditions. Such additional conditions on 2-cocycles
are used in the next section for constructions of cocycle invariants for
surface ribbons.

###  4.1. Reversibility and additivity for TSD operations

######  Definition 4.1.

Let $(X,T)$ be a set with a ternary operation $T$. We say that $(X,T)$
satisfies the idempotency condition if $T(w,x,x)=w$ for all $w,x\in X$. We say
that $(X,T)$ satisfies the reversibility condition if $T(T(w,x,y),y,x)=w$ for
all $w,x,y\in X$. We say that $(X,T)$ satisfies the additivity condition if
$T(T(w,x,y),y,z)=T(w,x,z)$ for all $w,x,y,z\in X$.

######  Remark 4.2.

Additivity and idempotency conditions imply the reversibility condition by
setting $x=z$.

######  Remark 4.3.

The reversibility and additivity conditions are used in Lemmas 3.2 and 5.2 for
the well-definedness of the fundamental heap and colorings under the YI and IY
moves.

######  Example 4.4.

Direct computations show that any group heap satisfies reversibility and
additivity conditions.

######  Example 4.5.

Let $X$ be a module over ${\mathbb{Z}}[t^{\pm 1},s]$ and define a ternary
operation by $T(x,y,z)=tx+sy+(1-t-s)z$. It is computed that $T$ is a TSD
operation (e.g., [ESZcascade]). Then direct calculations show that $X$ does
not satisfy the reversibility and additivity conditions in general. Thus these
conditions can be used to detect non-heap TSD operations.

###  4.2. Reversibility and additivity for TSD 2-cocycles

######  Definition 4.6.

Let $X$ be a group heap, and let $A$ be an abelian group. A 2-cocycle $\psi\in
Z^{2}_{\rm SD}(X,A)$ is said to be nondegenerate if $\psi(x,y,y)=0$ for all
$x,y\in X$ [ESZheap]. A 2-cocycle $\psi\in Z^{2}_{\rm SD}(X,A)$ is said to
satisfy the reversibility condition if it holds that

$\psi(w,x,y)+\psi(wx^{-1}y,y,x)=0$

for all $w,x,y\in X$. A 2-cocycle $\psi$ is said to satisfy the additivity
condition if it holds that

$\psi(w,x,y)+\psi(wx^{-1}y,y,z)=\psi(w,x,z)$

for all $w,x,y,z\in X$.

######  Remark 4.7.

If a 2-cocycle $\psi\in Z^{2}_{\rm SD}(X,A)$ is nondegenerate, then the
additivity condition on $\psi$ implies the reversibility condition.

The reversibility and additivity conditions ensure well-definedness of the
cocycle invariant defined in the next section.

Direct computations imply the following.

######  Lemma 4.8.

Let $X$ be a group heap and $A$ an abelian group. Then any 2-coboundary
$\delta f$, $f\in Z^{1}_{\rm SD}(X,A)$, satisfies the reversibility and
additivity conditions. Furthermore, linear combinations of reversible and
additive 2-cocycles are reversible and additive, respectively.

In [CS], it was shown that certain identities satisfied by a quandle induce
subcomplexes. Similarly, it is expected that Lemma 4.8 extends to higher
dimensions to form corresponding subcomplexes. We pose the following
definition for dimension $2$.

######  Definition 4.9.

Let $X$ denote a heap and let $H^{2}_{\rm SD}(X,A)$ denote the self-
distributive second cohomology group of $X$. Let $Z^{2}_{\rm RA}(X,A)$ denote
the subgroup of $Z^{2}_{\rm SD}(X,A)$ of $2$-cocycles that are reversible and
additive. As an application of Lemma 4.8, the quotient $H^{2}_{\rm
RA}(X,A):=Z^{2}_{\rm RA}(X,A)/B^{2}_{\rm SD}(X,A)$ is a well defined subgroup
of $H^{2}_{\rm SD}(X,A)$, which we call the reversible and additive second
cohomology group of $X$, or RA cohomology for short, and similarly, the RA
cocycle group for $Z^{2}_{\rm RA}(X,A)$.

Direct computations show the following.

######  Proposition 4.10.

Let $(X,T)$ be a TSD set satisfying reversibility and additivity conditions,
$A$ an abelian group, and let $\psi:X^{3}\rightarrow A$ be a function. Define
a ternary operation on $X\times A$ by
$\hat{T}((x,a),(y,b),(z,c)):=(T(x,y,z),a+\psi(x,y,z))$. Then, $\hat{T}$
defines a TSD structure on $X\times A$ if and only if $\psi$ is a TSD
$2$-cocycle. Moreover, $\hat{T}$ satisfies reversibility and additivity
condition if and only if $\psi$ satisfies the reversibility and additivity
conditions.

In order to obtain 2-cocycles with reversibility and additivity conditions, we
review constructions of cocycles from [SZframedlinks]. For completeness we
include a proof of the first construction, while defer the reader to
[SZframedlinks] for a proof of the second.

######  Lemma 4.11.

([SZframedlinks]) Let
${\mathbb{Z}}_{n}=\langle\,\zeta\mid\zeta^{n}=1\,\rangle$ be the cyclic group
of order $n$ in multiplicative notation with a generator $\zeta$. Let
$\phi_{i}=\sum_{x\in{\mathbb{Z}}_{n}}[\sum_{j=0}^{n-1}\chi_{(x,\zeta^{j},\zeta^{j+i})}]$,
$i=1,\ldots,n-1$, where $\chi$ denotes the characteristic function. Then
$\phi_{i}$ is a nondegenerate $2$-cocycle, $\phi_{i}\in C^{2}_{\rm
NDH}({\mathbb{Z}}_{n},{\mathbb{Z}})$, for all $i=1,\ldots,n-1$. Moreover,
reducing coefficients modulo $n$, we obtain $\phi_{i}\in C^{2}_{\rm
NDH}({\mathbb{Z}}_{n},{\mathbb{Z}}_{n})$ for all $i$.

###### Proof.

For a fixed $i$, the 2-cocycle $\phi_{i}$ vanishes for 2-chains $(x,u,v)\in
C_{2}^{\rm NDH}(X,{\mathbb{Z}})$ if $v\neq ur^{i}$. Hence if $v\neq ur^{i}$,
then the last two terms of $(*)$, $-\phi(x,u,v)+\phi(xy^{-1}z,u,v)$, both
vanish. If $v=ur^{i}$, then both terms are $1$ and cancel. Hence we focus on
the first two terms.

Let $v=u\zeta^{k}$, then the first two terms of $(*)$ are
$\phi(x,\zeta^{j},\zeta^{j+m})-\phi(x\zeta^{k},\zeta^{j+k},\zeta^{j+m+k})$ for
some $j,m\in{\mathbb{Z}}_{n}$. If $m=i$, then both terms are 1 and cancel. If
$m\neq i$, then both vanish. Hence $(*)$ holds. ∎

######  Lemma 4.12.

([SZframedlinks])  Let $D_{n}$ be the dihedral group of order $2n$ generated
by a rotation $\zeta$ and reflection $a$ with a relation $a\zeta
a=\zeta^{-1}$. Let $\psi_{i}=\sum_{x\in
D_{n}}[\sum_{j=0}^{n-1}(\chi_{(x,\zeta^{j},\zeta^{j+i})}+\chi_{(x,a\zeta^{-j},a\zeta^{-j-i})})]$,
$i=1,\ldots,n-1$. Then $\psi_{i}$ is a nondegenerate $2$-cocycle, $\psi_{i}\in
C^{2}_{\rm NDH}(D_{n},{\mathbb{Z}})$, for all $i=1,\ldots,n-1$. Moreover,
reducing coefficients modulo $n$, we obtain $\psi_{i}\in C^{2}_{\rm
NDH}({\mathbb{Z}}_{n},{\mathbb{Z}}_{n})$ for all $i$.

######  Lemma 4.13.

Let $\vec{a}=(a_{0},a_{1},\ldots,a_{n-1})\in({\mathbb{Z}}_{n})^{n}$, where we
set $a_{0}=0$. Let $\phi_{\vec{a}}=\sum_{i=0}^{n-1}a_{i}\phi_{i}$ and
$\psi_{\vec{a}}=\sum_{i=0}^{n-1}a_{i}\psi_{i}$, where $\phi_{i}$ and
$\psi_{i}$ are cocycles in Lemma 4.11 and 4.12 with coefficients modulo $n$,
respectively, and we formally set $\phi_{0}=\psi_{0}\equiv 0$. Then
$\phi_{\vec{a}}$ and $\psi_{\vec{a}}$ satisfy the reversibility and additivity
conditions if and only if $a_{k+\ell}=a_{k}+a_{\ell}$ for all $k,\ell$, where
$k,\ell$ are taken modulo $n$.

###### Proof.

For $\phi$, any triplet $(x,y,z)\in({\mathbb{Z}}_{n})^{3}$ can be written as
$(x,\zeta^{j},\zeta^{j+i})$ for some $i,j\in{\mathbb{Z}}_{n}$. Since
$\phi(x,\zeta^{j},\zeta^{j+i})=a_{i}$, one computes that the additivity

$\phi(x,\zeta^{j},\zeta^{j+k})+\phi(x\zeta^{j},\zeta^{j+k},\zeta^{j+k+\ell})=\phi(x,\zeta^{j},\zeta^{j+k+\ell})$

is equivalent to $a_{k}+a_{\ell}=a_{k+\ell}$, where the subscripts are
considered modulo $n$. The invertibility
$\phi(x,\zeta^{j},\zeta^{j+k})=-\phi(x\zeta^{\ell},\zeta^{j+k},\zeta^{j})$ is
equivalent to $a_{-k}=-a_{k}$, which follows from the equation
$a_{k}+a_{\ell}=a_{k+\ell}$. Similar arguments apply to $\psi$. ∎

######  Remark 4.14.

The preceding lemma implies that a cocycle for ${\mathbb{Z}}_{n}$ and $D_{n}$
are determined by the value of $a_{1}\in{\mathbb{Z}}_{n}$. It is proved in
[SZframedlinks] that $\phi$ and $\psi$ are non-trivial in $H^{2}_{\rm SD}(X)$
and generally linearly independent. From Examples 5.12 and 5.13 in that
article, one also sees that if $\vec{a}\neq 0$, the cocycles $\phi_{\vec{a}}$
and $\psi_{\vec{a}}$ are nontrivial as well.

Let $R$ be a ring considered with abelian heap operation with respect to its
additive structure. In [SZframedlinks] it was shown that the $2$-cochains
$\psi_{(a,b,c)}(x,y,z)=(ax+b(z-y)+c)(z-y)$ are nontrivial $2$-cocycles for any
choice of $a,b,c\in R$ with $a\neq 0$. In fact, it turns out that
$\psi_{(a,b,c)}-\psi_{(a,b,0)}$ is a coboundary and, therefore, we will omit
the index $c$ in the rest of the article, and simply write $\psi_{(a,b)}$ for
$\psi_{(a,b,0)}$.

######  Lemma 4.15.

The cocycles $\psi_{(a,b)}$ are reversible and additive if and only if $a=2b$.

###### Proof.

Since each $\psi_{(a,b)}$ is non-degenerate, it is enough to show that
$\psi_{(a,b)}$ is additive. The additive condition

$\psi_{(a,b)}(w,x,y)+\psi_{(a,b)}(w-x+y,y,z)=\psi_{(a,b)}(w,x,z)$

becomes

$(aw+b(y-x))(y-x)+(a(w-x+y)+b(z-y))(z-y)=(aw+b(z-x))(z-x).$

This is readily seen to hold for all $w,x,y,z$ if and only if $a=2b$. ∎

We set $\psi_{b}:=\psi_{(2b,b)}$ to denote the reversible and additive
cocycles of Lemma 4.15.

###  4.3. Mutually distributive RA cocycles

Let $X$ be a group heap and let $A$ be an abelian group. Suppose that
$\psi_{1}$ and $\psi_{2}$ are reversible and additive $2$-cocycles. If the
pair $(\psi_{1},\psi_{2})$ is mutually distributive (as defined in Section
2.3), then we say that $(\psi_{1},\psi_{2})$ is a reversible and additive
mutually distributive pair, or RA mutually distributive pair for short.

######  Example 4.16.

Let $\psi_{b}$ and $\psi_{d}$ denote two RA $2$-cocycles of Lemma 4.15. Then
in order to verify whether they are mutually distributive, it is enough, by
symmetry, to show the equality

$\psi_{b}(x,y,z)+\psi_{d}(x-y+z,u,v)=\psi_{b}(x-u+v,y-u+v,z-u+v)+\psi_{d}(x,u,v).$

Using the definition of $\psi_{b}$ and $\psi_{d}$ we see that this holds for
all $x,y,z,u,v$ if and only if $2(b-d)=0$.

######  Definition 4.17.

Let $X$ be a group heap and let $A$ an abelian group. A $2$-cocycle $\psi$ is
said to be separable if it satisfies the $2$-cocycle condition pairwise, as
follows:

$\displaystyle\psi(xy^{-1}y,u,v)$ $\displaystyle=$ $\displaystyle\psi(x,u,v),$
$\displaystyle\psi(x,y,z)$ $\displaystyle=$
$\displaystyle\psi(xu^{-1}v,yu^{-1}v,zu^{-1}v)$

for all $x,y,u,v\in X$. In other words, $\psi$ is separable if and only if
$\psi$ and the zero cocycle are mutually distributive.

######  Example 4.18.

The cocycles $\phi_{i}$ of Lemma 4.11 are separable, as a direct computation
shows. Since $\phi=\sum_{i=1}^{n-1}i\phi_{i}$ is additive and reversible as
well, it follows that $\phi$ and the zero cocycle are RA mutually distributive
cocycles. Similarly, the cocycles $\psi_{i}$ of Lemma 4.12 are separable and,
therefore, $\psi=\sum_{i=1}^{n-1}i\psi_{i}$ is RA mutually distributive with
the zero cocycle.

######  Remark 4.19.

If $\bar{\psi}=(\psi_{i})_{i=1}^{n}$ is a sequence of separable cocycles, then
they are pairwise mutually distributive.

##  5\. Colorings and cocycle invariants of ribbon graphs

A coloring of a surface ribbon diagram by a heap is defined by assigning
elements of the heap to double arcs as follows, in a manner similar to quandle
coloring, and cocycle invariants are also similarly defined as in [CJKLS]. In
this section we give such definitions, realizations of surface ribbons with
non-trivial invariant values, and an application to non-trivial cohomology.

###  5.1. Colorings

First we define and examine colorings of surface ribbon diagrams by heaps.

######  Definition 5.1.

Let $X$ be a heap. Let $D$ be a surface ribbon diagram and ${\cal A}$ the set
of doubled arcs. A coloring  of $D$ by $X$ is a map ${\cal C}:{\cal
A}\rightarrow X$ that satisfies the coloring condition as depicted in Figure 2
(A) and (C), where $(z,w)=(xu^{-1}v,yu^{-1}v)$.

From the definition we obtain the following by checking the moves. The proof
parallels that of Lemma 3.2.

######  Lemma 5.2.

The sets of colorings of two surface ribbon diagrams are in bijection between
each move listed in Figure 4.

In particular, the number of colorings of a surface ribbon diagram by a finite
heap $X$ is an invariant of a surface ribbon, that does not depend on the
choice of a diagram, and is denoted by ${\rm Col}_{X}(S)$. Similarly to
[CJKLS, SZframedlinks], the set of colorings of a surface ribbon $S$ by a heap
$X$ can be considered as the set of heap homomorphisms from $h(S)$ to $X$.
Although the fundamental heap was defined by group presentations, these
homomorphisms need not be group homomorphism; assigning a single color to all
arcs that is not the identity element is a heap homomorphism but not a group
homomorphism.

We observe that from the definition, if $x=y$ at a crossing as in Figure 2
(A), then we have $z=w$. Consequently, if $x=y$ (the two colors are equal) at
one pair of arcs of a ribbon, then the entire ribbon (band) has this property.
In this situation we say that this is a monochromatic ribbon. We also note
that if the over-arc is a monochromatic ribbon, then the colors of the under-
arc in Figure 2 (A) satisfy $x=z$ and $y=w$, i.e. a monochromatic overpassing
ribbon does not change the colors of the under-arc.

######  Example 5.3.

Let $S=B_{1}^{n}\natural B_{2}^{m}$ be the surface obtained by concatenating
$n$ trivial handles and $m$ crossed handle pairs, as in Example 3.9. Then no
restriction on the assignment of colors arises for any $X$. The number of
colors of $S$ by $X$ is $|X|^{n+m+1}$.

######  Example 5.4.

Let $S=D(m_{1},\ldots,m_{n})$ denote the surface of Example 3.10, where we
take the connected sum of $n$ copies of looped ribbons, with $m_{j}$ twists
for $j=1,\ldots,n$, constructed by attaching copies of Figure 10 vertically.
The $j^{\rm th}$ looped ribbon is denoted by $H^{m_{j}}$. We order the
boundary components by taking $b_{0}$ to be the base of $S$ (the components
that contains the top through bottom outside curve in standard position), and
$b_{1},\ldots,b_{n}$ are the inner boundary components of the handles in the
order they appear in the standard position, from top to bottom. The colorings
of $S$ by the cyclic group $X={\mathbb{Z}}_{m}$ (taken here in multiplicative
notation with a generator $\zeta$ as in Lemma 4.11) are determined as follows.

The ribbon $H^{m_{j}}$ has two boundary curves, the outer component belonging
to $b_{0}$ and the inner component $b_{j}$. The colors assigned are $y$ for
the outer component (base) $b_{0}$ (that corresponds to the top and bottom
arcs labeled by $y$ in Figure 10), and $x_{j}$ for the inner component
$b_{j}$, that corresponds to $x$ in the figure, where $j$ represents that
$b_{j}$ is at the $j^{\rm th}$ handle $H^{m_{j}}$. Let
$\alpha_{j}:=x_{j}^{-1}y$. Then the coloring condition for $H^{m_{j}}$ is
$\alpha_{j}^{m_{j}}=1$ from Example 3.10. If $y=\zeta^{t}$ and
$x_{j}=\zeta^{t_{j}}$, then $\alpha_{j}^{m_{j}}=1$ is equivalent to
$\zeta^{m_{j}(t-t_{j})}=1$, that is, $m_{j}(t-t_{j})\equiv 0$ modulo $m$. Set
$d_{j}:={\rm gcd}(m,m_{j})$. For each arbitrary $y$, there are $d_{i}$
solutions to the equation for $x_{i}$, namely given by
$t-t_{j}=s_{j}\frac{m}{d_{j}}$, for $s_{j}=0,\ldots,d_{j}-1$. The total number
of colorings is $|X|\sum_{j}d_{j}$.

###  5.2. Cocycle invariants with respect to the boundary curves

In this subsection we consider heap cocycle invariants of surface ribbons. Let
$X$ be a heap and $D$ a surface ribbon diagram. Then a 2-cocycle invariant is
defined in a manner similar to the quandle 2-cocycle invariant as follows.

Figure 21. Orientations and labeling across double arcs

The orientations of boundary curves of a surface ribbon diagram $D$ are
defined as depicted in Figure 21. The curves of a ribbon are oriented in
antiparallel directions, in such a way that are consistent with the
counterclockwise orientation of the complementary regions as depicted.

######  Definition 5.5.

Let $X$ be a heap and $A$ an abelian group with multiplicative notation. Let
$\bar{\psi}=(\psi_{1},\ldots,\psi_{n})$ be a sequence of pairwise mutually
distributive RA $2$-cocycles of $X$ with coefficient group $A$, so that
$\psi_{i}\in Z^{2}_{\rm RA}(X,A)$ for all $i=1,\ldots,n$ and each pair is
mutually distributive. The 2-cocycle heap invariant of a surface ribbon $S$
with respect to $\bar{\psi}=(\psi_{1},\ldots,\psi_{n})$ is defined as follows.

Let $D$ be an oriented diagram of a surface ribbon $S$. Let
$S=S_{1}\cup\cdots\cup S_{n}$ denote the connected components of $S$, and
$D=D_{1}\cup\cdots\cup D_{n}$ be the corresponding oriented diagram. We
decorate $D_{i}$ with the $2$-cocycle $\psi_{i}$ for each $i$.

For each $D_{i}$, we order the connected components of its boundary; let
$b_{i}^{j}$ be the ordered boundary components of $D_{i}$. We fix a base point
in each $b_{i}^{j}$ and order the arcs of $D_{i}$ following the orientation of
$b_{i}^{j}$, as well as the crossings where $b_{i}^{j}$ underpasses. Let
$\tau_{i}^{j}(1),\ldots,\tau_{i}^{j}(r)$ denote the crossings where
$b_{i}^{j}$ underpasses in this order.

We define, for a given coloring $\mathcal{C}$,
$\Psi_{\bar{\psi}}^{i}(\mathcal{C},D)=\otimes_{j}\prod_{k}B^{\bar{\psi}}(\mathcal{C},\tau_{i}^{j}(k))$,
where the tensor product runs over each boundary component of $D_{i}$, the
product runs over all the crossings where boundary components $b_{i}^{j}$
underpasses, and at each crossing we use the cocycle that decorates the
overpassing connected component:
$B^{\bar{\psi}}(\mathcal{C},\tau_{i}^{j}(k))=\psi_{\ell(i,j,k)}(x,y,z)$, where
$\ell(i,j,k)$ is the number assigned to the overpass at $\tau_{i}^{j}(k)$, $x$
is the color assigned to the undrpass arc right before $\tau_{i}^{j}(k)$,
$(y,z)$ is the pair of colors assigned to the overpass that appear in this
order (cf. Figure 3). The invariant values are considered equivalent up to
permutations of tensor factors of each $\Psi_{\bar{\psi}}^{i}(\mathcal{C},S)$,
to allow renumbering boundary components. Thus the value is regarded as an
element of the symmetric algebra $S({\mathbb{Z}}[A])$ (in fact its subspace of
degree being the number of boundary components), but we also take a tensor
form $x_{1}\otimes\cdots\otimes x_{k}$ as invariant values, regarding it as a
representative of elements of $S({\mathbb{Z}}[A])$, so that tensors of
permutations of $x_{j}$s are considered equal. Then we set

$\Psi_{\bar{\psi}}(S)=\sum_{\mathcal{C}}(\Psi_{\bar{\psi}}^{1}(\mathcal{C},S),\ldots,\Psi_{\bar{\psi}}^{n}(\mathcal{C},S)),$

where each entry of this formal vector corresponds to one connected component
of $S$, and the sum refers to each entry of the vector component-wise.

Figure 22. Invariance under the Reidemeister type II move Figure 23.
Invariance under the cancelation move

The following is a special case of labeled homology defined in [ESZcascade]
restricted to RA cocycles.

######  Definition 5.6.

Let $X$ be a heap and $A$ an abelian group with multiplicative notation. Let
$\bar{\psi}=(\psi_{1},\ldots,\psi_{n})$ be a sequence of pairwise mutually
distributive RA $2$-cocycles of $X$ with coefficient group $A$.

We call $\bar{\psi}$ a coboundary if $\psi_{i}$ are coboundaries
simultaneously, that is, there is a 1-cochain $f\in C^{1}_{\rm SD}(X,A)$ such
that $\psi_{i}=\delta f$ for all $i=1,\ldots,n$. Two sequences
$\bar{\psi}=(\psi_{i})_{i=1}^{n}$ and
$\bar{\psi}^{\prime}=(\psi_{i}^{\prime})_{i=1}^{n}$ are called cohomologous if
$\bar{\psi}-\bar{\psi}^{\prime}$ is a coboundary. The equivalence classes by
this relation of cohomologous is called the cohomology class $[\bar{\psi}]$,
and they form an abelian group by component addition. The group of
cohomologous classes of $\bar{\psi}$ is denoted by $\bar{H}^{2,n}_{RA}(X,A)$.

The following is proved by arguments similar to those found in [CJKLS], with
the only difference that we need to take into consideration that different
cocycles may decorate different overpassing connected components.

######  Theorem 5.7.

The $2$-cocycle heap invariant is indeed an invariant of surface ribbons.
Moreover, a labeled 2-coboundary yields an integer multiple of the vector with
trivial tensors in its entries (with an appropriate number of entries, and an
appropriate number of tensor products in each entry). The cocycle invariant
$\Psi_{\bar{\psi}}(S)$ depends only on the labeled cohomology class
$[\bar{\psi}]\in\bar{H}^{2,n}_{\rm RA}(X,A)$.

###### Proof.

The invariance is proved by checking Reidemeister moves. More than one
component of overpassing ribbons appears only in the type III move, so that it
is delayed to the last, and the remaining cases are checked for a single
cocycle assigned on overpassing ribbons. The invariance under the Reidemeister
type II move (RII in Figure 4) follows from the reversibility condition of
2-cocycles as depicted in Figure 22. Observe that this is done for a single
under-arc and, therefore, it shows invariance of $\Psi_{\psi}(S)$ with respect
to each boundary component. The IH move does not involve cocycles and keeps
$\Psi$ unchanged. The cancelation move (CL in Figure 4) follows from the
reversibility condition and the equality $\psi(x,y,z)=\psi(x,z,zy^{-1}z)$
which is obtained by setting $y=u$ and $z=v$ in the 2-cocycle condition $(*)$
and changing variables. The cancelations of 2-cocycles under the CL moves are
depicted in Figure 23, where $+$ indicates pairs of terms that cancel. It is
clear that the canceling terms are paired with respect to different arcs and,
consequently, weights corresponding to different boundary components remain
unchanged. The invariance under YI move and IY move follow from the additivity
condition and reversibility condition, respectively, and depicted in Figures
24 and 25. Observe that Figure 24 refers to a single boundary component that
is slid beneath a fat vertex, while Figure 25 is obtained from the single arc
Reidemeister move II of Figure 22 relative to arcs $y$ and $u,v$.

Figure 24. Labels of arcs behind a trivalent vertex

Lastly we check the type III move, refer to Figure 3. In the figure, there are
two overpassing ribbons, the middle one $R_{1}$ labeled by $(x,y)$ at the top,
and the top one $R_{2}$ labeled by $(u,v)$. Assume that $R_{1}$ and $R_{2}$
belong to distinct connected components of the given surface ribbon $S$, and
assigned two cocycles $\psi_{1}$ and $\psi_{2}$, respectively. Then the ribbon
crossings are assigned cocycle values as indicated in the figure, and the
equality of the LHS and RHS is exactly the definition of mutual distributivity
in Section 2.3. If $R_{1}$ and $R_{2}$ belong to the same component, then the
equality follows from the original 2-cocycle condition. It follows that
$\Psi_{\psi}(S)$ is well defined.

Let $S$ be a surface ribbon diagram. Then colored boundary components
represent 2-cycles of $Z_{2}^{\rm SD}(X,{\mathbb{Z}})$. Let $S_{\mathcal{C}}$
be a colored diagram, and $b_{\mathcal{C}}$ be one of the boundary components
of $S_{\mathcal{C}}$. Let $\tau$ be a crossing of $b_{\mathcal{C}}$ where it
goes under a ribbon colored by $(y,z)$ in this order, changing the color from
$x$ to $xy^{-1}z$. If the cocycle assigned to the overpassing ribbon is
$\psi_{i}$, then the weight assigned to $\tau$ is $\psi_{i}(x,y,z)$. Suppose
$\bar{\psi}$ is a coboundary, then $\psi_{i}=\delta f$ for some $f\in
C^{1}_{\rm SD}(X,A)$, so that $\psi_{i}(x,y,z)=\delta
f(x,y,z)=f(x)-f(xy^{-1}z)$. Assign $f(x)$ to the arc colored by $x$ near
$\tau$, and $-f(xy^{-1}z)$ to the arc colored by $xy^{-1}z$. Since
$b_{\mathcal{C}}$ is a closed curve, these assigned values cancel at the both
ends of each arc. (This argument is similar to that of [CJKLS].) Hence the
tensor factor corresponding to $b_{\mathcal{C}}$ is trivial,
$e\otimes\cdots\otimes e$, for the multiplicative identity $e$ of $A$. Then
$\Psi_{\psi}(S)$ depends only on the cohomology class $[\bar{\psi}]$. In
particular, if $\Psi_{\psi}(S)$ is non-trivial, then $[\bar{\psi}]\neq 0$. ∎

Figure 25. Labels of arcs over a trivalent vertex

######  Remark 5.8.

We observe that if $\psi=\delta f$, then the integer that is the coefficient
of $e\otimes\cdots\otimes e$ is the number of colorings of $S$ by $X$. This
is, in fact, a direct consequence of the proof of the theorem, since for a
given $\delta f$ each coloring determines a copy of the trivial vector.

######  Example 5.9.

Let $S=B_{1}^{n}\natural B_{2}^{m}$ be the surface obtained by connecting $n$
single trivial handles and $m$ crossed handle pairs, as in Example 3.9. There
are $m+1$ components of boundary curves. Let $X$ be a heap and let $\psi$
denote an additive and reversible $2$-cocycle with coefficients in $A$. Then,
since each crossing contributes trivially to each boundary component, and the
coloring conditions are trivially satisfied, we have
$\Psi_{\psi}(S)=|X|^{n+m+1}\cdot e^{\otimes(m+1)}$, where $e$ is the unit of
$A$.

We consider an example where nontrivial contributions arise.

Figure 26. Two unlinked annuli linking a third annulus

######  Example 5.10.

Let $S=S_{1}\cup S_{2}\cup S_{3}$ denote the surface consisting of three
annuli where $S_{2}$ and $S_{3}$ link $S_{1}$ but are mutually unlinked, as
depicted in Figure 26. Let $X=\mathbb{Z}_{n}$ in multiplicative notation
generated by $\zeta$, $A=\mathbb{Z}_{n}$ also in multiplicative notation
generated by $g$, and let $\phi$ denote the RA $2$-cocycle of Example 4.18
which is separable, and therefore mutually distributive with the zero cocycle.
Let us take the triple of pairwise mutually distributive cocycles
$(0,\phi,0)$, where $S_{1}$ and $S_{3}$ are decorated by the zero cocycles,
and $\phi$ decorates $S_{2}$. Colors assigned to arcs are represented by $x,y$
for $S_{1}$, $z,w$ for $S_{2}$ and $u,v$ for $S_{3}$ as depicted. Observe that
the coloring conditions on $S_{2}$ and $S_{3}$ immediately imply that $x=y$.
Set $z=\zeta^{r}$, $w=\zeta^{s}$, $u=\zeta^{p}$ and $v=\zeta^{q}$. On $S_{1}$,
the coloring condition for the component labeled $x$ is $xz^{-1}wu^{-1}v=x$,
which is equivalent to $\zeta^{s-r+q-p}=1$, and the $y$ component has the same
condition. For each arbitrary choice of $r,s,p$ there exists a unique $q$ such
that $\zeta^{s-r+q-p}=1$. Therefore, also considering that $x=y$ can be chosen
freely, it follows that there are $n^{4}$ colorings.

From the definition of the cocycle invariant we have that, setting
$\bar{\phi}=(0,\phi,0)$, $\Psi_{\bar{\phi}}(S)$ is a sum of six tensor
products, the first two entries corresponding to the two boundary components
of $S_{1}$ and, similarly, the other entries corresponding to the boundary
components of $S_{2}$ and $S_{3}$. For each coloring, the Boltzmann weight on
the boundary components of $S_{2}$ and $S_{3}$ are always trivial, since
$\phi$ is non-degenerate and $x=y$. The Boltzmann weights on $S_{1}$ are
identical for both boundary components and are determined by
$z^{-1}w=\zeta^{s-r}$. The definition of $\psi=\sum_{i=0}^{n-1}i\psi_{i}$ is
written multiplicatively $\psi=\prod_{i=0}^{n-1}(\psi_{i})^{i}$, and
$\psi_{i}(x,\zeta^{r},\zeta^{s})=g^{i}$ if and only if $i=s-r$ for any $x$,
where we have that $g$ is a multiplicative generator of $A$. Hence
$\psi(x,z,w)=g^{i}$ for all $x$ and $z=\zeta^{r}$, $w=\zeta^{s}$ and $i=s-r$.
For each given $i$, there are $n$ choices for $r$ and $s$ such that $s-r=i$,
and for each choice of $(r,s)$, there are $n$ choices for $p$ and $q$ such
that $\zeta^{s-r+q-p}=1$, and independently there are $n$ choices for $x$.
Thus we obtain $\Psi_{\bar{\phi}}(S)=n^{3}\sum_{i=0}^{n-1}(g^{i})^{\otimes
2}\otimes e^{\otimes 4}$.

From Theorem 3.4 and the preceding example, we have the following.

######  Corollary 5.11.

We have $\bar{H}^{2,n}_{\rm RA}({\mathbb{Z}}_{m},{\mathbb{Z}}_{m})\neq 0$ for
all $m>1$ and $n\geq 3$.

We present below a realization result for surface ribbons with non-trivial
invariant.

Figure 27. Linking two rings

######  Proposition 5.12.

The following statements hold.

  * (A)

For every $n\geq 3$ and pairs $(g_{i},b_{i})$ with $g_{i}\geq 0$ and
$b_{i}\geq 0$ for each $i=1,\ldots,n$, there exists an $n$-component non-split
surface ribbon $S=S_{1}\cup\cdots\cup S_{n}$ with nontrivial cocycle invariant
for some group heap $X$ and coefficient group $A$, such that $S_{i}$ has genus
$g_{i}$ and $b_{i}$ boundary components.

  * (B)

Let $X$ be a heap, $A$ an abelian group and let
$\bar{\psi}=(\psi_{1},\ldots,\psi_{n})$ be an $n$-tuple of mutually
distributive (non-degenerate) cocycles. Suppose that $S=S_{1}\cup\cdots\cup
S_{n}$ is an $n$-component surface ribbon, where $S_{i}$ has genus $g_{i}$ and
$b_{i}$ boundary components, such that $\Psi_{\bar{\psi}}(S)$ is nontrivial.
Then there exists a surface ribbon with nontrivial invariant with $n^{\prime}$
connected components and $(g^{\prime}_{i},b^{\prime}_{i})$ for
$i=1,\ldots,n^{\prime}$ such that:

    * (i)

$n^{\prime}\geq n+2$, $g^{\prime}_{i}\geq g_{i}$, $b^{\prime}_{i}\geq b_{i}$
with $g^{\prime}_{n+1},g^{\prime}_{n+2}\geq 0$ and
$b^{\prime}_{n+1},b^{\prime}_{n+2}\geq 0$;

    * (ii)

$n^{\prime}\geq n$, $b_{i}^{\prime}\geq b_{i}$, and there exists $j$ such that
$g_{j}^{\prime}>g_{j}$, and for all $i\neq j$, $g_{i}^{\prime}\geq g_{i}$.

###### Proof.

Let $S=S_{1}\cup\cdots\cup S_{n}$ be a non-split surface ribbon with $n$
connected components. Let $X$ be a heap and $A$ an abelian group in general.
Suppose that $\Psi_{\bar{\psi}}(S)$ is nontrivial for some choice of $n$
decorating mutually distributive separable cocycles
$\bar{\psi}=(\psi_{1},\ldots,\psi_{n})$. Since $S$ is non-split, none of the
components is a disk, hence $H_{1}(S_{i})\neq 0$ so that there is a nontrivial
handle for all $i$. Let us assume further the condition that:

(**) There is a coloring ${\mathcal{C}}$ of $S$ such that it is monochromatic
on $S_{j}$ and $\Psi_{\bar{\psi}}(S)$ is non-trivial.

Let $S^{\prime}=S\cup R_{1}\cup\cdots\cup R_{m}$ be the surface ribbon
obtained from $S$ by linking annular rings $R_{1},\ldots,R_{m}$ on $S_{j}$ (as
$S_{2}$ and $S_{3}$ linking $S_{1}$ in Figure 26). Let
${\mathcal{C}}^{\prime}$ be a coloring of $S^{\prime}$ obtained by extending
${\mathcal{C}}$ with monochromatic colors on $R_{1},\ldots,R_{m}$. Let
$\bar{\psi}^{\prime}$ be obtained from $\bar{\psi}$ by appending $m$ zero
cocycles decorating each ribbon ring $R_{k}$. Then we have that the invariant
$\Psi_{\bar{\psi}^{\prime}}(S^{\prime})$ is nontrivial. It follows that for
each $k\geq n$ there is a non-split surface ribbon with $k$ connected
components whose corresponding cocycle invariant is nontrivial.

Let $S=S_{1}\cup\cdots\cup S_{n}$ be a surface with nontrivial cocycle
invariant satisfying the same conditions as above. Let us denote by
$(g_{i},b_{i})$ the genus and the number of boundary components. By adding
trivial bands to $S_{i}$ as in Figure 9 (A), we can increase the number of
connected components $b_{i}$ of $S_{i}$ arbitrarily. Similarly, by attaching
trivial torus band pairs as in Figure 9 (B), we can increase the genus $g_{i}$
arbitrarily. Applying Remark 5.13 we see that in both cases the invariant does
not change and, therefore, it is nontrivial under either procedure.

The surface of Example 5.10 satisfies the required condition (**) with $n=3$
and $X={\mathbb{Z}}_{\ell}=A$. Moreover, for $i=1,2,3$ we have that
$(g_{i},b_{i})=(0,2)$. From the preceding argument, then, it follows that
there exists a non-split $k$-component surface ribbon with nontrivial cocycle
invariant for every prescribed choice of $n\geq 3$ and for each choice of $n$
pairs $(g_{i},b_{i})$ with $g_{i}\geq 0$ and $b_{i}\geq 2$. This completes the
proof of (A).

Let us now prove (B). For the statement of (i), as before, we have that $S$ is
nontrivial and at least one connected component contributes a non-trivial
value to $\Psi_{\bar{\psi}}(S)$. Suppose this is $S_{j}$. Since $S_{j}$ has a
handle, we add two ribbon surfaces as in Figure 27 and denote the resulting
surface by $S^{\prime}$. Let us decorate both added surfaces by some (non-
degenerate by hypothesis) $\psi_{i}$. Denote $\bar{\psi}^{\prime}$ the
$(n+2)$-tuple obtained by adding two $\psi_{i}$ to $\bar{\psi}$. Since
$\Psi_{\bar{\psi}}(S)$ is nontrivial, there is a coloring whose corresponding
Boltzmann weight is nontrivial. Let us indicate by $x,y$ the corresponding
colors of the arcs of $S_{j}$. Then, taking in Figure 27 $z=w=u=1$ and
$v=x^{-1}y$, we obtain a coloring of $S^{\prime}$, whose associated Boltzmann
weight is nontrivial. Since this summand in the cocycle invariant is not
canceled by other weights, it follows that
$\Psi_{\bar{\psi}^{\prime}}(S^{\prime})$ is nontrivial. Observe that
$S^{\prime}$ has $n+2$ connected components, it has $(g_{i},b_{i})$ unchanged
with respect to $S$ for all $i=1,\ldots,n$, and it has $g_{n+1}=g_{n+2}=0$,
$b_{n+1}=b_{n+2}=2$. Moreover, it satisfies the hypothesis $(**)$ of part (A),
with respect to the $(n+1)^{\rm st}$ component. We can apply part (A) to
complete the proof of (i). To prove part (ii), let $S_{j}$ be a connected
component of $S$ with a nontrivial handle. Observe that we can add to $S_{j}$
a torus band pair where the rightmost foot is monochromatic, augmenting the
genus of $S$ by one unit. Now we can proceed as in (A) to complete. ∎

######  Remark 5.13.

Let us now consider the cocycle invariant under boundary connected sum. Let
$S_{1}=S_{1}^{1}\cup\cdots\cup S_{1}^{n}$ and $S_{2}=S_{2}^{1}\cup\cdots\cup
S_{2}^{m}$, be surface ribbons with cocycle invariants
$\Psi_{\bar{\psi}}(S_{1})$ and $\Psi_{\bar{\phi}}(S_{2})$, respectively, for
some additive and reversible cocycle tuples of mutually distributive cocycles
$\bar{\psi},\bar{\phi}\in\bar{Z}^{2}_{\rm RA}(X,A)$ where $X$ is a heap, and
$A$ is an abelian group. Suppose that $\psi_{r}=\phi_{s}$, and that the
corresponding connected component $S^{r}_{1}$ has $b$ boundary components,
while $S^{s}_{2}$ has $b^{\prime}$. When we perform the boundary connected sum
of $S_{1}$ and $S_{2}$, with respect to $S_{1}^{r}$ and $S_{2}^{s}$, this
construction is applied to two boundary components, which we assume being the
$i^{\rm th}$ and $j^{\rm th}$ ones for $S^{r}_{1}$ and $S^{s}_{2}$,
respectively. In this situation, the connected components of $S_{1}\natural
S_{2}$ are ordered as $(S_{1}^{r}\natural
S_{2}^{s},S_{1}^{1},\ldots,\hat{S}_{1}^{i},\ldots,S_{1}^{n},S_{2}^{1},\ldots,\hat{S}_{2}^{j},\ldots
S_{2}^{m})$, where the symbol $\hat{\ }$ indicates omission of the component.
The cocycles decorating the connected components are arranged in the tuple
$\bar{\psi}\cdot_{rs}\bar{\phi}:=(\psi_{i},\psi_{1},\ldots,\hat{\psi}_{i}\ldots,\psi_{n},\phi_{1},\ldots,\hat{\phi}_{i}\ldots,\phi_{m})$,
where $\psi_{i}=\phi_{j}$ by assumption. Observe that the cocycle invariant
associated to each connected component different from $S_{1}^{r}$ and
$S_{2}^{s}$ remains unaltered from this procedure. Therefore, we can focus on
the computation of the tensor component relative to $S_{1}^{r}\natural
S_{2}^{s}$. For simplicity, we omit referring to the components that are
unchanged, and for $x\in X$, we denote by $\mathcal{C}_{1}(x)$, the colorings
of $S_{1}^{r}$, and similarly for $S_{2}^{s}$, where $x$ is the color assigned
to the arc $b_{1}$ and $b_{2}$ of Figure 12 (A). Certain choices of $x$ might
not admit colorings $\mathcal{C}_{i}(x)$ for either value of $i$, depending on
the surface ribbon that is being considered. Let us define an element of the
symmetric algebra $S({\mathbb{Z}}[A])$ as follows. If we have that

$\Psi^{r}_{\bar{\psi}}(S_{1})=\sum_{\mathcal{C}_{1}}\otimes_{t}\prod_{k_{1}}B_{x_{t}(k_{1})}(\mathcal{C}_{1},\tau_{t}(k_{1}))\quad{\rm
and}\quad\Psi^{s}_{\bar{\phi}}(S_{2})=\sum_{\mathcal{C}_{2}}\otimes_{l}\prod_{k_{2}}B_{x_{l}(k_{2})}(\mathcal{C}_{2},\tau_{l}(k_{2}))$

we set

$\displaystyle\Psi^{r}_{\bar{\psi}}(S_{1})\cdot_{ij}\Psi^{s}_{\bar{\phi}}(S_{2})$
$\displaystyle:=$
$\displaystyle\sum_{x}\sum_{\mathcal{C}_{1}(x),\mathcal{C}_{2}(x)}\left[\prod_{k_{1}}B_{x_{i}(k_{1})}(\mathcal{C}_{1}(x),\tau_{i}(k_{1}))\cdot\prod_{\ell}B_{x_{j}(k_{2})}(\mathcal{C}_{2}(x),\tau_{j}(k_{2}))\right]$
$\displaystyle\otimes_{t\neq
i}\prod_{k_{1}}B_{x_{t}(k_{1})}(\mathcal{C}_{1}(x),\tau_{t}(k_{1}))\otimes_{l\neq
j}\prod_{k_{2}}B_{x_{l}(k_{2})}(\mathcal{C}_{2}(x),\tau_{l}(k_{2}))),$

where $\tau$’s and $x_{t}(k_{1})$, $x_{l}(k_{2})$ follow the same conventions
of Definition 5.5. In other words,
$\Psi^{r}_{\bar{\psi}}(S_{1})\cdot_{ij}\Psi^{s}_{\bar{\phi}}(S_{2})$ is
obtained from the cocycle invariants by juxtaposing the tensors corresponding
to all the boundary components different from $i$ and $j$, while these latter
entries are multiplied together to give a single entry in the tensor product.
The sum runs over all colorings $\mathcal{C}_{1}$ of $S^{r}_{1}$ and
$\mathcal{C}_{2}$ of $S^{s}_{2}$ assigning the same value $x$ to the arcs
$b_{1}$ and $b_{2}$, and then $x$ is taken over all elements of $X$. By
convention, if such a coloring does not exist, the corresponding summand is
zero.

A coloring of a ribbon surface $S$ by a heap $X$ is the same as a heap
morphism from the fundamental heap $h(S)$ to $X$. Applying Proposition 3.13 we
see that morphisms from the free product of the reduced heaps of $S_{1}$ and
$S_{2}$ induce morphisms of $h(S_{1}\natural S_{2})$, and therefore colorings
of $S_{1}\natural S_{2}$ by $X$. However, there may be colorings that do not
arise in this way. They are colorings such that the arcs corresponding to
$b_{1}$ and $b_{2}$ have distinct colorings, that correspond to
$x_{1}^{\prime}$ and $x_{2}^{\prime}$ in the proof of Proposition 3.13, that
do not come from colorings of $S_{1}$ and $S_{2}$, as discussed in the proof
below in more details. We say that these are the residual colorings of
$S_{1}\natural S_{2}$. They are characterized by the fact that they do not
factor through the free product $\hat{h}(S_{1})*\hat{h}(S_{2})$. We examine
the (hypothetical) invariant factors if there is a non-empty residual
colorings.

Let $S_{1}$ and $S_{2}$ be as above, and let $\Psi^{r}_{\bar{\psi}}(S_{1})$
and $\Psi^{s}_{\bar{\phi}}(S_{2})$ be their cocycle invariants componentes
relative to $S_{1}^{r}$ and $S_{2}^{s}$, respectively. Suppose that
$S_{1}\natural S_{2}$ is the boundary connected sum along the $i^{\rm th}$
boundary component of $S^{r}_{1}$ and the $j^{\rm th}$ boundary component of
$S^{s}_{2}$. Then we have

$\Psi^{k}_{\bar{\psi}\cdot_{rs}\bar{\phi}}(S_{1}\natural
S_{2})=\begin{cases}\Psi^{r}_{\bar{\psi}}(S_{1})\cdot_{ij}\Psi^{s}_{\bar{\phi}}(S_{2})+\sum_{\mathcal{C}^{*}}\otimes_{r=1}^{b+b^{\prime}-1}\prod_{d}B_{x_{r}(d)}(\mathcal{C}^{*},\tau_{r}(d))\
\ {\rm if}\ \ k=1\\\ \Psi^{k-1}_{\bar{\psi}}(S_{1})\ \ {\rm if}\ \ 1\neq k\leq
i-1\\\ \Psi^{k}_{\bar{\psi}}(S_{1})\ \ {\rm if}\ \ i\leq k\leq n\\\
\Psi^{k}_{\bar{\phi}}(S_{2})\ \ {\rm if}\ \ n+1\leq k\leq n+j-1\\\
\Psi^{k-1}_{\bar{\phi}}(S_{2})\ \ {\rm if}\ \ n+j\leq k\leq n+m\end{cases}$

where $\mathcal{C}^{*}$ denotes the residual colorings of $S_{1}\natural
S_{2}$, and the ordering of the boundary components is as described above.

Let us now prove this claim. As observed above, the invariants relative to
connected components different from $S_{1}^{r}$ and $S_{2}^{s}$ remain
unchanged, with the only difference being the numbering of the connected
component they refer to. This is the same as in the formula stated above. Let
us consider $S^{r}_{1}\natural S^{s}_{2}$ as in Figure 12 (B), where $b_{1}$
and $b_{2}$ of the figure correspond to the $i^{\rm th}$ boundary component of
$S_{1}$ and the $j^{\rm th}$ boundary component of $S_{2}$, respectively. We
call these components $b^{1}_{i}$ and $b^{2}_{j}$. Colorings of $S_{1}\natural
S_{2}$ arise from the fundamental heap as follows. For each choice of $x\in
X$, one colors the upper arc of the connecting band of Figure 12 (B) with $x$
and, proceeding counterclockwise, goes along $b_{1}$. The arc outgoing from
$S^{r}_{1}$, which is the lower arc of the band in Figure 12 (B), has now
picked a color $x\Phi_{1}$ which will enter $S^{s}_{2}$. After proceeding
along $b^{2}_{j}$ in $S_{2}$ we have a color $x\Phi_{1}\Phi_{2}$ which forces
a condition $\Phi_{1}\Phi_{2}=1$. This is the extra relation of Proposition
3.13. Denoting $\mathcal{C}_{1}(x)$ and $\mathcal{C}_{2}(x)$ colorings of
$S_{1}$ and $S_{2}$ such that $x$ is the color assigned to the arcs of
$b^{1}_{i}$ and $b^{2}_{j}$, we see that they give rise to a coloring of
$S_{1}\natural S_{2}$ since $\mathcal{C}_{1}(x)$ and $\mathcal{C}_{2}(x)$
would both map $\Phi_{1}$ and $\Phi_{2}$ to $1$. These colorings factor
through the free product $\hat{h}(S_{1})*\hat{h}(S_{2})$ since $\Phi_{i}$ are
relators of $\hat{h}(S_{i})$. The contribution of these colorings to the
cocycle invariant is obtained by multiplying the tensor entries corresponding
to the boundary components $b^{1}_{i}$ and $b^{2}_{j}$. In fact, since they
are now connected through a band, they constitute a single boundary component
and ordering the arcs counterclockwise, as in the preceding considerations
regarding the colorings, one obtains a product of Boltzmann weights that has
all the crossings of $S_{1}$ first, and all the crossings of $S_{2}$
afterwards. The other boundary components are not modified by this procedure.
We obtain, letting $x$ vary in $X$, the term
$\Psi^{r}_{\bar{\psi}}(S_{1})\cdot_{ij}\Psi^{s}_{\bar{\phi}}(S_{2})$ in the
statement. The contribution given by colorings that do not map $\Phi_{i}$ to
$1$, but that map $\Phi_{1}\Phi_{2}$ to $1$, is due to the residual colorings
by definition. This is the second summand appearing in the formula above.

We observe that the residual colorings of a boundary connected sum of surfaces
are automatically trivial if $\Gamma$ of Proposition 3.13 is an isomorphism.
Therefore, if $\Gamma$ is monic, it would follow that the formula in Remark
5.13 (for the $r$ component) would reduce to the product of cocycle invariants
$\cdot_{ij}$. It is not clear, at this point, whether $\Gamma$ is always an
isomorphism or not.

Although it is beyond the scope of this paper, it is desirable to investigate
further properties of the fundamental heap and cocycle invariants of surface
ribbons, and relations to other invariants of surfaces.

Acknowledgements. MS was supported in part by NSF DMS-1800443. EZ was
supported by the Estonian Research Council via the Mobilitas Pluss scheme,
grant MOBJD679.

## References

  * [1]
Bar-NatanDrorFulmanJasonKauffmanLouis H.An elementary proof that all spanning
surfaces of a link are tube-equivalentJ. Knot Theory RamificationsJournal of
Knot Theory and its Ramifications77873–8791998@article{BFK, author = {Bar-
Natan, Dror}, author = {Fulman, Jason}, author = {Kauffman, Louis H.}, title =
{An elementary proof that all spanning surfaces of a link are tube-
equivalent}, journal = {J. Knot Theory Ramifications}, fjournal = {Journal of
Knot Theory and its Ramifications}, volume = {7}, number = {7}, pages =
{873–879}, year = {1998}}

  * [3]
Quandle cohomology and state-sum invariants of knotted curves and
surfacesCarterJ.
ScottJelsovskyDanielKamadaSeiichiLangfordLaurelSaitoMasahicoTransactions of
the American Mathematical Society355103947–39892003@article{CJKLS, title =
{Quandle cohomology and state-sum invariants of knotted curves and surfaces},
author = {Carter, J. Scott}, author = {Jelsovsky, Daniel}, author = {Kamada,
Seiichi}, author = {Langford, Laurel}, author = {Saito, Masahico}, journal =
{Transactions of the American Mathematical Society}, volume = {355}, number =
{10}, pages = {3947–3989}, year = {2003}}

  * [5]
CarterScottIshiiAtsushiSaitoMasahicoTanakaKokoroHomology for quandles with
partial group operationsPacific J. Math.Pacific Journal of
Mathematics2872017119–48@article{CIST, author = {Carter, Scott}, author =
{Ishii, Atsushi}, author = {Saito, Masahico}, author = {Tanaka, Kokoro}, title
= {Homology for quandles with partial group operations}, journal = {Pacific J.
Math.}, fjournal = {Pacific Journal of Mathematics}, volume = {287}, year =
{2017}, number = {1}, pages = {19–48}}

  * [7]
CarterJ. ScottLebedVictoriaYangSeung YeopA prismatic classifying
spaceNonassociative mathematics and its applicationsContemp.
Math.72143–68Amer. Math. Soc., Providence, RI2019@article{CLY, author =
{Carter, J. Scott}, author = {Lebed, Victoria}, author = {Yang, Seung Yeop},
title = {A prismatic classifying space}, booktitle = {Nonassociative
mathematics and its applications}, series = {Contemp. Math.}, volume = {721},
pages = {43–68}, publisher = {Amer. Math. Soc., Providence, RI}, year =
{2019}}

  * [9]
CarterJ. ScottSaitoMasahicoSatohShinRibbon concordance of surface-knots via
quandle cocycle invariantsJ. Aust. Math. Soc.801131–1472006@article{CSS,
author = {Carter, J. Scott}, author = {Saito, Masahico}, author = {Satoh,
Shin}, title = {Ribbon concordance of surface-knots via quandle cocycle
invariants}, journal = {J. Aust. Math. Soc.}, volume = {80}, number = {1},
pages = {131–147}, year = {2006}}

  * [11]
ClarkW. EdwinSaitoMasahicoQuandle identities and homologyKnots, links, spatial
graphs, and algebraic invariants, Contemp. Math., Amer. Math. Soc.,
Providence, RIContemp. Math.68923–35Amer. Math. Soc., Providence,
RI2017@article{CS, author = {Clark, W. Edwin }, author = {Saito, Masahico},
title = {Quandle identities and homology}, journal = {Knots, links, spatial
graphs, and algebraic invariants, Contemp. Math., Amer. Math. Soc.,
Providence, RI}, series = {Contemp. Math.}, volume = {689}, pages = {23–35},
publisher = {Amer. Math. Soc., Providence, RI}, year = {2017}}
ElhamdadiMohamedGreenMatthewMakhloufAbdenacerTernary distributive structures
and quandlesKyungpook Math. J.56201611–27ISSN 1225-6951Review
MathReviewsDocument@article{EGM, author = {Elhamdadi, Mohamed}, author =
{Green, Matthew}, author = {Makhlouf, Abdenacer}, title = {Ternary
distributive structures and quandles}, journal = {Kyungpook Math. J.}, volume
= {56}, date = {2016}, number = {1}, pages = {1–27}, issn = {1225-6951},
review = {\MR{3609661}}, doi = {10.5666/KMJ.2016.56.1.1}} Heap and ternary
self-distributive
cohomologyElhamdadiMohamedSaitoMasahicoZappalaEmanueleCommunications in
Algebra1–242021Taylor & Francis@article{ESZheap, title = {Heap and ternary
self-distributive cohomology}, author = {Elhamdadi, Mohamed}, author = {Saito,
Masahico}, author = {Zappala, Emanuele}, journal = {Communications in
Algebra}, pages = {1–24}, year = {2021}, publisher = {Taylor \& Francis}}
ElhamdadiMohamedSaitoMasahicoZappalaEmanueleHigher arity self-distributive
operations in cascades and their cohomologyJournal of Algebra and Its
Applications DocumentLinkhttps://doi.org/10.1142/S0219498821501164
@article{ESZcascade, author = {Elhamdadi, Mohamed}, author = {Saito,
Masahico}, author = {Zappala, Emanuele}, title = {Higher arity self-
distributive operations in Cascades and their cohomology}, journal = {Journal
of Algebra and Its Applications}, volume = { {}}, number = { {}}, year = {},
doi = {10.1142/S0219498821501164}, url =
{https://doi.org/10.1142/S0219498821501164 }, eprint =
{https://doi.org/10.1142/S0219498821501164 }}

  * [16]
FennRogerRourkeColinRacks and links in codimension twoJournal of Knot Theory
and Its Ramifications141992343–406@article{FR, author = {Fenn, Roger}, author
= {Rourke, Colin}, title = {Racks and Links in Codimension Two}, journal =
{Journal of Knot Theory and Its Ramifications}, volume = {1}, number = {4},
date = {1992}, pages = {343–406}}

  * [18]
GreenMatthew J.Generalizations of quandles and their cohomologiesPhD
dissertation, University of South Florida2018ISSN Review
Document@article{Green, author = {Green, Matthew J.}, title = {Generalizations
of Quandles and their cohomologies}, journal = {PhD dissertation, University
of South Florida}, volume = {{}}, date = {2018}, number = {{}}, pages = {{}},
issn = {{}}, review = {{}}, doi = {{}}}

  * [20]
IshiiAtsushiMoves and invariants for knotted handlebodiesAlgebr. Geom.
Topol.Algebraic \& Geometric Topology8200831403–1418@article{Ishii, author =
{Ishii, Atsushi}, title = {Moves and invariants for knotted handlebodies},
journal = {Algebr. Geom. Topol.}, fjournal = {Algebraic \& Geometric
Topology}, volume = {8}, year = {2008}, number = {3}, pages = {1403–1418}}

  * [22]
IshiiAtsushiMatsuzakiShosakuMuraoTomoA multiple group rack and oriented
spatial surfacesJ. Knot Theory RamificationsJournal of Knot Theory and its
Ramifications29202072050046, 20@article{IMM, author = {Ishii, Atsushi}, author
= {Matsuzaki, Shosaku}, author = {Murao, Tomo}, title = {A multiple group rack
and oriented spatial surfaces}, journal = {J. Knot Theory Ramifications},
fjournal = {Journal of Knot Theory and its Ramifications}, volume = {29}, year
= {2020}, number = {7}, pages = {2050046, 20}}

  * [24]
A classifying invariant of knots, the knot quandleJoyceDavidJournal of Pure
and Applied Algebra23137–651982Elsevier@article{Joyce, title = {A classifying
invariant of knots, the knot quandle}, author = {Joyce, David}, journal =
{Journal of Pure and Applied Algebra}, volume = {23}, number = {1}, pages =
{37–65}, year = {1982}, publisher = {Elsevier}}

  * [26]
LebedVictoriaQualgebras and knotted 3-valent graphsFund. Math.Fundamenta
Mathematicae23020152167–204@article{Lebed, author = {Lebed, Victoria}, title =
{Qualgebras and knotted 3-valent graphs}, journal = {Fund. Math.}, fjournal =
{Fundamenta Mathematicae}, volume = {230}, year = {2015}, number = {2}, pages
= {167–204}}

  * [28]
The betti numbers of some finite racksLitherlandR.A.NelsonSamJournal of Pure
and Applied Algebra1782187–2022003Elsevier@article{LitherlandNelson, title =
{The Betti numbers of some finite racks}, author = {Litherland, R.A.}, author
= {Nelson, Sam}, journal = {Journal of Pure and Applied Algebra}, volume =
{178}, number = {2}, pages = {187–202}, year = {2003}, publisher = {Elsevier}}

  * [30]
MatsuzakiShosakuA diagrammatic presentation and its characterization of non-
split compact surfaces in the 3-spherePreprint,
arXiv:1905.031592019@article{Matsu, author = {Matsuzaki, Shosaku}, title = {A
diagrammatic presentation and its characterization of non-split compact
surfaces in the 3-sphere}, journal = {Preprint, arXiv:1905.03159}, year =
{2019}}

  * [32]
Distributive groupoids in knot theoryMatveevSergei
VladimirovichMatematicheskii Sbornik161178–881982Russian Academy of Sciences,
Steklov Mathematical Institute of Russian …@article{Mat, title = {Distributive
groupoids in knot theory}, author = {Matveev, Sergei Vladimirovich}, journal =
{Matematicheskii Sbornik}, volume = {161}, number = {1}, pages = {78–88}, year
= {1982}, publisher = {Russian Academy of Sciences, Steklov Mathematical
Institute of Russian~…}}

  * [34]
McAteeJenelleSilverDaniel S.WilliamsSusan G.Coloring spatial graphsJ. Knot
Theory RamificationsJournal of Knot Theory and its
Ramifications1020011109–120@article{MSW, author = {McAtee, Jenelle}, author =
{Silver, Daniel S.}, author = {Williams, Susan G.}, title = {Coloring spatial
graphs}, journal = {J. Knot Theory Ramifications}, fjournal = {Journal of Knot
Theory and its Ramifications}, volume = {10}, year = {2001}, number = {1},
pages = {109–120}}

  * [36]
Local biquandles and niebrzydowski’s tribracket
theoryNelsonSamOshiroKanakoOyamaguchiNatsumiTopology and its
Applications258474–5122019Elsevier@article{NOO, title = {Local biquandles and
Niebrzydowski's tribracket theory}, author = {Nelson, Sam}, author = {Oshiro,
Kanako}, author = {Oyamaguchi, Natsumi}, journal = {Topology and its
Applications}, volume = {258}, pages = {474–512}, year = {2019}, publisher =
{Elsevier}}

  * [38]
On some ternary operations in knot theoryNiebrzydowskiMaciejFundamenta
Mathematicae2014225259–276@article{Nie1, title = {On some ternary operations
in knot theory}, author = {Niebrzydowski, Maciej}, journal = {Fundamenta
Mathematicae}, year = {2014}, volume = {225}, pages = {259–276}}

  * [40]
Homology of ternary algebras yielding invariants of knots and knotted
surfacesNiebrzydowskiMaciejAlgebraic & Geometric
Topology2052337–23722020Mathematical Sciences Publishers@article{Nie2, title =
{Homology of ternary algebras yielding invariants of knots and knotted
surfaces}, author = {Niebrzydowski, Maciej}, journal = {Algebraic \& Geometric
Topology}, volume = {20}, number = {5}, pages = {2337–2372}, year = {2020},
publisher = {Mathematical Sciences Publishers}}

  * [42]
@book{Rolf}

  * AUTHOR = Rolfsen, Dale, TITLE = Knots and links, NOTE = Mathematics Lecture Series, No. 7, PUBLISHER = Publish or Perish, Inc., Berkeley, Calif., YEAR = 1976, PAGES = ix+439,
  * [44]
Fundamental heap for framed links and ribbon cocycle invariantsauthor=Zappala,
EmanueleSaito, MasahicoPreprint, arXiv:2011.036842021@article{SZframedlinks,
title = {Fundamental heap for framed links and ribbon cocycle invariants},
author = {{Saito, Masahico} author={Zappala, Emanuele}}, journal = {Preprint,
arXiv:2011.03684}, year = {2021}}

  * [46]
SaitoMasahicoZappalaEmanueleBraided frobenius algebras from certain hopf
algebrasarXiv::2102.09593, to appear in: Journal of Algebra and its
Applications2021@article{SZbraidedFrob, author = {Masahico Saito}, author =
{Emanuele Zappala}, title = {Braided Frobenius algebras from certain Hopf
algebras }, journal = {arXiv::2102.09593, to appear in: Journal of Algebra and
its Applications}, year = {2021}}

  * [48]
@article{Scott-Wall}

  * AUTHOR = Scott, Peter, author=Wall, Terry, TITLE = Topological methods in group theory, NOTE = London Mathematical Society Lecture Note Series, No. 36, Homological Group Theory, PUBLISHER = Cambridge University Press, YEAR = 1979, PAGES = 137–203,
  * [50]
Groups defined by periodic paired relationsVinbergErnest BorisovichSbornik:
Mathematics188912691997IOP Publishing@article{Vin, title = {Groups defined by
periodic paired relations}, author = {Vinberg, Ernest Borisovich}, journal =
{Sbornik: Mathematics}, volume = {188}, number = {9}, pages = {1269}, year =
{1997}, publisher = {IOP Publishing}} Non-associative algebraic structures in
knot theoryZappalaEmanuelePh.D. dissertation, University of South
Florida2020@article{EZ, title = {Non-Associative Algebraic Structures in Knot
Theory}, author = {Zappala, Emanuele}, journal = {Ph.D. dissertation,
University of South Florida}, year = {2020}}

  * [53]