# Some properties of grid homology for MOY graphs

Hajime Kubota

###### Abstract.

Grid homology for MOY graphs is immediately defined from grid homology for
transverse spatial graphs developed by Harvey and O’Donnol [4]. We studied
some properties of grid homology for MOY graphs such as the oriented skein
relation, the effect of contracting an edge, and the effect of uniting
parallel edges with the same orientation.

##  1\. Introduction

Grid homology is the combinatorial link Floer homology developed by Manolescu,
Ozsváth, Szabó, Thurston [6]. Knot/link Floer homology is widely studied
because it has various applications to low-dimensional topology. For example,
knot Floer homology detects the genus of a knot [9]. Knot Floer homology is
called the categorification of the Alexander polynomial since the graded Euler
characteristic coincides with the Alexander polynomial.

Harvey and O’Donnol extended grid homology to transverse spatial graphs [4].
Roughly, a transverse spatial graph is a spatial graph with a separation for
incoming edges and outgoing edges at each vertex. For a transverse spatial
graph $f\colon G\to S^{3}$, Harvey and O’Donnol defined the extended grid
homology $HFG^{-}(f)$ as a relatively bigraded
$\mathbb{F}[U_{1},\dots,U_{V}]$-modules and $\widehat{HFG}(f)$ as a relatively
bigraded $\mathbb{F}$-vector space, where $V$ is the number of vertices of $f$
and $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$. A $relatively\ bigraded$ means that a
module or vector space with absolute $\mathbb{Z}$-valued Maslov grading and
relative $H_{1}(S^{3}-f(G))$-valued Alexander grading. They showed that the
hat version of their grid homology is the combinatorial version of sutured
Floer homology [4, Theorem 6.6]. As a corollary, the graded Euler
characteristic of their hat version coincides with the torsion invariant
$T(E(f),\gamma(f))\in\mathbb{Z}[H_{1}(E(f))]$ of Friedl, Juhász, and Rasmussen
[3], where $(E(f),\gamma(f))$ is the sutured manifold determined by $f$.
Harvey and O’Donnol defined the Alexander polynomial for transverse spatial
graphs as the torsion invariant $T(E(f),\gamma(f))$.

Harvey and O’Donnol also mentioned that their grid homology is the same as
Heegaard Floer homology for some balanced bipartite graphs defined by Bao [1]
if the graphs have no sinks, sources, or cut edges. For a transverse spatial
graph $f$, we can take the balanced bipartite spatial graph $f^{\prime}$
uniquely. Because $f^{\prime}$ is also a transverse spatial graph, we can take
a (good) grid diagram $g^{\prime}$ representing $f^{\prime}$. Bao pointed out
that if we regard $g^{\prime}$ as a special Heegaard diagram, her chain
complex coincides with the grid chain complex of $g^{\prime}$.

Roughly, transverse spatial graphs with a certain edge coloring which is
called balanced coloring are called MOY graphs. Combining balanced coloring
with the extended grid homology for transverse spatial graphs, we can quickly
define grid homology for MOY graphs, we denoted by $HF^{-}(f,\omega)$ or
$\widehat{HF}(f,\omega)$. The Alexander grading of our grid homology takes
values in $\mathbb{Z}$ because we can take the homomorphism $\omega\colon
H_{1}(S^{3}-f(G))\to\mathbb{Z}$ determined by the balanced coloring of $f$. We
observed grid homology for MOY graphs and its graded Euler characteristic
$\Delta_{(f,\omega)}(t)\in\mathbb{Z}[t^{\pm 1}]$.

Our main goal is to study some properties of grid homology for MOY graphs such
as the oriented skein relation, the effect of contracting an edge, and the
effect of uniting parallel edges with the same orientation.

###  1.1. MOY graphs

An oriented spatial graph $f$ is the embedding image of an abstract directed
graph in $S^{3}$. A transverse spatial graph is an oriented spatial graph such
that there is a small disk $D\subset S^{3}$ that separates the incoming edges
and the outgoing edges for each vertex (Figure 1). In this paper, we always
assume that every transverse spatial graph is sourceless and sinkless, which
means that each vertex has both incoming edges and outgoing edges.

######  Definition 1.1.

We denote the set of edges of $f$ by $E(f)$ and the set of vertices of $f$ by
$V(f)$. For a vertex $v$, let $\mathrm{In}(v)$ be the set of edges incoming to
$v$ and $\mathrm{Out}(v)$ be the set of edges outgoing to $v$.

  1. (1)

A balanced coloring $\omega$ for $f$ is a map $E(f)\to\mathbb{Z}$ satisfying
$\sum_{e\in\mathrm{In}(v)}\omega(e)=\sum_{e\in\mathrm{Out}(v)}\omega(e)$ for
each $v\in V(f)$

  2. (2)

An MOY graph $(f,\omega)$ is a transverse spatial graph with a balanced
coloring.

######  Remark 1.2.

Mellor, Kong, Lewald, and Pigrish [7] define a balanced spatial graph as a
pair of a spatial graph and a balanced coloring, which is an MOY graph without
the transverse condition. It is confusing that Vance[11] and the author
paper[5] defined a balanced spatial graph, which is a transverse spatial graph
satisfying that the number of incoming edges equals the number of outgoing
edges at each vertex. Note that a balanced spatial graph of Vance and the
author is a special case of an MOY graph.

###  1.2. Main results

Let $W(i)$ be a two-dimensional graded vector space
$W(i)\cong\mathbb{F}_{0,0}\oplus\mathbb{F}_{-1,-i}$, where
$\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$. For a bigraded $\mathbb{F}$-vector space
$X$, $X\llbracket a,b\rrbracket$ denotes a shift of a bigraded
$\mathbb{F}$-vector space such that $X\llbracket
a,b\rrbracket_{d,s}=X_{d+a,s+b}$. Then,

$X\otimes W(i)\cong X\oplus X\llbracket 1,i\rrbracket.$

Figure 1. A small disk at a vertex of a transverse spatial graph Figure 2. An
oriented skein triple of MOY graphs. Edges with the same label correspond to
each other.

######  Definition 1.3.

Let $f,f_{+},f_{-},f_{0}$ be four transverse spatial graphs as in Figure 2 and
$\omega$ be a balanced coloring for $f$. We call a 4-tuple
$(f,f_{+},f_{0},\omega)$ or $(f,f_{-},f_{0},\omega)$ an oriented skein triple
of MOY graphs if the balanced colorings of $f_{+},f_{-},f_{0}$ are naturally
determined by $\omega$ and if
$\sum_{s=1}^{i}\omega(a_{s})=\sum_{s=1}^{j}\omega(b_{s})=\sum_{s=1}^{k}\omega(c_{s})=\sum_{s=1}^{l}\omega(d_{s})$.

The following theorem implies the existence of exact sequences of grid
homology $\widehat{HF}(f,\omega)$ for oriented skein triple of MOY graphs.

######  Theorem 1.4.

(1) Suppose that $(f,f_{+},f_{0},\omega)$ is an oriented skein triple of MOY
graphs as in Figure 2. Let $\omega_{+},\omega_{0}$ be balanced colorings for
$f_{+},f_{0}$ determined by $\omega$ and let $m=\sum_{s=1}^{i}\omega(a_{s})$.
Then, there is an exact triangle as absolute Maslov graded, relative Alexander
graded $\mathbb{F}$-vector spaces;

$\textstyle{\widehat{HF}(f_{0},\omega_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1,0)}$$\textstyle{\widehat{HF}(f_{+},\omega_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,0)}$$\textstyle{\widehat{HF}(f,\omega)\otimes
W(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1,0)}$

(2) Suppose that $(f,f_{-},f_{0},\omega)$ is an oriented skein triple of MOY
graphs as in Figure 2. Let $\omega_{-},\omega_{0}$ be balanced colorings for
$f_{-},f_{0}$ determined by $\omega$ and let $m=\sum_{s=1}^{i}\omega(a_{s})$.
Then, there is an exact triangle as absolute Maslov graded, relative Alexander
graded $\mathbb{F}$-vector spaces;

$\textstyle{\widehat{HF}(f_{-},\omega_{-})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1,0)}$$\textstyle{\widehat{HF}(f_{0},\omega_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,0)}$$\textstyle{\widehat{HF}(f,\omega)\otimes
W(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1,0)}$

######  Definition 1.5.

For a graph grid diagram $g$, let $\Delta_{g}(t)$ be the graded Euler
characteristic

$\Delta_{g}(t)=\sum_{d,s\in\mathbb{Z}}(-1)^{d}\mathrm{dim}_{\mathbb{F}}\widehat{HF}(g,\omega)_{d,s}\cdot
t^{s}.$

This is an invariant for transverse spatial graphs up to multiplication by
$\pm t^{\pm n}$ because $\widehat{HF}(g)$ is independent of the choice of $g$
representing $f$, up to shift of Alexander grading. We denote by
$\Delta_{(f,\omega)}(t)$.

######  Corollary 1.6.

(1) Let $(f,f_{+},f_{0},\omega)$ be an oriented skein triple of MOY graphs and
$m=\sum_{s=1}^{i}\omega(a_{s})$. Then, there is a pair of integers
$(m_{1},m_{2})$ such that

$t^{m_{1}}\cdot\Delta_{(f_{0},\omega_{0})}(t)-t^{m_{2}}\cdot\Delta_{(f_{+},\omega_{+})}(t)\doteq(1-t^{m})\cdot\Delta_{(f,\omega)}(t),$

where $\doteq$ means equal up to multiplication by $\pm t^{\pm n}$.

(2) Let $(f,f_{-},f_{0},\omega)$ be an oriented skein triple of MOY graphs and
$m=\sum_{s=1}^{i}\omega(a_{s})$. Then, there is a pair of integers
$(m_{1},m_{2})$ such that

$t^{m_{1}}\cdot\Delta_{(f_{0},\omega_{0})}(t)-t^{m_{2}}\cdot\Delta_{(f_{-},\omega_{-})}(t)\doteq(1-t^{m})\cdot\Delta_{(f,\omega)}(t),$

######  Remark 1.7.

We need a pair of integers $(m_{1},m_{2})$ because $\Delta_{(f,\omega)}(t)$ is
defined up to multiplication by $\pm t^{m}$, equivalently,
$\widehat{HF}(f,\omega)$ is an invariant as relativity Alexander graded vector
space.

Figure 3. Left: graphs in Theorem 1.8. Right: contracting an edge of Theorem
1.9

######  Theorem 1.8.

Let $(f,\omega)$ be an MOY graph. Suppose that $f$ has two parallel edges
$e_{1},e_{2}$ with the same orientation. Let $(f^{\prime},\omega^{\prime})$ be
an MOY graph obtained by replacing $e_{1},e_{2}$ with one edge whose weight is
$\omega(e_{1})+\omega(e_{2})$ and the orientation of the new edge is the same
as the orientation of $e_{1},e_{2}$ (See the left part of Figure 3). Let $V$
be the number of vertices of $f,f^{\prime}$. Then,

$\displaystyle HF^{-}(f,\omega)$ $\displaystyle\cong
HF^{-}(f^{\prime},\omega^{\prime}),$ $\displaystyle\widehat{HF}(f,\omega)$
$\displaystyle\cong\widehat{HF}(f^{\prime},\omega^{\prime}),$

as absolute Maslov graded, relative Alexander graded
$\mathbb{F}[U_{1},\dots,U_{V}]$-modules and as absolute Maslov graded,
relative Alexander graded $\mathbb{F}$-vector spaces, respectively.

######  Theorem 1.9.

Let $(f,\omega)$ be an MOY graph. Let $e$ be an edge of $f$ which is not a
loop. Suppose that at either of its endpoints, $e$ is the only outgoing edge
or incoming edge. Let $(f^{\prime},\omega^{\prime})$ be an MOY graph such that
$f^{\prime}$ is obtained from $f$ by contracting $e$ and $\omega^{\prime}$ is
naturally determined by $\omega$ (See the right part of Figure 3). Let $V$ be
the number of vertices of $f^{\prime}$. Then,

$\displaystyle HF^{-}(f,\omega)$ $\displaystyle\cong
HF^{-}(f^{\prime},\omega^{\prime}),$ $\displaystyle\widehat{HF}(f,\omega)$
$\displaystyle\cong\widehat{HF}(f^{\prime},\omega^{\prime})\otimes
W(\omega(e)).$

as absolute Maslov graded, relative Alexander graded
$\mathbb{F}[U_{1},\dots,U_{V}]$-modules and as absolute Maslov graded,
relative Alexander graded $\mathbb{F}$-vector spaces, respectively.

##  2\. Grid homology for MOY graphs

###  2.1. Background in bigraded chain complexes

######  Definition 2.1.

Let $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$.

  * •

A bigraded $\mathbb{F}[U_{1},\dots,U_{n}]$-module $M$ is an
$\mathbb{F}[U_{1},\dots,U_{n}]$-module which has a splitting
$M=\bigoplus_{d,s\in\mathbb{Z}}M_{d,s}$ as a $\mathbb{F}$-module so that
$V_{i}$ sends $M_{d,s}$ to $M_{d-2,s-\omega(i)}$ for all $i=1,\dots,n$, where
$\omega(i)$’s are some integers.

  * •

A bigraded $\mathbb{F}[U_{1},\dots,U_{n}]$-module homomorphism $f\colon
M\rightarrow M^{\prime}$ is a homomorphism between two bigraded
$\mathbb{F}[U_{1},\dots,U_{n}]$-modules which sends $M_{d,s}$ to
$M^{\prime}_{d,s}$ for all $d,s\in\mathbb{Z}$.

  * •

$\mathbb{F}[U_{1},\dots,U_{n}]$-module homomorphism $f\colon M\rightarrow
M^{\prime}$ is a homogeneous of degree $(m,t)$ if it sends $M_{d,s}$ to
$M^{\prime}_{d+m,s+t}$ for all $d,s\in\mathbb{Z}$.

  * •

$(C,\partial)$ is a bigraded chain complex over
$\mathbb{F}[U_{1},\dots,U_{n}]$ if $C$ is bigraded
$\mathbb{F}[U_{1},\dots,U_{n}]$-module and if
$\mathbb{F}[U_{1},\dots,U_{n}]$-module homomorphism $\partial\colon
C\rightarrow C$ satisfies $\partial\circ\partial=0$ and is homogeneous of
degree $(-1,0)$.

Throughout this paper, we often consider chain maps, chain homotopy
equivalences, etc. as bigraded $\mathbb{F}[U_{1},\dots,U_{n}]$-module
homomorphisms.

###  2.2. The definition of grid homology for MOY graphs

This section provides an overview of grid homology for MOY graphs. Roughly,
the Alexander grading of our grid homology is obtained by taking homomorphism
$\omega\colon H_{1}(S^{3}-f(G))\to\mathbb{Z}$ determined by the balanced
coloring and collapsing $H_{1}(S^{3}-f(G))$-valued Alexander grading of Harvey
and O’Donnol into $\mathbb{Z}$. So the almost definitions are the same as
those of Harvey and O’Donnol.

A planar graph grid diagram $g$ is a $n\times n$ grid of squares some of that
are decorated with an $X$\- or $O$\- (sometimes $O{}^{*}$-) markings with the
following conditions.

  1. (i)

There is just one $O$ or $O{}^{*}$ on each row and column.

  2. (ii)

There is at least one $X$ on each row and column.

  3. (iii)

$O$’s (or $O{}^{*}$’s) and $X$’s do not share the same square.

We denote the set of $O$-markings and $O^{*}$-markings by $\mathbb{O}$ and the
set of $X$-markings by $\mathbb{X}$ and use the labeling of markings as
$\\{O_{i}\\}_{i=1}^{n}$ and $\\{X_{j}\\}_{j=1}^{m}$.

Graph grid diagrams realize spatial graphs by connecting the $O$\- (or
$O^{*}$)- markings and $X$-markings by horizontal or vertical segments in each
row and column, and assuming that the vertical segments always cross above the
horizontal segments. $O^{*}$-markings correspond to vertices, $O$\- and
$X$-markings to interior of edges of a transverse spatial graph.

Harvey and O’Donnol showed that any two graph grid diagrams representing the
same spatial graph are connected by a finite sequence of the graph grid moves.
The graph grid moves are the following three moves$\colon$

  * •

Cyclic permutation (Figure 6) permuting the rows or columns cyclically.

  * •

Commutation’ (Figure 6) permuting two adjacent columns satisfying the
following condition; vertical line segments $\textrm{LS}_{1},\textrm{LS}_{2}$
on the torus such that (1) $\mathrm{LS}_{1}\cup\mathrm{LS}_{2}$ contains all
the $X$’s and $O$’s in the two adjacent columns, (2) the projection of
$\mathrm{LS}_{1}\cup\mathrm{LS}_{2}$ to a single vertical circle $\beta_{i}$
is $\beta_{i}$ , and (3) the projection of their endpoints,
$\partial(\mathrm{LS}_{1})\cup\partial(\mathrm{LS}_{2})$, to a single
$\beta_{i}$ is precisely two points. Permuting two rows is defined in the same
way.

  * •

(de-)stabilization’ (Figure 6) let $g$ be an $n\times n$ graph grid diagram
and choose an $X$-marking. Then $g^{\prime}$ is called a stabilization’ of $g$
if it is an $(n+1)\times(n+1)$ graph grid diagram obtained by adding a new row
and column next to the $X$-marking of $g$, moving the $X$-marking to next
column, and putting new one $O$-marking just above the $X$-marking and one
$X$-marking just upper left of the $X$-marking. The inverse of stabilization
is called destabilization.

Figure 4. Cyclic permutation and commutation’, gray lines are
$\mathrm{LS}_{1}$ and $\mathrm{LS}_{2}$

Figure 5. Stabilization’

Figure 6. An example of a state and a rectangle

A toroidal graph grid diagram is a graph grid diagram that we think it as a
diagram on the torus obtained by identifying edges in a natural way. We write
the horizontal and vertical circles, which separate squares as
$\boldsymbol{\alpha}=\\{\alpha_{i}\\}_{i=1}^{n}$ and
$\boldsymbol{\beta}=\\{\beta_{j}\\}_{j=1}^{n}$. We assume that every toroidal
diagram is oriented naturally.

A state $\mathbf{x}$ of $g$ is a bijection
$\boldsymbol{\alpha}\rightarrow\boldsymbol{\beta}$. We denote by
$\mathbf{S}(g)$ the set of states of $g$. We describe a state as $n$ points on
the graph grid diagram (Figure 6).

For $\mathbf{x,y}\in S$(g), a domain $\psi$ from $\mathbf{x}$ to $\mathbf{y}$
is a formal sum of the closure of squares, which is satisfying the following
conditions;

  * •

$\psi$ is divided by $\boldsymbol{\alpha}\cup\boldsymbol{\beta}$

  * •

$\partial(\partial_{\alpha}\psi)=\mathbf{y}-\mathbf{x}$ and
$\partial(\partial_{\beta}\psi)=\mathbf{x}-\mathbf{y}$, where
$\partial_{\alpha}\psi$ is the portion of the boundary of $\psi$ in the
horizontal circles $\alpha_{1}\cup\dots\cup\alpha_{n}$ and
$\partial_{\beta}\psi$ is the portion of the boundary of $\psi$ in the
vertical ones.

A domain $\psi$ is positive if the coefficient of any square is nonnegative.
Here, we always assume that any domain is positive. Let $\pi(\mathbf{x,y})$
denote the set of positive domains from $\mathbf{x}$ to $\mathbf{y}$.

Consider $\mathbf{x,y\in S}(g)$ satisfying that $|\mathbf{x\cap y}|=n-2$. An
rectangle $r$ from $\mathbf{x}$ to $\mathbf{y}$ is a domain that satisfies
that $\partial r$ is the union of four segments. A rectangle $r$ is empty if
$\mathbf{x}\cap\mathrm{Int}(r)=\mathbf{y}\cap\mathrm{Int}(r)=\phi$. Let
$\mathrm{Rect}^{\circ}(\mathbf{x,y})$ be the set of empty rectangles from
$\mathbf{x}$ to $\mathbf{y}$.

######  Definition 2.2.

For a graph grid diagram $g$ representing $f$ with balanced coloring $\omega$,
a weight $\omega_{g}\colon\mathbb{O}\cup\mathbb{X}\to\mathbb{Z}$ is a map
naturally determined by $\omega$ as follows;

  * •

$\omega_{g}(O_{i})=\omega(e)$ if $O_{i}$ corresponds to the interior of the
edge $e$.

  * •

$\omega_{g}(X_{j})=\omega(e)$ if $X_{j}$ corresponds to the interior of the
edge $e$.

  * •

$\omega_{g}(O_{i})=\sum_{e\in\mathrm{In}(v)}\omega(e)=\sum_{e\in\mathrm{Out}(v)}\omega(e)$
if $O_{i}$ corresponds to the vertex $v$, where $\mathrm{In}(v)$ is the set of
edges that go into $v$ and $\mathrm{Out}(v)$ is the set of edges that go out
of $v$.

We abbreviate $\omega_{g}$ to $\omega$ as long as there is no confusion.

The grid chain complex $(CF^{-}(g,\omega),\partial^{-})$ is a module over
$\mathbb{F}[U_{1},\dots,U_{n}]$ freely generated by $\mathbf{S}(g)$ and the
$U_{i}$’s are formal variables corresponding to the $O_{i}$’s in $g$. The
differential $\partial^{-}$ is defined as counting empty rectangles by

$\partial^{-}(\mathbf{x})=\sum_{\mathbf{y}\in\mathbf{S}(g)}\left(\sum_{\\{r\in\mathrm{Rect}^{\circ}(\mathbf{x,y})|r\cap\mathbb{X}=\phi\\}}U_{1}^{O_{1}(r)}\cdots
U_{n}^{O_{n}(r)}\right)\mathbf{y},$

where $O_{i}(r)=1$ if $r$ contains $O_{i}$ and $O_{i}(r)=0$ otherwise for
$i=1,\dots,n$.

There are two gradings for $CF^{-}(g)$, the Maslov grading and the Alexander
grading. A planar realization of toroidal diagram $g$ is a planar figure
obtained by cutting the toroidal diagram $g$ along $\alpha_{i}$ and
$\beta_{j}$ for some $i$ and $j$, and putting on
$[0,n)\times[0,n)\in\mathbb{R}^{2}$ in a natural way. For two points
$(a_{1},a_{2}),(b_{1},b_{2})\subset\mathbb{R}^{2}$, let
$(a_{1},a_{2})<(b_{1},b_{2})$ if $a_{1}<b_{1}$ and $a_{2}<b_{2}$. For two sets
of finitely points $A,B\subset\mathbb{R}^{2}$, let $\mathcal{I}(A,B)$ be the
number of pairs $a\in A,b\in B$ with $a<b$ and let
$\mathcal{J}(A,B)=(\mathcal{I}(A,B)+\mathcal{I}(B,A))/2$.

######  Definition 2.3.

For $\mathbf{x\in S}(g)$, the Maslov grading $M(\mathbf{x})$ and the Alexander
grading $A(\mathbf{x})$ are defined by

(2.1) $\displaystyle M(\mathbf{x})$
$\displaystyle=\mathcal{J}(\mathbf{x}-\mathbb{O},\mathbf{x}-\mathbb{O})+1,$
(2.2) $\displaystyle A(\mathbf{x})$
$\displaystyle=\mathcal{J}(\mathbf{x},\sum_{j=1}^{m}\omega(X_{j})\cdot
X_{j}-\sum_{i=1}^{n}\omega(O_{i})\cdot O_{i}).$

These two gradings are extended to the whole of $CF^{-}(g,\omega)$ by

(2.3) $\displaystyle M(U_{i})=-2,\ A(U_{i})=-\omega(O_{i})\ (i=1,\dots,n).$

Alexander grading is defined as counting pairs with multiplying weights of
markings.

######  Remark 2.4.

Our Alexander grading is obtained by taking the homomorphism
$H_{1}(S^{3}-f(G))\to\mathbb{Z}$, which sends the meridian of each edge to the
assigned integer determined by $\omega$, and the rest of the definitions are
the same as the definitions of Harvey and O’Donnol. So the invariance of our
grid homology is naturally inherited from the arguments of Harvey and
O’Donnol.

Also, note that the Alexander grading is not well-defined as a toroidal
diagram, however, relative Alexander grading
$A^{rel}(\mathbf{x,y})=A(\mathbf{x})-A(\mathbf{y})$ is well-defined. It is
easy to verify that the differential $\partial^{-}$ drops Maslov grading by
one and preserves Alexander grading, using the same argument such as
Proposition 4.18[4] and Proposition 3.7,[11]. So we can get this:

######  Proposition 2.5.

$(CF^{-}(g,\omega),\partial^{-})$ is an absolute Maslov graded, relative
Alexander graded chain complex.

Suppose that $O$-markings are labeled so that $O_{1},\dots,O_{V}$ are
$O^{*}$-markings and $O_{V+1},\dots,O_{n}$ are $O$-markings. Let $\mathcal{U}$
be the minimal subcomplex of $CF^{-}(g,\omega)$ containing
$U_{1}CF^{-}(g,\omega)\cup\dots\cup U_{V}CF^{-}(g,\omega)$.
$(\widehat{CF}(g,\omega),\widehat{\partial})$ is an absolute Maslov graded,
relative Alexander graded chain complex over $\mathbb{F}$-vector space
obtained by letting $\widehat{CF}(g,\omega)=CF^{-}(g,\omega)/\mathcal{U}$ and
$\widehat{\partial}$ be the map induced by $\partial^{-}$.

######  Definition 2.6.

$HF^{-}(g,\omega)$ and $\widehat{HF}(g,\omega)$ are the homology of
$(CF^{-}(g,\omega),\partial^{-})$ and
$(\widehat{CF}(g,\omega),\widehat{\partial})$, respectively.

Note that $HF^{-}(g,\omega)$ is regarded as an absolute Maslov graded,
relative Alexander graded $\mathbb{F}[U_{1},\dots,U_{V}]$-module, and
$\widehat{HF}(g,\omega)$ is an absolute Maslov graded and relative Alexander
graded $\mathbb{F}$-vector space.

######  Proposition 2.7.

$HF^{-}(g,\omega)$ and $\widehat{HF}(g,\omega)$ are invariants for transverse
balanced-colored spatial graphs.

###### Proof.

Our chain complexes are obtained from the chain complexes of Harvey and
O’Donnol by taking some homomorphism $H_{1}(S^{3}-f(G))\to\mathbb{Z}$. The
only difference between these two versions are domains of Alexander grading.
So the invariances are inherited. ∎

######  Remark 2.8.

$\widehat{CF}(g,\omega)$ can be regarded as absolute Maslov graded, relative
Alexander graded $\mathbb{F}$-vector space with basis
$\\{U_{V+1}^{k_{V+1}}\cdots U_{n}^{k_{n}}\cdot\mathbf{x}|k_{i}\geq
0,\mathbf{x\in S}(g)\\}$ and the differential

$\widehat{\partial}(\mathbf{x})=\sum_{\mathbf{y}\in\mathbf{S}(g)}\left(\sum_{\\{r\in\mathrm{Rect}^{\circ}(\mathbf{x,y})|r\cap\mathbb{X}=r\cap\mathbb{O}^{*}=\phi\\}}U_{V+1}^{O_{V+1}(r)}\cdots
U_{n}^{O_{n}(r)}\right)\mathbf{y},$

where $\mathbb{O}^{*}$ is the set of $O^{*}$-markings.

##  3\. The proofs for the main Theorem 1.4

######  Lemma 3.1 (Lemma A.3.10,[10]).

Let $C$, $C^{\prime}$, and $C^{\prime\prime}$ be three bigraded chain
complexes, and $f\colon C\to C^{\prime}$ and $g\colon C^{\prime}\to
C^{\prime\prime}$ are chain maps that are homogeneous of degrees $(a,p)$ and
$(b,q)$ respectively. Then there is a chain map
$\Phi\colon\mathrm{Cone}(g)\to\mathrm{Cone}(f)$ which is homogeneous of degree
$(-b-1,-q)$ and whose induced map on homology fits into an exact triangle

$\textstyle{H(\mathrm{Cone}(g))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-b-1,-q)}$$\textstyle{H(\mathrm{Cone}(g))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(b,q)}$$\textstyle{H(\mathrm{Cone}(g\circ
f))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,0)}$

where the pair of integers indicate shifts of bigradings.

Let $(f,f_{+},f_{0},\omega)$ be an oriented skein triple of MOY graphs. Choose
graph grid diagrams $(g,g_{+},g_{0})$, take a point $c_{+}$ in $g_{+}$ and
$c_{0}$ in $g_{0}$, and label the $X$-markings so that each $X$-marking
corresponds to the weight of each edge in Figure 2. Let
$m=\sum_{n=1}^{i}\omega_{X}(a_{n})=\sum_{n=1}^{j}\omega_{X}(j_{n})=\sum_{n=1}^{k}\omega_{X}(c_{n})=\sum_{n=1}^{l}\omega_{X}(d_{n})$.
Here $\omega_{+},\omega_{0}$ denotes the weights determined by $\omega$.

Figure 7. Graph grid diagrams representing $f$,$f_{+}$, and $f_{0}$

We will decompose the set of states $\mathbf{S}(g_{+})$ as the disjoint union
$\mathbf{I}(g_{+})\cup\mathbf{N}(g_{+})$, where
$\mathbf{I}(g_{+})=\\{\mathbf{x\in S}(g_{+})|c\in\mathbf{x}\\}$ and
$\mathbf{N}(g_{+})=\\{\mathbf{x\in S}(g_{+})|c\notin\mathbf{x}\\}$. Let
$I_{+}$ and $N_{+}$ be a subspace of $\widehat{CF}(g_{+},\omega_{+})$ as

$\displaystyle I_{+}$ $\displaystyle=\\{U_{V+1}^{k_{V+1}}\cdots
U_{n}^{k_{n}}\mathbf{x}\in\widehat{CF}(g_{+})|\mathbf{x}\in I(g_{+})\\},$
$\displaystyle N_{+}$ $\displaystyle=\\{U_{V+1}^{k_{V+1}}\cdots
U_{n}^{k_{n}}\mathbf{x}\in\widehat{CF}(g_{+})|\mathbf{x}\in N(g_{+})\\}$

This decomposition gives a decomposition of
$\widehat{CF}(g_{+},\omega_{+})=I_{+}\oplus N_{+}$ as a vector space. Then we
can write the differential on $\widehat{CF}(g_{+},\omega_{+})$ as

$\widehat{\partial}=\begin{pmatrix}\partial_{I}^{I}&0\\\
\partial_{I}^{N}&\partial_{N}^{N}\end{pmatrix},$

where, for instance, $\partial_{I}^{N}$ is a part of the differential such
that it is a map from $I_{+}$ to $N_{+}$. So $(N_{+},\partial_{N}^{N})$ is a
subcomplex of $(\widehat{CF}(g_{+},\omega_{+}),\widehat{\partial})$, and
$(I_{+},\partial_{I}^{I})$ is a quotient complex. By the definition of
algebraic cone, we see that
$\widehat{CF}(g_{+},\omega_{+})\cong\mathrm{Cone}(\partial_{I}^{N}\colon(I_{+},\partial_{I}^{I})\to(N_{+},\partial_{N}^{N}))$.

We can define $I_{0}$ and $N_{0}$ for $\widehat{CF}(g_{0})$ in the same way.
In this case, the differential can be written as

$\begin{pmatrix}\partial_{I}^{I}&\partial_{N}^{I}\\\
0&\partial_{N}^{N}\end{pmatrix},$

and
$\widehat{CF}(g_{0},\omega_{0})\cong\mathrm{Cone}(\partial_{N}^{I}\colon(I_{0},\partial_{I}^{I})\to(N_{0},\partial_{N}^{N}))$.

Consider the bijection $c_{+}\colon\mathbf{S}(g)\to\mathbf{I}(g_{+})$ defined
as $\mathbf{x}\mapsto\mathbf{x}\cup\\{c_{+}\\}$. Using 2.1 and 2.2, simple
calculations show that $c_{+}$ preserves Maslov grading and increases
Alexander grading by $m$. Also Consider the bijection
$c_{0}\colon\mathbf{I}(g_{0})\to\mathbf{S}(g)$ defined as
$\mathbf{x}\cup\\{c_{0}\\}\mapsto\mathbf{x}$. This map increases Maslov
grading by 1 and preserves Alexander grading.

The followings are easy.

######  Lemma 3.2 (Lemma 5.2.5,[10]).

  1. (1)

$c_{+}$ induces an isomorphism $\widehat{CF}(g,\omega)\cong I_{+}$ between
absolute Maslov graded, relative Alexander graded chain complexes.

  2. (2)

$c_{0}$ induces an isomorphism $I_{0}\cong\widehat{CF}(g,\omega)$ between
absolute Maslov graded, relative Alexander graded chain complexes.

###### Proof.

The bijection $c_{+}$ induces a homomorphism on chain complexes because there
is a natural bijection between $\\{U_{V+1},\dots,U_{n}\\}$ on $g$ and
$\\{U_{V+1},\dots,U_{n}\\}$ on $g_{+}$. It is easy to check that the induced
map by $c_{+}$ is a chain map. The induced map is an isomorphism because there
is a bijection between rectangles connecting states of $\mathbf{S}(g)$ and
states of $\mathbf{I}(g_{+})$.

$c_{0}$ induces an isomorphism in the same way. ∎

Consider the bijection $\alpha\colon\mathbf{N}(g_{+})\to\mathbf{N}(g_{0})$
defined as $\mathbf{x}\mapsto\mathbf{x}$. This map increases Maslov grading by
1 and preserves Alexander grading. The following lemma can be shown in the
same way as Lemma 3.2.

######  Lemma 3.3.

$\alpha$ induces an isomorphism $N_{+}\cong N_{0}$ between absolute Maslov
graded, relative Alexander graded chain complexes.

###### proof of theorem 1.4.

Now we have a diagram such as

$\textstyle{I_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{I}^{N}}$$\textstyle{N_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{N_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{N}^{I}}$$\textstyle{I_{0}.}$

Using Lemma 3.1, we can get the following exact triangle:

$\textstyle{H(\mathrm{Cone}(\partial_{N}^{I}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,0)}$$\textstyle{H(\mathrm{Cone}(\alpha\circ\partial_{I}^{N}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1,0)}$$\textstyle{H(\mathrm{Cone}(\partial_{N}^{I}\circ\alpha\circ\partial_{I}^{N}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,0)}$

As mentioned above,
$H(\mathrm{Cone}(\partial_{N}^{I}))\cong\widehat{HF}(g,\omega)$. By Lemma 3.3,
we have $H(\mathrm{Cone}(\alpha\circ\partial_{I}^{N}))\cong
H(\mathrm{Cone}(\partial_{I}^{N}))\llbracket-1,0\rrbracket\cong\widetilde{HF}(g_{+},\omega_{+})\llbracket-1,0\rrbracket$.

For any state $\mathbf{x}\in I_{+}$,
$(\partial_{N}^{I}\circ\alpha\circ\partial_{I}^{N})(\mathbf{x})=0$ because the
composite domain $r_{1}*r_{2}$ where $r_{1}$ is appear in $\partial_{N}^{I}$
and $r_{2}$ in $\partial_{I}^{N}$ must be an annulus, so some $X$-markings are
contained in $r_{1}$ or $r_{2}$.

So we have
$H(\mathrm{Cone}(\partial_{N}^{I}\circ\alpha\circ\partial_{I}^{N}))\cong
H(\mathrm{Cone}(0\colon I_{+}\to I_{0}))$. The composite map
$\partial_{N}^{I}\circ\alpha\circ\partial_{I}^{N}$ is homogeneous of degree
$(-1,0)$. So using lemma 3.2, we see that $H(\mathrm{cone}(0\colon I_{+}\to
I_{0}))\cong H(I_{+})\oplus H(I_{0})\cong\widehat{HF}(g,\omega)\llbracket
0,-m\rrbracket\oplus\widehat{HF}(g,\omega)\llbracket
1,0\rrbracket\cong\widehat{HF(g,\omega)}\otimes W(m)\llbracket
0,-m\rrbracket$. Then we get the following exact triangle:

$\textstyle{\widehat{HF}(g_{0},\omega_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1,0)}$$\textstyle{\widehat{HF}(g_{+},\omega_{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1,0)}$$\textstyle{\widehat{HF}(g,\omega)\otimes
W(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1,0)}$

Figure 8. Graph grid diagrams representing $f$,$f_{0}$, and $f_{-}$

Let $(f,f_{-},f_{0})$ be an oriented skein triple of MOY graphs. The exact
triangle of $(f,f_{0},f_{-})$ can be obtained in the same way;

First, choose the graph grid diagrams $(g,g_{0},g_{-})$ as in Figure 8.

Second, consider bijections $c^{\prime}_{0}\colon\mathbf{S}(g)\to I_{0}$,
$c_{-}\colon I_{-}\to\mathbf{S}(g)$, and $\beta\colon N_{0}\to N_{-}$ in the
same way as $c_{+}$, $c_{0}$, and $\alpha$, respectively. $c^{\prime}_{0}$ is
homogeneous of degree $(0,m)$, $c_{-}$ is $(1,0)$, and $\beta$ is $(1,-m)$.
These three maps induce isomorphisms on homology.

Finally, the exact triangle is obtained from the following diagram;

$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{I}^{N}}$$\textstyle{N_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{N_{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{N}^{I}}$$\textstyle{I_{-}.}$

∎

##  4\. The proofs for the main Theorem 1.8 and Theorem 1.9

First, we show Theorem 1.8 and Theorem 1.9 as corollaries of the following
proposition.

Figure 9. Graph grid diagrams in Proposition 4.1

######  Proposition 4.1.

Let $g,g^{\prime}$ are two graph grid diagrams as in Figure 9. $g^{\prime}$ is
obtained from $g$ by destabilization’-like move. The weight $\omega^{\prime}$
for $g^{\prime}$ is naturally determined by the weight $\omega$ for $g$.
Assume that the following conditions are satisfied:

  * •

$g$ is a $(n+1)\times(n+1)$ graph grid diagram.

  * •

None of the $X$-markings on the two rows along the horizontal circle
$\alpha_{i}$ line up vertically.

  * •

None of the $X$-markings on the two columns along the vertical circle
$\beta_{j}$ line up horizontally.

  * •

$A=O$ or $A=O^{*}$.

  * •

there is just one $X$-marking in either row or column containing $A$.

Then, there is a quasi-isomorphism between $CF^{-}(g,\omega)$ and
$CF^{-}(g^{\prime},\omega^{\prime})$ as absolute Maslov graded, relative
Alexander graded chain complexes over $\mathbb{F}[U_{1},\dots,U_{n}]$,

######  Remark 4.2.

Since $g$ is a $(n+1)\times(n+1)$ graph grid diagram, $CF^{-}(g,\omega)$ is
defined as an absolute Maslov graded, relative Alexander graded chain complex
over $\mathbb{F}[U_{1},\dots,U_{n+1}]$, where $U_{n+1}$ are related to the
$O$-marking at $A$. The homology of this chain complex is an invariant for
transverse spatial graphs when we regard $HF^{-}(g,\omega)$ as
$\mathbb{F}[U_{1},\dots,U_{V}]$-module, where $V$ is the number of vertices of
$f$.

Figure 10. Graph grid diagrams in the proof of Theorem 1.8

###### Proof of theorem 1.8.

First, we assume that the parallel two edges are connecting two different
vertices. Take a pair of a graph grid diagram and a weight $(g,\omega)$
representing $(f,\omega)$ such that the parallel two edges appear in the
$2\times 2$ block as in Figure 10. Then $g^{\prime}$ appearing in Figure 9
represents $f^{\prime}$ and the weight $\omega^{\prime}$ for $g^{\prime}$
naturally determined by $(g,\omega)$ represents the balanced coloring for
$f^{\prime}$. We get $HF^{-}(g,\omega)\cong
HF^{-}(g^{\prime},\omega^{\prime})$ by using Proposition 4.1 as $A=O$
directly. The quasi-isomorphism $CF^{-}(g,\omega)\to
CF^{-}(g^{\prime},\omega^{\prime})$ induces the quasi-isomorphism
$\widehat{CF}(g,\omega)=\frac{CF^{-}(g,\omega)}{U_{1}=\dots=U_{V}=0}\to\widehat{CF}(g^{\prime},\omega^{\prime})=\frac{CF^{-}(g^{\prime},\omega^{\prime})}{U_{1}=\dots=U_{V}=0}$,
so we get
$\widehat{HF}(g,\omega)\cong\widehat{HF}(g^{\prime},\omega^{\prime})$.

Figure 11. Proof of Theorem 1.8, when the parallel two edges are loops. The
graphs of the left part is obtained by contracting the edge $e_{0}$.

Next, we assume that the two parallel edges $e,e^{\prime}$ of $f$ are loops at
$v$. Take a new balanced transverse spatial graph $(f_{1},\omega_{1})$ such
that $f$ is obtained from $f_{1}$ by contracting an edge $e_{0}$ connecting
$v$ and the new vertex $v^{\prime}$ as in Figure 11, and $\omega_{1}$ is
naturally determined by $\omega$. By Theorem 1.9, $HF^{-}(f,\omega)\cong
HF^{-}(f_{1},\omega_{1})$ as $\mathbb{F}[U_{1},\dots,U_{V}]$-module, where
$U_{V}$ is related to $v$ and $U_{V+1}$ is related to $v^{\prime}$ (note that
$HF^{-}(f_{1})$ is a $\mathbb{F}[U_{1},\dots,U_{V+1}]$-module). Let
$(f_{2},\omega_{2})$ be a balanced transverse spatial graph obtained by
unifying the two parallel edges $e_{1},e^{\prime}_{1}$ determined by
$e,e^{\prime}$. Since $e_{1},e^{\prime}_{1}$ are not loops,
$HF^{-}(f_{1},\omega_{1})\cong HF^{-}(f_{2},\omega_{2})$ as
$\mathbb{F}[U_{1},\dots,U_{V+1}]$-module. $f^{\prime}$ is obtained from
$f_{2}$ by contracting $e_{0}$. Thus $HF^{-}(f_{2},\omega_{2})\cong
HF^{-}(f^{\prime},\omega^{\prime})$ as $\mathbb{F}[U_{1},\dots,U_{V}]$-module.
Note that $U_{V}$ and $U_{V+1}$ are chain homotopic because of Proposition
4.21, [4]. ∎

###### Proof of theorem 1.9.

If $e$ is the only incoming edge of $v$, take a graph grid diagram $g$
representing $f$ such that the $O^{*}$-marking related to $v$ is located at
$A$ as in Figure 9. If $e$ is the only outgoing edge of $v$, take a graph grid
diagram $g$ representing $f$ such that the $O^{*}$-marking related to $v$ is
located at $A$ as in Figure 9. We get $HF^{-}(g,\omega)\cong
HF^{-}(g^{\prime},\omega^{\prime})$ by using Proposition 4.1 as $A=O^{*}$
directly.

Next, we show the hat version carefully. Number the $O^{*}$-markings in
$g^{\prime}$ as $O_{1},\dots,O_{V}$. Since $g$ has one more $O^{*}$-marking,
number $O^{*}$-markings in $g$ as $O_{1},\dots,O_{V},O_{n+1}$. By the
definition of the hat chain complexes,
$\widehat{CF}(g,\omega)=\frac{CF^{-}(g,\omega)}{U_{1}=\dots=U_{V}=U_{n+1}=0}$
and
$\widehat{CF}(g^{\prime},\omega^{\prime})=\frac{CF^{-}(g^{\prime},\omega^{\prime})}{U_{1}=\dots=U_{V}=0}$.
The quasi-isomorphism $CF^{-}(g,\omega)\to CF^{-}(g^{\prime},\omega^{\prime})$
induces the quasi-isomorphism
$\frac{CF^{-}(g,\omega)}{U_{1}=\dots=U_{V}=0}\to\widehat{CF}(g^{\prime},\omega^{\prime})$.
Lemma 4.21,[4] shows that $U_{n+1}$ is null-homotopic as homogeneous of degree
$(-1,-m)$. Using Lemma 4.31,[4], we see that
$\widehat{HF}(g,\omega)\cong\widehat{HF}(g^{\prime},\omega^{\prime})\otimes
W(m)$ ∎

###### Proof of Proposition 4.1.

The basic idea is the same as the idea of the proof of stabilization
invariance (Section 5.2.3,[10]). The same argument works because we assume
that there is just one $X$-marking in either row or column containing $A$.
Take the intersection point $\alpha_{i}\cap\beta_{j}$ as $c$. Label the
markings around $c$ as in the right part of Figure 12 and the $O$-marking in
the column containing $X_{j+1}$ as $O_{n}$.

Figure 12. The graph grid diagram in the proof of Theorem 1.9 and the labels
of the markings around $c$

Now, we assume that there is just one $X$-marking in the $row$ containing $A$.

First, we introduce some notations like in the previous section. The set of
states $\mathbf{S}(g)$ are decomposed as the disjoint union
$\mathbf{I}(g)\cup\mathbf{N}(g)$. Let $N$ (respectively, $I$) be the submodule
of $CF^{-}(g,\omega)$ generated by $\mathbf{N}(g)$ (respectively,
$\mathbf{I}(g)$). As in the previous section, $N$ is a subcomplex of
$CF^{-}(g,\omega)$. Using the representation of the differential of
$CF^{-}(g,\omega)$ as a matrix, we see that
$CF^{-}(g,\omega)\cong\mathrm{Cone}(\partial_{I}^{N}\colon(I,\partial_{I}^{I})\to(N,\partial_{N}^{N}))$.
The bijection between $\mathbf{I}(g)$ and $\mathbf{S}(g^{\prime})$ induces the
isomorphism $c\colon I\to CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]$ as
$\mathbb{F}[U_{1},\dots,U_{n+1}]$-modules, where
$CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]=CF^{-}(g^{\prime},\omega^{\prime})\otimes_{\mathbb{F}[U_{1},\dots,U_{n}]}\mathbb{F}[U_{1},\dots,U_{n+1}]$.
By simple computations, $c$ is homogeneous of degree $(0,-A(\\{c\\}))$.

Second, we define the homotopy operators. Consider the map
$\mathcal{H}^{-}_{O}\colon I\to N$ and $\mathcal{H}^{-}_{X}\colon N\to I$ as

$\displaystyle\mathcal{H}^{-}_{O}(\mathbf{x})$
$\displaystyle=\sum_{\mathbf{y\in
N}(g)}\sum_{\\{r\in\mathrm{Rect}^{\circ}(\mathbf{x,y})|r\cap\mathbb{X}=\phi,O_{n+1}\in
r\\}}U_{1}^{O_{1}(r)}\cdots U_{n}^{O_{n}(r)}\cdot\mathbf{y},$
$\displaystyle\mathcal{H}^{-}_{X}(\mathbf{x})$
$\displaystyle=\sum_{\mathbf{y\in
I}(g)}\sum_{\\{r\in\mathrm{Rect}^{\circ}(\mathbf{x,y})|r\cap\mathbb{X}=\\{X_{j+1}\\}\\}}U_{1}^{O_{1}(r)}\cdots
U_{n}^{O_{n}(r)}\cdot\mathbf{y}.$

######  Lemma 4.3.

  * •

$\mathcal{H}^{-}_{X}$ is homogeneous of degree $(-1,-\omega(O_{n+1}))$ and
$\mathcal{H}_{O}$ is homogeneous of degree
$(1,\omega(X_{j+1}))=(1,\omega(O_{n+1}))$.

  * •

$\mathcal{H}_{X}$ and $\mathcal{H}_{O}$ are chain homotopy equivalences.

###### Proof.

The proof is the same as Lemma 5.2.19,[10]. Consider
$\mathcal{H}^{-}_{O,X}\colon N\to N$ as

$\mathcal{H}^{-}_{O,X}(\mathbf{x})=\sum_{\mathbf{y\in
N}(g)}\sum_{\\{r\in\mathrm{Rect}^{\circ}(\mathbf{x,y})|r\cap\mathbb{X}=\\{X_{j+1}\\},O_{n+1}\in
r\\}}U_{1}^{O_{1}(r)}\cdots U_{n}^{O_{n}(r)}\cdot\mathbf{y}.$

Then, it is sufficient to verify that
$\mathcal{H}^{-}_{X}\circ\mathcal{H}^{-}_{O}=\mathrm{Id}_{I}$ and
$\mathcal{H}^{-}_{O}\circ\mathcal{H}^{-}_{X}+\mathcal{H}^{-}_{O,X}\circ\partial_{N}^{N}+\partial_{N}^{N}\circ\mathcal{H}^{-}_{O,X}=\mathrm{Id}_{N}$.
The former equation is straightforward, and the latter equation holds by Lemma
5.2.19,[10]. ∎

We see the following commutative diagram:

$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{I}^{N}}$$\scriptstyle{c}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c\circ\mathcal{H}^{-}_{X}}$$\textstyle{CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]\llbracket
1,\omega(O_{n+1})\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]}$

where $U=U_{n+1}-U_{n}$ if there is just one $X$-marking in the column
containing $X_{j+1}$ and $U=U_{n+1}$ otherwise.

######  Lemma 4.4.

A commutative diagram of bigraded $\mathcal{R}$-module chain complexes

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\phi}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{D^{\prime}}$

induces a bigraded chain map $\Phi\colon\mathrm{Cone}(f)\to\mathrm{Cone}(g)$.
If $\phi$ and $\phi^{\prime}$ are quasi-isomorphisms, then $\Phi$ is a quasi-
isomorphism.

Using this lemma, we see that
$H(\mathrm{Cone}(\partial_{I}^{N}))\cong\mathrm{Cone}(U)$. Since
$H(\mathrm{Cone}(\partial_{I}^{N}))\cong HF^{-}(g,\omega)$, it is sufficient
to show that $\mathrm{Cone}(U)\cong HF^{-}(g^{\prime},\omega^{\prime})$.

By the definition of $CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]$, we get
$H(CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}])\cong
H(CF^{-}(g^{\prime},\omega^{\prime}))[U_{n+1}]$. Hence the injective map
$U\colon CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]\to
CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}])$ induces the injective map
$U\colon HF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]\to
HF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}])$ on homology. Consider a long
exact sequence of a mapping cone $\mathrm{Cone}(U)$, we get the following
short exact sequence of absolute Maslov, relative Alexander graded
$\mathbb{F}[U_{1},\dots,U_{n+1}]$-modules:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{HF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{HF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H(\mathrm{Cone}(U))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Therefore we see that
$\frac{HF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]}{\mathrm{Im}U}\cong
H(\mathrm{Cone}(U))$. It is natural that
$\frac{HF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]}{\mathrm{Im}U}\cong
HF^{-}(g^{\prime},\omega^{\prime})$ as absolute Maslov, relative Alexander
graded $\mathbb{F}[U_{1},\dots,U_{n}]$-modules. Combining these equations, we
get $HF^{-}(g,\omega)\cong HF^{-}(g^{\prime},\omega^{\prime})$ as absolute
Maslov, relative Alexander graded $\mathbb{F}[U_{1},\dots,U_{n}]$-modules. ∎

######  Remark 4.5.

If there is just one $X$-marking in the $column$ containing $A$, the same
result is shown by a small modification. Define $\mathcal{H}^{-}_{X}\colon
N\to I$ as

$\mathcal{H}^{-}_{X}(\mathbf{x})=\sum_{\mathbf{y\in
I}(g)}\sum_{\\{r\in\mathrm{Rect}^{\circ}(\mathbf{x,y})|r\cap\mathbb{X}=\\{X_{j}\\}\\}}U_{1}^{O_{1}(r)}\cdots
U_{n}^{O_{n}(r)}\cdot\mathbf{y}.$

$\mathcal{H}^{-}_{X}$ is homogeneous of degree
$(1,\omega(X_{j}))=(1,\omega(O_{n+1}))$. We see that $\mathcal{H}^{-}_{O}$ and
$\mathcal{H}^{-}_{X}$ are chain homotopy equivalences by using chain homotopy
$\mathcal{H}^{-}_{O,X}$ defined as

$\mathcal{H}^{-}_{O,X}(\mathbf{x})=\sum_{\mathbf{y\in
N}(g)}\sum_{\\{r\in\mathrm{Rect}^{\circ}(\mathbf{x,y})|r\cap\mathbb{X}=\\{X_{j}\\},O_{n+1}\in
r\\}}U_{1}^{O_{1}(r)}\cdots U_{n}^{O_{n}(r)}\cdot\mathbf{y}.$

Then, there is a commutative diagram

$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{I}^{N}}$$\scriptstyle{c}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c\circ\mathcal{H}^{-}_{X}}$$\textstyle{CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]\llbracket
1,\omega(O_{n+1})\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{CF^{-}(g^{\prime},\omega^{\prime})[U_{n+1}]}$

where $U=U_{n+1}-U_{n}$ if there is just one $X$-marking in the row containing
$X_{j}$ and $U=U_{n+1}$ otherwise.

##  5\. Applications

######  Lemma 5.1.

Let $(f,\omega)$ be an MOY graph and $\omega_{n}$ be a balanced coloring for
$f$ defined as $\omega_{n}(e)=n\cdot\omega(e)$ for any edge $e$. Then there is
a natural isomorphism $A_{n}\colon HF^{-}(f,\omega)\cong
HF^{-}(f,\omega_{n})$, where $A_{n}$ is a multiplication of Alexander grading
by $n$.

###### Proof.

There is a natural isomorphism $a_{n}\colon CF^{-}(g,\omega)\to
CF^{-}(g,\omega_{n})$ defined by $\mathbf{x}\mapsto\mathbf{x}$. Using (2.2),
$a_{n}$ preserves Maslov grading and multiples Alexander grading by $n$. ∎

######  Proposition 5.2.

Let $(f,\omega)$ be an MOY graph and $(f^{n},\omega^{n})$ be the MOY graph
obtained from $(f,\omega)$ by replacing each edge with $n$ parallel copies of
the edge with the same orientation and weight as the original edge. Then there
is a natural isomorphism $A_{n}\colon HF^{-}(f,\omega)\cong
HF^{-}(f^{n},\omega^{n})$, where $A_{n}$ is a multiplication of Alexander
grading by $n$.

###### Proof.

Combine Theorem 1.8 and Lemma 5.1. ∎

This Proposition is the partial extension of [7, Theorem 4.5].

##  6\. Further problems

Bao and Wu [2] researched an Alexander polynomial for MOY graphs. They defined
Alexander polynomial using Kauffman state sum $\langle D,\omega\rangle$ for a
given graph diagram $D$ of $f$ with balanced coloring $\omega$. They
summarized that the state sum $\langle D,\omega\rangle$ is an elaboration of
the Alexander polynomial of Harvey and O’Donnol. In their paper [2], Bao and
Wu defined the Alexander polynomial as $\langle
D,\omega\rangle/(t^{-1/2}-t^{1/2})^{V-1}$, where $V$ is the number of vertices
of the graph. Their Alexander polynomial of MOY graphs satisfies some
relations that are proposed by Murakami-Ohtsuki-Yamada [8]. So the natural
question is whether these relations are shown in the framework of grid
homology.

Balanced spatial graphs researched by Vance[11] and Kubota[5] is a special
case of MOY graphs. They defined the tau invariant and the Upsilon invariant
for balanced spatial graphs respectively. By the definitions, it is
straightforward that the tau invariant and the Upsilon invariant are extended
to MOY graphs. Then the question is what information about MOY graphs do these
invariants contain?

##  7\. Acknowledgement

I am grateful to my supervisor, Tetsuya Ito, for helpful discussions and
corrections. This work was supported by JST, the establishment of university
fellowships towards the creation of science technology innovation, Grant
Number JPMJFS2123.

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contain $T$, and let $\mathcal{C}_{2}$ be the set of all unital maximal
subrings that do not. Let $M\in\mathcal{C}_{2}$, and suppose that
$J\not\subseteq M$. Then, by Lemma 2.6, $M=S^{\prime}\oplus I$ for some
$S^{\prime}\in\mathscr{S}(R)$ and some maximal subideal $I\subseteq J$. But
then, $T\subseteq S^{\prime}\subseteq M$, a contradiction. So, each subring in
$\mathcal{C}_{2}$ contains $J$. Use a bar to denote passage to $R/J\cong S$.
By Lemma 8.7, $\overline{M}$ is Type $\Pi_{2}$ with $\overline{S}_{i}$ linked
to $\overline{S}_{j}$ for some $1\leqslant i<j\leqslant p$, and this is true
for every $M\in\mathcal{C}_{2}$.

From here, we will show that for each $a\in\widehat{T}\cong R/T$, there exists
$\alpha\in T$ such that $a+\alpha$ is in a maximal subring containing $T$. Let
$a\in\widehat{T}$, and write $a$ as $a=\alpha_{p}+b$, where $\alpha_{p}\in
S_{p}$ and $b\in\big{(}\bigoplus_{i=p+1}^{N}S_{i}\big{)}\oplus J$. Note that
$\alpha_{p}$ is equal to an element of $\mathbb{F}_{p}$. Let
$\alpha_{1},\ldots,\alpha_{p-1}$ be all of the elements of $\mathbb{F}_{p}$
that are not equal to $\alpha_{p}$. Considering $\alpha_{i}$ as an element of
$S_{i}$ for each $1\leqslant i\leqslant p-1$, we take
$\alpha:=\sum_{i=1}^{p-1}\alpha_{i}\in T$.

Certainly, $a+\alpha$ lies in some unital maximal subring $M$ of $R$. Suppose
that $M\in\mathcal{C}_{2}$. As noted above, $\overline{M}$ is Type $\Pi_{2}$
with $\overline{S}_{i}$ linked to $\overline{S}_{j}$ for some $1\leqslant
i<j\leqslant p$. Since the only automorphism of $\mathbb{F}_{p}$ is the
identity, this means that $\overline{\alpha_{i}}=\overline{\alpha_{j}}$.
However, this is impossible, because $\alpha_{1},\ldots,\alpha_{p}$ are all
distinct elements of $\mathbb{F}_{p}$. Thus, $a+\alpha$ lies in some subring
in $\mathcal{C}_{1}$.

To complete the proof, let $\mathcal{C}$ be a minimal cover of $R$ by unital
subrings, and let $\mathcal{A}=\mathcal{C}\cap\mathcal{C}_{1}$. Then, each
subring in $\mathcal{A}$ contains $T$, and their images in $R/T$ cover the
$R/T\cong\widehat{T}$. Hence,
$\sigma_{u}(R/T)\leqslant|\mathcal{A}|\leqslant\sigma_{u}(R)$, which
contradicts the fact that $R$ is $\sigma_{u}$-elementary. ∎

######  Proposition 8.9.

Let $R$ be as described prior to Lemma 8.5, and let $S$ be as in (8.6). Then,

$R=\Big{(}\bigoplus_{i=1}^{p}S_{i}\Big{)}\oplus\Big{(}\bigoplus_{1\leqslant
i,j\leqslant p}J_{ij}\Big{)}.$

###### Proof.

By Proposition 8.8, we know that $J_{ij}=\\{0\\}$ for all $1\leqslant
i\leqslant p$ and $p+1\leqslant j\leqslant N$. Decompose $R$ as $R=R_{1}\oplus
R_{2}$, where

$\displaystyle R_{1}$
$\displaystyle=\Big{(}\bigoplus_{i=1}^{p}S_{i}\Big{)}\oplus\Big{(}\bigoplus_{1\leqslant
i,j\leqslant p}J_{ij}\Big{)},\text{ and }$ $\displaystyle R_{2}$
$\displaystyle=\Big{(}\bigoplus_{i=p+1}^{N}S_{i}\Big{)}\oplus\Big{(}\bigoplus_{p+1\leqslant
i,j\leqslant N}J_{ij}\Big{)}.$

We claim that all unital maximal subrings of $R$ respect this decomposition.
That is, if $M$ is a unital maximal subring of $R$, then either $M=M_{1}\oplus
R_{2}$ with $M_{1}$ maximal in $R_{1}$, or $M=R_{1}\oplus M_{2}$ with $M_{2}$
maximal in $R_{2}$. When $J\not\subseteq M$, this can be shown just as in the
proof of Proposition 7.2. When $J\subseteq M$, $M=T\oplus J$ for some maximal
subring $T$ of $S$. In this case, the only way that $M$ would not respect the
direct sum decomposition of $R$ is if $T$ were Type $\Pi_{2}$ with $S_{i}$
linked to $S_{j}$ for some $1\leqslant i\leqslant p$ and $p+1\leqslant
j\leqslant N$. However, this is impossible, because
$S_{j}\not\cong\mathbb{F}_{p}$ for all $p+1\leqslant j\leqslant N$.

Since every maximal subring of $R$ has the form $M_{1}\oplus R_{2}$ or
$R_{1}\oplus M_{2}$, we can apply Lemma 2.3, which yields
$\sigma_{u}(R)=\min\\{\sigma_{u}(R_{1}),\sigma_{u}(R_{2})\\}$. Since $R$ is
$\sigma_{u}$-elementary, this means that either $R=R_{1}$ or $R=R_{2}$.
Because only $R_{1}$ has $\bigoplus_{i=1}^{p}\mathbb{F}_{p}$ as a residue
ring, we conclude that $R=R_{1}$. ∎

######  Proposition 8.10.

Let $R$ be a $\sigma_{u}$-elementary ring that is not $\sigma$-elementary, and
such that $R\not\cong\bigoplus_{i=1}^{p+1}\mathbb{F}_{p}$. Then, $R\cong
A(1,2,2)$.

###### Proof.

By Lemma 8.5, $R$ has no residue ring isomorphic to
$\bigoplus_{i=1}^{p+1}\mathbb{F}_{p}$. By Proposition 8.9,
$R=\big{(}\bigoplus_{i=1}^{p}\mathbb{F}_{p}\big{)}\oplus J$, and
$J\neq\\{0\\}$ because $R$ is coverable by unital subrings. By Proposition
8.8, $J_{ii}=\\{0\\}$ for all $1\leqslant i\leqslant p$. Moreover, by Theorem
5.1, there is a unique pair $(i,j)$ with $i\neq j$ such that
$J_{ij}\neq\\{0\\}$, and $\lambda_{ij}=1$.

Thus, $S_{i}\oplus S_{j}\oplus J_{ij}\cong A(1,p,p)$, which has unital
covering number $p+1$. By Lemma 4.2, $A(1,p,p)$ occurs as a residue ring of
$R$. Thus, $p+\binom{p}{2}\leqslant\sigma_{u}(R)\leqslant p+1$, which forces
$p=2$ and $\sigma_{u}(R)=3$. Since $R$ is $\sigma_{u}$-elementary, we must
have $R=S_{i}\oplus S_{j}\oplus J_{ij}\cong A(1,2,2)$. ∎

###### Proof of Theorem 1.6.

Apply Proposition 8.3, Lemma 8.5, and Proposition 8.10. The covering numbers
for $\bigoplus_{i=1}^{p+1}\mathbb{F}_{p}$ and $A(1,2,2)$ were found in Lemma
8.2. ∎

##  9\. Bounds on the number of integers that are covering numbers of rings

This section is dedicated to the proof of Corollary 1.9. Recall that

$\mathscr{E}(N):=\\{m:m\leqslant N,\sigma(R)=m\text{ for some ring }R\\}.$

Let $\log x$ denote the binary (base 2) logarithm. Our goal is to show that
$|\mathscr{E}(N)|$ is bounded above by $cN/\log N$ for some positive constant
$c$. Rather than attempting to find an optimal value for $c$, we will be
content with a value that is easy to verify.

Let $\pi$ be the prime counting function, which counts the number of primes
less than or equal to a given positive number. The following bounds on
$\pi(x)$ will be useful in later estimates.

######  Lemma 9.1.

For all $x>1$, $\pi(x)<2x/\log x$. For all $x>5$, $x/\log x<\pi(x)$.

###### Proof.

Let $\ln x$ denote the natural logarithm. By [29, Corollary 1],
$\pi(x)<(1.25506x)/(\ln x)$ when $x>1$, and $\pi(x)>x/\ln(x)$ when $x\geqslant
17$. The remaining cases for the lower bound follow from inspection. ∎

Let $\Pi$ be the prime power counting function, i.e.,

$\Pi(x):=|\\{m\in\mathbb{N}:2\leqslant m\leqslant x\text{ and }m\text{ is a
prime power}\\}|.$

Using Lemma 9.1, it is not difficult to get an upper bound on $\Pi(x)$.

######  Lemma 9.2.

Let $x>1$. Then, $\Pi(x)<8x/\log x$.

###### Proof.

Let $\ell:=\lfloor\log x\rfloor$. Then,

$\Pi(x)=\sum_{r=1}^{\ell}\pi(x^{1/r})=\pi(x)+\sum_{r=2}^{\ell}\pi(x^{1/r})\leqslant\pi(x)+(\ell-1)\pi(x^{1/2}).$

Applying the upper bound for $\pi(x)$ from Lemma 9.1 gives

$\Pi(x)\leqslant\dfrac{2x}{\log
x}+(\ell-1)\dfrac{2x^{1/2}}{(1/2)\log(x)}\leqslant\dfrac{2x}{\log
x}+4x^{1/2}.$

The result now follows from the inequality $x^{1/2}<1.5x/\log x$, which holds
for all $x>1$. ∎

Using Lemma 9.2, we can provide bounds on the number of integers that are
expressible in terms of the formulas given in the statement of Theorem 1.3.

######  Lemma 9.3.

Let $N>1$ be a natural number.

  1. (1)

The number of integers $m$, $2\leqslant m\leqslant N$, that can be expressed
as $\tau(q)\nu(q)+d\binom{\tau(q)}{2}$ for some prime power $q=p^{d}$ is less
than $8N/\log N$.

  2. (2)

The number of integers $m$, $2\leqslant m\leqslant N$, that can be expressed
as $q+1$ for some prime power $q=p^{d}$ is less than $8N/\log N$.

###### Proof.

Both parts follow from the fact that there is one such integer $m$ of the
desired form for each prime power. Hence, the number of such integers $m$ is
bounded above by $\Pi(N)$, which is less than $8N/\log N$ by Lemma 9.2. ∎

######  Lemma 9.4.

Let $N>1$ be a natural number. The number of integers $m$, $2\leqslant
m\leqslant N$, that can be expressed as

$\frac{1}{a}\prod_{k=1,\\\ a\nmid k}^{n-1}(q^{n}-q^{k})+\sum_{k=1,\\\ a\nmid
k}^{\lfloor n/2\rfloor}{n\choose k}_{q},$

where $q$ is some prime power, $n\geqslant 2$, $a$ is the smallest prime
divisor of $n$, and ${n\choose k}_{q}$ is the $q$-binomial coefficient is less
than $40N/\log N$.

###### Proof.

Let

$P:=\frac{1}{a}\prod_{k=1,\\\ a\nmid k}^{n-1}(q^{n}-q^{k})\;\text{ and
}\;S:=\sum_{k=1,\\\ a\nmid k}^{\lfloor n/2\rfloor}{n\choose k}_{q}.$

We first note that $P+S\geqslant\tfrac{1}{2}q^{2}$. This is clear, because
when $n=2$, we have $P+S=\tfrac{1}{2}(q^{2}+q+2)$, and when $n\geqslant 3$,
$S\geqslant\binom{n}{1}_{q}\geqslant q^{2}$. Hence,
$\tfrac{1}{2}q^{2}\leqslant N$, from which we conclude that
$q\leqslant\sqrt{2N}$, and there at most $\Pi(\sqrt{2N})$ choices for $q$.

Next, by Lemma 3.31 we have $P\geqslant q^{n(n-(n/a)-1)}$, so

$N\geqslant P+S>q^{n(n-(n/a)-1)}\geqslant q^{n(n-2)/2}\geqslant
q^{(n-2)^{2}/2}.$

It follows that $(n-2)^{2}<2\log N$ and $n<2+\sqrt{2\log N}$. Since
$n\geqslant 2$, there are at most $1+\sqrt{2\log N}$ choices for $n$ for a
given fixed $q$. Since there is exactly one integer expressible as $P+S$ for
each pair $(q,n)$, the number of such integers $m$, $2\leqslant m\leqslant N$,
having this form is bounded above by

$\Pi(\sqrt{2N})\cdot(1+\sqrt{2\log N})\leqslant\frac{8\sqrt{2N}(1+\sqrt{2\log
N})}{(1/2)\log(2N)}<\frac{16\sqrt{2N}(1+\sqrt{2\log N})}{\log N}.$

Since $\sqrt{2x}(1+\sqrt{2\log x})<2.5x$ for all $x>1$, we obtain the desired
bound. ∎

######  Lemma 9.5.

Let $N>1$ be a natural number. The number of integers $m$, $2\leqslant
m\leqslant N$, that can be expressed as

$q^{n}+{n\choose d}_{q_{1}}+\omega(d),$

where $q$ and $q_{1}$ are prime powers, $q=q_{1}^{d}$, $n\geqslant 3$, and
$d<n$ is less than $72N/\log N$.

###### Proof.

First, since $q^{3}\leqslant q^{n}<N$, there are at most $\Pi(\sqrt[3]{N})$
choices for $q$. Moreover, since $2^{n}\leqslant q^{n}<N$ and $d<n$, there are
at most $\log N$ choices for $n$ for a fixed $q$, and at most $\log N$ choices
for $d$ given a fixed $n$. Hence, the number of integers expressible in the
desired form is bounded above by the number of triples $(q,n,d)$. The number
of such triples is in turn bounded above by

$\Pi(\sqrt[3]{N})(\log N)^{2}<\frac{8\sqrt[3]{N}(\log N)^{2}}{(1/3)\log
N}<\frac{72N}{\log N},$

where the last inequality follows from using $\sqrt[3]{x}(\log x)^{2}<3x$ for
all $x>1$. ∎

We now have what we need to prove Corollary 1.9.

###### Proof of Corollary 1.9.

The upper bound of $128N/\log N$ follows from Lemmas 9.3, 9.4, and 9.5. For
the lower bound, we note that by Theorem 1.3 (2), every integer of the form
$q+1$, where $q$ is a prime power, is a covering number of a ring with unity.
Thus, $|\mathscr{E}(N)|\geqslant\Pi(N)-1$, and for $N\geqslant 5$ we have
$\Pi(N)-1\geqslant\pi(N)>N/\log N$. ∎

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