Quandles versus symmetric quandles for oriented links

A quandle [10, 15], see also [6], is a set $X$ with a binary operation $*:X\times X\to X$ satisfying the following axioms.

• (Q1)

  For any $x\in X$, $x*x=x$.

• (Q2)

  For any $x,y\in X$, there exists a unique element $z\in X$ such that $z*y=x$.

• (Q3)

  For any $x,y,z\in X$, $(x*y)*z=(x*z)*(y*z)$.

The unique element $z$ in the second axiom is denoted by $x*y^{-1}$. For a quandle $X$, when we specify the quandle operation $*$, we also denote by $(X,*)$ the quandle $X$.

The proof of the next lemma is straightforward, and we leave it to the reader.

Carter et al. [2] defined homology groups of quandles, which are deeply related to homology groups of racks due to Fenn et al. [7], and various versions are introduced so far, see [1, 3]. In this paper, we adopt the definition of the homology group of a quandle $X$ with an $X$-set $Y$. We note that when $Y$ is a singleton, the homology group concides with that defined in [2], and when $Y=X$, the homology group coincides with that defined in [3].

The associated group of a quandle $(X,*)$ is

where we note that any element of $X$ can be regarded as an element of $G_{X}$ through the map $X\to G_{X};x\mapsto x$. An $X$-set is a set $Y$ equipped with a right action by the associated group $G_{X}$. In this paper, we use the same symbol for the operation of a given quandle $X$ and the action on a given $X$-set $Y$ by $G_{X}$, that is, we denote by $r*g$ the image of an element $r\in Y$ acted on by $g\in G_{X}$.

Let $(X,*)$ be a quandle and $Y$ an $X$-set. The same proof as in Lemma 2.1 works for the next lemma, and we leave it to the reader.

Let $C_{n}(X)_{Y}$ be the free abelian group generated by $(n+1)$-tuples $(r,x_{1},\ldots,x_{n})\in Y\times X^{n}$ when $n$ is a positive integer, and let $C_{n}(X)_{Y}=\{0\}$ otherwise. The boundary homomorphism $\partial_{n}:C_{n}(X)_{Y}\to C_{n-1}(X)_{Y}$ is defined by

for $n>1$ and $\partial_{n}=0$ otherwise, where $\hat{x_{i}}$ represents that $x_{i}$ is removed. Then $C_{*}(X)_{Y}=\{C_{n}(X)_{Y},\partial_{n}\}_{n\in\mathbb{Z}}$ is a chain complex.

Let $D_{n}(X)_{Y}$ be the subgroup of $C_{n}(X)_{Y}$ generated by the elements of

Then $D_{*}(X)_{Y}=\{D_{n}(X)_{Y},\partial_{n}\}_{n\in\mathbb{Z}}$ is a subchain complex of $C_{*}(X)_{Y}$. Let $C_{n}^{\rm Q}(X)_{Y}=C_{n}(X)_{Y}/D_{n}(X)_{Y}$, and we denote by $\partial_{n}^{\rm Q}$ the induced boundary homomorphism $\partial_{n}^{\rm Q}:C_{n}^{\rm Q}(X)_{Y}\to C_{n-1}^{\rm Q}(X)_{Y}$. The quotient chain complex $C_{*}^{\rm Q}(X)_{Y}=\{C_{n}^{\rm Q}(X)_{Y},\partial_{n}^{\rm Q}\}_{n\in\mathbb{Z}}$ leads to the quandle homology group of $X$ and $Y$ defined by $H_{n}^{\rm Q}(X)_{Y}={\rm Ker}\partial_{n}^{\rm Q}/{\rm Im}\partial_{n+1}^{\rm Q}$.

For an abelian group $A$, we define the chain complex $C_{*}^{\rm Q}(X;A)_{Y}=\{C_{n}^{\rm Q}(X)_{Y}\otimes A,\partial_{n}^{\rm Q}\otimes{\rm id}\}_{n\in\mathbb{Z}}$. The homology groups are denoted by $H_{*}^{\rm Q}(X;A)_{Y}$. Note that when $A=\mathbb{Z}$, $H_{*}^{\rm Q}(X;\mathbb{Z})_{Y}$ represents $H_{*}^{\rm Q}(X)_{Y}$. We also define the cochain group $C_{\rm Q}^{n}(X;A)_{Y}$ and the coboundary homomorphism $\delta_{\rm Q}^{n}:C_{\rm Q}^{n}(X;A)_{Y}\to C_{\rm Q}^{n+1}(X;A)_{Y}$ by $C_{\rm Q}^{n}(X;A)_{Y}={\rm Hom}(C_{n}^{\rm Q}(X)_{Y},A)$ and $\delta_{\rm Q}^{n}(\theta)=\theta\circ\partial^{\rm Q}_{n+1}$, respectively. The quandle cohomology group with coefficients in $A$ is defined by $H_{\rm Q}^{n}(X;A)_{Y}={\rm Ker}\delta_{\rm Q}^{n}/{\rm Im}\delta_{\rm Q}^{n-1}$. We denote by $Z_{\rm Q}^{n}(X;A)_{Y}$ and $B_{\rm Q}^{n}(X;A)_{Y}$ the cocycle group ${\rm Ker}\delta_{\rm Q}^{n}$ and the coboundary group ${\rm Im}\delta_{\rm Q}^{n-1}$, respectively.

A link is a disjoint union of circles embedded in $\mathbb{R}^{3}$. Two links are said to be equivalent if there exists an orientation-preserving self-homeomorphism of $\mathbb{R}^{3}$ which maps one link onto the other. A diagram of a link is its image, via a generic projection from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$, equipped with the height information around the crossings. The height information of a diagram of a link is given by removing regular neighborhoods of the lower crossing points, and thus, the diagram is regarded as a disjoint union of connected compact parts. We call each connected component an arc of the diagram. The $2$-dimensional space $\mathbb{R}^{2}$ is separated into several connected regions by a diagram. We call each connected region a complementary region of the diagram.

Let $(X,*)$ be a quandle. Let $(D,o)$ be an oriented link diagram, that is, a pair of a link diagram $D$ and an orientation $o$ of $D$. In this paper, we often use normal orientations of arcs for representing the orientation $o$ of $D$ as in Figure 1, where for an arc $\alpha$ of $D$, the pair of the orientation $o$ and the normal orientation $n_{\alpha}$ of the arc always forms the right-handed orientation of $\mathbb{R}^{2}$.

An $X$-coloring of $(D,o)$ is an assignment of an element of $X$ to each arc satisfying the following condition.

• •

  Suppose that two arcs $\alpha_{1}$ and $\alpha_{2}$ which are under-arcs at a crossing $\chi$ are labeled by $x_{1}$ and $x_{2}$, respectively, and the over arc, say $\alpha_{3}$, of $\chi$ is labeled by $x_{3}$. We assume that the normal orientation of $\alpha_{3}$ points from $\alpha_{1}$ to $\alpha_{2}$. Then $x_{1}*x_{3}=x_{2}$ holds. See the left of Figure 2.

We denote by ${\rm Col}_{X}(D,o)$ the set of $X$-colorings of $(D,o)$. We then have the following lemma.

This implies that $\#{\rm Col}_{X}(D,o)$, that is the cardinality of ${\rm Col}_{X}(D,o)$, is an invariant for oriented links, and hence, we also denote it by $\#{\rm Col}_{X}(L,o)$ and call it the quandle coloring number of $(L,o)$ when $X$ is finite, where $(L,o)$ is an oriented link which $(D,o)$ represents. We note that in this paper, when a diagram $(D,o)$ represents an oriented link, we use the same symbol $o$ for the orientation of the oriented link.

Let $Y$ be an $X$-set. An $X_{Y}$-coloring of $(D,o)$ is an $X$-coloring of $(D,o)$ with an assignment of an element of $Y$ to each complementary region of $D$ satisfying the following condition.

• •

  Suppose that two complementary regions $\gamma_{1}$ and $\gamma_{2}$ are adjacent over an arc $\alpha$, and $\gamma_{1}$, $\gamma_{2}$ and $\alpha$ are labeled by $r_{1}$, $r_{2}$ and $x$, respectively. We assume that the normal orientation of $\alpha$ points from $\gamma_{1}$ to $\gamma_{2}$. Then $r_{1}*x=r_{2}$ holds. See the right of Figure 2.

We denote by ${\rm Col}_{X_{Y}}(D,o)$ the set of $X_{Y}$-colorings of $(D,o)$. Then we have the following lemma.

This implies that $\#{\rm Col}_{X_{Y}}(D,o)$, that is the cardinality of ${\rm Col}_{X_{Y}}(D,o)$, is an invariant for oriented links.

In this subsection, we review the definition of a quandle cocycle invariant of an oriented link. We remark that several variations for quandle cocycle invariants are introduced so far, see [2, 3], and we adopt the definition of a shadow quandle cocycle invariant with a quandle $X$ and an $X$-set $Y$. We note that when $Y$ is a singleton, the shadow quandle cocycle invariant coincides with the quandle cocycle invariant with $X$ defined in [2], and when $Y=X$, the shadow quandle cocycle invariant coincides with that defined in [3].

Let $(X,*)$ be a quandle and $Y$ an $X$-set. Let $(D,o)$ be a diagram of an oriented link $(L,o)$ and $C$ an $X_{Y}$-coloring of $(D,o)$. For each crossing $\chi$ of $D$, we extract a weight $w_{\chi}$ as follows. Let $\alpha_{1}$ and $\alpha_{2}$ denote the under-arc and the over-arc of $\chi$, respectively, such that the normal orientation of $\alpha_{2}$ points from $\alpha_{1}$, and $\gamma$ the complementary region of $D$ around $\chi$ which the normal orientations of $\alpha_{1}$ and $\alpha_{2}$ point from. We suppose that $\gamma$, $\alpha_{1}$ and $\alpha_{2}$ are labeled by $r$, $x_{1}$ and $x_{2}$, respectively. Then the weight of the crossing $\chi$ is $w_{\chi}=\varepsilon(r,x_{1},x_{2})$, where $\varepsilon=+$ if $\chi$ is positive and $\varepsilon=-$ if $\chi$ is negative, see Figure 3. We note that a crossing $\chi$ is positive if the pair of the normal orientations of the over-arc and an under-arc of $\chi$ matches the right-handed orientation of $\mathbb{R}^{2}$, and $\chi$ is negative otherwise. Let $W^{X_{Y}}((D,o),C)$ denote the sum of the weights of all the crossings, and then, $W^{X_{Y}}((D,o),C)$ is a $2$-cycle of $C_{*}^{\rm Q}(X)_{Y}$. Define

as a multi-set, where a multi-set is a collection of elements which may appear more than once.

For an abelian group $A$, let $\theta:C_{2}^{\rm Q}(X)_{Y}\to A$ be a quandle $2$-cocycle. We define the multi-set $\Phi_{\theta}^{X_{Y}}(D)$ by

We also denote these invariants by $\mathcal{W}^{X_{Y}}(L,o)$ and $\Phi_{\theta}^{X_{Y}}(L,o)$, respectively, and call them the quandle homology invariant of $(L,o)$ by $X$ and $Y$ and the quandle cocycle invariant of $(L,o)$ by $X,Y$ and $\theta$, respectively.

A symmetric quandle [11], see also [13], is a pair $(X,\rho)$ of a quandle $(X,*)$ and a good involution $\rho$ of $X$, where an involution $\rho:X\to X$ is said to be good if

for any $x,y\in X$. For a symmetric quandle $(X,\rho)$, when we specify the symmetric quandle operation $*$, we also denote by $(X,\rho,*)$ the symmetric quandle $(X,\rho)$.

Kamada [11], see also [13] in detail, defined (co)homology groups of symmetric quandles. In this subsection, we review the definition of a symmetric quandle homology group.

The associated group of a symmetric quandle $(X,\rho,*)$ is

An $(X,\rho)$-set is a set $Y$ equipped with a right action of the associated group $G_{(X,\rho)}$. In this paper, we use the same symbol for the operation of a given symmetric quandle $(X,\rho)$ and the action on a given $(X,\rho)$-set $Y$ by $G_{(X,\rho)}$, that is, we denote by $r*g$ the image of an element $r\in Y$ acted on by $g\in G_{(X,\rho)}$.

Let $(X,\rho,*)$ be a symmetric quandle and $Y$ be an $(X,\rho)$-set. The chain group $C_{n}(X)_{Y}$, the boundary homomorphism $\partial_{n}:C_{n}(X)_{Y}\to C_{n-1}(X)_{Y}$, and the subgroup $D_{n}(X)_{Y}$ of $C_{n}(X)_{Y}$ are defined as in Subsection 2.2.

Let $D_{n}(X,\rho)_{Y}$ be the subgroup of $C_{n}(X)_{Y}$ generated by the elements of

Then $D_{*}(X,\rho)_{Y}=\{D_{n}(X,\rho)_{Y},\partial_{n}\}_{n\in\mathbb{Z}}$ is a subchain complex of $C_{*}(X)_{Y}$. Let $C_{n}^{\rm SQ}(X,\rho)_{Y}=C_{n}(X)_{Y}/(D_{n}(X)_{Y}+D_{n}(X,\rho)_{Y})$, and we denote by $\partial_{n}^{\rm SQ}$ the induced boundary homomorphism $\partial_{n}^{\rm SQ}:C_{n}^{\rm SQ}(X,\rho)_{Y}\to C_{n-1}^{\rm SQ}(X,\rho)_{Y}$. The quotient chain complex $C_{*}^{\rm SQ}(X,\rho)_{Y}=\{C_{n}^{\rm SQ}(X,\rho)_{Y},\partial_{n}^{\rm SQ}\}_{n\in\mathbb{Z}}$ leads to the symmetric quandle homology group of $(X,\rho)$ and $Y$ defined by $H_{n}^{\rm SQ}(X,\rho)_{Y}={\rm Ker}\partial_{n}^{\rm SQ}/{\rm Im}\partial_{n+1}^{\rm SQ}$.

For an abelian group $A$, we define the chain complex $C_{*}^{\rm SQ}(X,\rho;A)_{Y}=\{C_{n}^{\rm SQ}(X,\rho)_{Y}\otimes A,\partial_{n}^{\rm SQ}\otimes{\rm id}\}_{n\in\mathbb{Z}}$. The homology groups are denoted by $H_{*}^{\rm SQ}(X,\rho;A)_{Y}$. Note that when $A=\mathbb{Z}$, $H_{*}^{\rm SQ}(X,\rho;\mathbb{Z})_{Y}$ represents $H_{*}^{\rm SQ}(X,\rho)_{Y}$. We also define the cochain group $C_{\rm SQ}^{n}(X,\rho;A)_{Y}$ and the coboundary homomorphism $\delta_{\rm SQ}^{n}:C_{\rm SQ}^{n}(X,\rho;A)_{Y}\to C_{\rm SQ}^{n+1}(X,\rho;A)_{Y}$ by $C_{\rm SQ}^{n}(X,\rho;A)_{Y}={\rm Hom}(C_{n}^{\rm SQ}(X,\rho)_{Y},A)$ and $\delta_{\rm SQ}^{n}(\theta)=\theta\circ\partial^{\rm SQ}_{n+1}$, respectively. The symmetric quandle cohomology group with coefficients in $A$ is defined by $H_{\rm SQ}^{n}(X,\rho;A)_{Y}={\rm Ker}\delta_{\rm SQ}^{n}/{\rm Im}\delta_{\rm SQ}^{n-1}$, and we denote by $Z_{\rm SQ}^{n}(X,\rho;A)_{Y}$ and $B_{\rm SQ}^{n}(X,\rho;A)_{Y}$ the cocycle group ${\rm Ker}\delta_{\rm SQ}^{n}$ and the coboundary group ${\rm Im}\delta_{\rm SQ}^{n-1}$, respectively.

Let $(X,\rho,*)$ be a symmetric quandle and $D$ an (unoriented) link diagram. A semi-arc of $D$ is a connected component after removing the regular neighborhoods of the crossings of $D$.

An $(X,\rho)$-coloring of $D$ is the equivalence class of an assignment of a normal orientation and an element of $X$ to each semi-arc satisfying the following conditions.

• •

  Suppose that two semi-arcs coming from an over-arc of $D$ at a crossing are labeled by $x_{1}$ and $x_{2}$. If the normal orientations are coherent, then $x_{1}=x_{2}$, otherwise $x_{1}=\rho(x_{2})$. See the left upper half of Figure 4.

• •

  Suppose that two semi-arcs $\alpha_{1}$ and $\alpha_{2}$ which are under-arcs at a crossing $\chi$ are labeled by $x_{1}$ and $x_{2}$, respectively, and suppose that one of the semi-arcs coming from the over arc, say $\alpha_{3}$, of $\chi$ is labeled by $x_{3}$. We assume that the normal orientation of $\alpha_{3}$ points from $\alpha_{1}$ to $\alpha_{2}$. If the normal orientations of $\alpha_{1}$ and $\alpha_{2}$ are coherent, then $x_{1}*x_{3}=x_{2}$ holds, otherwise $x_{1}*x_{3}=\rho(x_{2})$ holds. See the left lower half of Figure 4.

Here the equivalence relation is generated by basic inversions which are operations of replacing the normal orientation of a semi-arc with the inverse one and the element $x$, assigned to the semi-arc, with $\rho(x)$, see Figure 5.

We denote by ${\rm Col}_{(X,\rho)}(D)$ the set of $(X,\rho)$-colorings of $D$. We then have the following lemma.

This implies that $\#{\rm Col}_{(X,\rho)}(D)$, that is the cardinality of ${\rm Col}_{(X,\rho)}(D)$, is an invariant for links, and hence, we also denote it by $\#{\rm Col}_{(X,\rho)}(L)$ and call it the symmetric quandle coloring number of $L$ when $X$ is finite, where $L$ is a link which $D$ represents.

Let $Y$ be an $(X,\rho)$-set. An $(X,\rho)_{Y}$-coloring of $D$ is an $(X,\rho)$-coloring of $D$ with an assignment of an element of $Y$ to each complementary region of $D$ satisfying the following condition.

• •

  Suppose that two complementary regions $\gamma_{1}$ and $\gamma_{2}$ are adjacent over a semi-arc $\alpha$, and $\gamma_{1}$, $\gamma_{2}$ and $\alpha$ are labeled by $r_{1}$, $r_{2}$ and $x$, respectively. We assume that the normal orientation of $\alpha$ points from $\gamma_{1}$ to $\gamma_{2}$. Then $r_{1}*x=r_{2}$ holds. See the right of Figure 4.

We denote by ${\rm Col}_{(X,\rho)_{Y}}(D)$ the set of $(X,\rho)_{Y}$-colorings of $D$. Then we have the following lemma.

This implies that $\#{\rm Col}_{(X,\rho)_{Y}}(D)$, that is the cardinality of ${\rm Col}_{(X,\rho)_{Y}}(D)$, is an invariant for links.

Let $(X,\rho,*)$ be a symmetric quandle and $Y$ an $(X,\rho)$-set. Let $D$ be a diagram of an (unoriented) link $L$ and $C$ an $(X,\rho)_{Y}$-coloring of $D$. For each crossing $\chi$ of $D$, we extract a weight $w_{\chi}$ as follows. For a crossing $\chi$ of $D$, there are four complementary regions of $D$ around $\chi$. We choose one of the regions, say $\gamma$, and call it the specified region for $\chi$. Let $\alpha_{1}$ and $\alpha_{2}$ denote the under-semi-arc and the over-semi-arc of $\chi$, respectively, which faces the specified region $\gamma$. By basic inversions, we may assume that the normal orientations $n_{1}$ and $n_{2}$ of $\alpha_{1}$ and $\alpha_{2}$, respectively, point from $\gamma$. Let $x_{1}$, $x_{2}$ and $r$ be the labels of $\alpha_{1}$, $\alpha_{2}$ and $\gamma$, respectively. Then the weight of the crossing $\chi$ is $w_{\chi}=\varepsilon(r,x_{1},x_{2})$, where if the pair $(n_{1},n_{2})$ of the normal orientations $n_{1}$ and $n_{2}$ assigned to $\alpha_{1}$ and $\alpha_{2}$, respectively, matches the right-handed orientation of $\mathbb{R}^{2}$, then $\varepsilon=+$, otherwise $\varepsilon=-$, see Figure 6.

Let $W^{(X,\rho)_{Y}}(D,C)$ denote the sum of the weights of all the crossings, and then, $W^{(X,\rho)_{Y}}(D,C)$ is a $2$-cycle of $C_{*}^{\rm SQ}(X,\rho)_{Y}$. Define

as a multi-set.

For an abelian group $A$, let $\theta:C_{2}^{\rm SQ}(X,\rho)_{Y}\to A$ be a symmetric quandle $2$-cocycle. We define the multi-set $\Phi_{\theta}^{(X,\rho)_{Y}}(D)$ by

We also denote these invariants by $\mathcal{W}^{(X,\rho)_{Y}}(L)$ and $\Phi_{\theta}^{(X,\rho)_{Y}}(L)$, respectively, and call them the symmetric quandle homology invariant of $L$ by $(X,\rho)$ and $Y$ and the symmetric quandle cocycle invariant of $L$ by $(X,\rho),Y$ and $\theta$, respectively.

Let $(X,*)$ be a quandle. We set two copies $X^{+1}$ and $X^{-1}$ of $X$ by

and let $D_{X}=X^{+1}\sqcup X^{-1}$. For $x^{\varepsilon}\in D_{X}$, we call $\varepsilon$ the sign of $x^{\varepsilon}$. Define a binary operation $\star:D_{X}\times D_{X}\to D_{X}$ by

We call the quandle $(D_{X},\star)$ the double of $X$.

Define a map ${\rho_{+\leftrightarrow-}}:D_{X}\to D_{X}$ by $\rho(x^{\varepsilon})=x^{-\varepsilon}$.

We call the symmetric quandle $(D_{X},{\rho_{+\leftrightarrow-}})$ the symmetric double of $X$.

Let $Y$ be an $X$-set. For the associated groups $G_{X}$ and $G_{(D_{X},{\rho_{+\leftrightarrow-}})}$, we define the group homomorphism $\pi:G_{(D_{X},{\rho_{+\leftrightarrow-}})}\to G_{X}$ by $\pi(x^{\varepsilon})=x^{\varepsilon}$ for $x^{\varepsilon}\in D_{X}$. Note that $x^{+1}$ represents $x$ in $G_{X}$ and $x^{-1}$ represents the inverse of $x$ in $G_{X}$. The next property can be proven by an easy verification as in Lemmas 2.1 and 2.5.

In this subsection, we show that the homology group of a given quandle and that of the symmetric double are isomorphic. This property is very important for applying symmetric quandle cocycle invariants for oriented (or unoriented orientable) links and saying that symmetric quandle cocycle invariants are a generalization of quandle cocycle invariants.

For simplicity, in this subsection, we omit parenthesis of operations or actions such as $x_{1}*x_{2}^{\varepsilon_{2}}*x_{3}^{\varepsilon_{3}}*\cdots*x_{n}^{\varepsilon_{n}}$ means $(\cdots((x_{1}*x_{2}^{\varepsilon_{2}})*x_{3}^{\varepsilon_{3}})*\cdots)*x_{n}^{\varepsilon_{n}}$, and $x_{1}\star x_{2}^{\varepsilon_{2}}\star x_{3}^{\varepsilon_{3}}\star\cdots\star x_{n}^{\varepsilon_{n}}$ means $(\cdots((x_{1}\star x_{2}^{\varepsilon_{2}})\star x_{3}^{\varepsilon_{3}})\star\cdots)\star x_{n}^{\varepsilon_{n}}$.

Let $(X,*)$ be a quandle, $(D_{X},{\rho_{+\leftrightarrow-}},\star)$ the symmetric double of $X$, and $Y$ an $X$-set. We note that by Lemma 4.5, $Y$ is regarded as a $(D_{X},{\rho_{+\leftrightarrow-}})$-set.

For each $(r,x_{1}^{\varepsilon_{1}},\ldots,x_{n}^{\varepsilon_{n}})\in Y\times D_{X}^{n}$, apply the following process (Step 0)-(Step 2), say the canonicalization, while $\{\delta_{1},\ldots,\delta_{n}\}\not=\{+1\}$.

• •

  (Step 0) Set the singed $(n+1)$-tuple $\delta(s,y_{1}^{\delta_{1}},\ldots,y_{n}^{\delta_{n}})\in\{\pm 1\}\times Y\times D_{X}^{n}$ by $(s,y_{1}^{\delta_{1}},\ldots,y_{n}^{\delta_{n}}):=(r,x_{1}^{\varepsilon_{1}},\ldots,x_{n}^{\varepsilon_{n}})$, $\delta:=+1$ and $i:=n$. Go to (Step 1).

• •

  (Step 1) If the $i$th sign $\delta_{i}$ is $-1$, replace $\delta(s,y_{1}^{\delta_{1}},\ldots,y_{n}^{\delta_{n}})$ with

  $\displaystyle-\delta(s*y_{i}^{\delta_{i}},(y_{1}*y_{i}^{\delta_{i}})^{\delta_{1}},\ldots,(y_{i-1}*y_{i}^{\delta_{i}})^{\delta_{i-1}},y_{i}^{-\delta_{i}},y_{i+1}^{\delta_{i+1}},\ldots,y_{n}^{\delta_{n}}),$

  (1)

  and then, denote by $\delta(s,y_{1}^{\delta_{1}},\ldots,y_{n}^{\delta_{n}})$ the replaced signed $(n+1)$-tuple (1). Go to (Step 2).

• •

  (Step 2) Set $i:=i-1$. Go to (Step 1).