Tensor products of quandles and 1-handles attached to surface-links

A quandle $X=(X,\ast,\overline{\ast})$ is a set $X$ with two binary operations $\ast$ and $\overline{\ast}$ satisfying conditions related to Reidemeister moves in knot theory (cf. [6, 14] and Section 2). A symmetric quandle is a quandle equipped with an involution called a good involution ([8, 9, 12]). For a given quandle $X$, there is a symmetric quandle $(D(X),\rho)$, called the symmetric double of $X$ (Example 2.3). Quandles and symmetric quandles are used in knot theory for construction of invariants of links and surface-links.

In this paper, we define the tensor product $X\otimes X$ of a quandle $X$. We give a method of computing the tensor product $D(X)\otimes D(X)$ of the symmetric double $D(X)$ from the tensor product $X\otimes X$ of $X$ (Theorem 3.4). Let $\tau:X\otimes X\to X\otimes X$ be the involution switching the components, i.e., $\tau(x,y)=(y,x)$. When $X$ is equipped with a good involution $\rho:X\to X$, we have another involution $\rho$ ($=\rho\otimes\rho$) $:X\otimes X\to X\otimes X$ with $\rho(x,y)=(\rho(x),\rho(y))$. We consider the quotient set $X\otimes X/\langle\tau\rangle$ ($X\otimes X/\langle\rho\rangle$ or $X\otimes X/\langle\tau,\rho\rangle$) of $X\otimes X$ by the equivalence relation generated by $\tau$ (by $\rho$ or by $\tau$ and $\rho$). It will be seen that the tensor product $X\otimes X$ itself and the quotient set $X\otimes X/\langle\tau\rangle$ (or $X\otimes X/\langle\tau,\rho\rangle$) of $X\otimes X$ are useful for studying $1$-handles attached to surface-links.

We compute the tensor product $R_{n}\otimes R_{n}$ for the dihedral quandle $R_{n}$ of every order $n$ (Theorems 5.1 and 5.3) in Section 5. Once we know the elements of $R_{n}\otimes R_{n}$, we can obtain the quotient set $R_{n}\otimes R_{n}/\langle\tau\rangle$ (Corollaries 5.2 and 5.4). For the symmetric double $(D(R_{n}),\rho)$ of $R_{n}$, using Theorem 3.4, we can compute $D(R_{n})\otimes D(R_{n})$ from $R_{n}\otimes R_{n}$ (Theorems 6.1 and 6.5), and then we obtain the quotient sets $D(R_{n})\otimes D(R_{n})/\langle\tau\rangle$, $D(R_{n})\otimes D(R_{n})/\langle\rho\rangle$ and $D(R_{n})\otimes D(R_{n})/\langle\tau,\rho\rangle$ in Section 6.

By a surface-link, we mean a closed surface embedded in $\mathbb{R}^{4}$. The surface may be disconnected or non-orientable. When it is orientable and oriented, it is called an oriented surface-link. A surface-link is called a surface-knot when it is connected. When $F$ is oriented, the knot quandle $Q(F)$ is defined (cf. [3, 6, 8, 9, 12, 14] and Example 2.5). For any surface-link $F$, the knot symmetric quandle $(\widetilde{Q}(F),\rho)$ is defined (cf. [8, 9, 12] and Example 2.6).

Using tensor products of quandles, we discuss classification and invariants of $1$-handles attached to surface-links.

F. Hosakawa and A. Kawauchi [5] studied unknotted surface-links and surgery along $1$-handles. They proved that any oriented surface-link can be transformed to an unknotted oriented surface-link by surgery along oriented $1$-handles. For non-orientable surface-links, an analoguos result was shown by the author in [7], that is, any non-orientable surface-link can be transformed to an unknotted non-orientable surface-link by surgery along $1$-handles. In this sense, surgery along a $1$-handle is often called an unknotting operation for surface-links. Surgery along a $1$-handle is also used to construct examples of surface-knots and surface-links.

J. Boyle [1] classified oriented $1$-handles attached to an oriented surface-knot $F$ up to strong equivalence in terms of the knot group $G(F):=\pi_{1}(\mathbb{R}^{4}\setminus F)$ and its peripheral subgroup. They are in one-to-one correspondence to double cosets of the knot group $G(F)$ by the peripheral subgroup. The author [10] extended Boyle’s argument to the case where $F$ is a surface-knot which may be unoriented or non-orientable. However the arguments in [1] and [10] using knot groups are not applied directly to surface-links which are not connected. Even for surface-knots, classifications of $1$-handles stated in terms of the knot groups given in [10] are not easy to discuss. The argument using tensor products discussed in this paper is simple and applicable to any surface-link which may be disconnected or non-orientable.

For a surface-link $F$, let $\mathcal{H}^{\rm s}(F)$ (or $\mathcal{H}^{\rm w}(F)$) denote the set of strong (or weak) equivalence classes of $1$-handles attached to $F$. (See Section 7 for the definition of strong/weak equivalence.)

This theorem is proved as Theorems 8.2 and 9.2.

When $F$ is an oriented surface-link, a $1$-handle attached to $F$ is called an oriented $1$-handle if the orientation is compatible with the orientation of $F$ (Definition 7.5). Let $\mathcal{H}^{\rm s}(F)^{\rm ori}$ (or $\mathcal{H}^{\rm w}(F)^{\rm ori}$) denote the set of strong (or weak) equivalence classes of oriented $1$-handles attached to $F$.

This theorem is proved as Theorems 8.4 and 9.4.

Theorems 1.1 and 1.2 give complete classifications of all $1$-handles and oriented $1$-handles in terms of the tensor products of quandles. When $\widetilde{Q}(F)$ is finite, the tensor product $\widetilde{Q}(F)\otimes\widetilde{Q}(F)$ and its quotient $\widetilde{Q}(F)\otimes\widetilde{Q}(F)/\langle\tau,\rho\rangle$ are finite and we can list the elements theoretically. However, when $\widetilde{Q}(F)$ is infinite or has a large cardinality, it is difficult to compute or list the elements of the tensor product and its quotient. In such a case, it is useful to construct an invariant. Theorems 1.1 and 1.2 can be used to construct invariants of $1$-handles as follows:

This paper is organized as follow: In Section 2 we recall quandles and symmetric quandles. In Section 3 the tensor product of a quandle is defined. For a quandle $X$, we give a method of computing the tensor product $D(X)\otimes D(X)$ of the symmetric double $D(X)$ from the tensor product $X\otimes X$ (Theorem 3.4). In Section 4 for a quandle $X$ or a symmetric quandle $(X,\rho)$, involutions $\tau$ and $\rho$ on the tensor product $X\otimes X$ and the quotients $X\otimes X/\langle\tau\rangle$, $X\otimes X/\langle\rho\rangle$ and $X\otimes X/\langle\tau,\rho\rangle$ are defined. In Section 5 the tensor product $R_{n}\otimes R_{n}$ of the dihedral quandle $R_{n}$ is computed. In Section 6 the tensor product $D(R_{n})\otimes D(R_{n})$ and its quotients for the symmetric double $D(R_{n})$ of $R_{n}$ are computed. In Section 7 the definition of strong/weak equivalence of $1$-handles is given. We recall the notion of chords attached to a surface-link and a relationship between $1$-handles and chords. In Section 8 we discuss strong equivalence classes of $1$-handles and show a bijection from the strong equivalence classes of $1$-handles (or oriented $1$-handles) attached to $F$ to the tensor product $\widetilde{Q}(F)\otimes\widetilde{Q}(F)$ (or $Q(F)\otimes Q(F)$) (Theorems 8.2 and 8.4). In Section 9 we discuss weak equivalence classes of $1$-handles and show a bijection from the weak equivalence classes of $1$-handles (or oriented $1$-handles) attached to $F$ to the quotient of the tensor product $\widetilde{Q}(F)\otimes\widetilde{Q}(F)/\langle\tau,\rho\rangle$ (or $Q(F)\otimes Q(F)/\langle\tau\rangle$) (Theorems 9.2 and 9.4). In Section 10 we give examples and discuss invariants of $1$-handles.

This work was supported by JSPS KAKENHI Grant Numbers JP19H01788 and JP17H06128.

A quandle is a set $X$ with two binary operations $\ast$ and $\overline{\ast}$ satisfying the following three conditions:

• (Q1)

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

• (Q2)

  For any $x,y\in X$, $(x\ast y)\,\overline{\ast}\,y=x$ and $(x\,\overline{\ast}\,y)\ast y=x$.

• (Q3)

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

This notion was introduced independently by Joyce [6] and Matveev [14]. The three conditions correspond to three basic moves on knot diagrams, called Reidemeister moves. A symmetric quandle is a pair $(X,\rho)$ of a quandle $X$ and an involution $\rho:X\to X$, called a good involution on $X$, satisfying the following conditions:

• (S1)

  For any $x,y\in X$, $\rho(x\ast y)=\rho(x)\ast y$,

• (S2)

  For any $x,y\in X$, $x\ast{\rho(y)}=x\,\overline{\ast}\,y$.

A symmetric quandle homomorphism $f:(X,\rho)\to(Y,\rho)$ means a quandle homomorphism $f:X\to Y$ with $f\circ\rho=\rho\circ f$. Refer to [8, 9, 12] for details on symmetric quandles and related topics.

We here give some examples of quandles and symmetric quandles.

Suppose that a group $G$ acts on a quandle $X_{1}$ from the right and on a quandle $X_{2}$ from the left; $x_{1}\cdot(gg^{\prime})=(x_{1}\cdot g)\cdot g^{\prime}$ and $(gg^{\prime})\cdot x_{2}=g\cdot(g^{\prime}\cdot x_{2})$. Consider the equivalence relation on $X_{1}\times X_{2}$ generated by

Let $X$ be a quandle. Let $F(X)$ be the free group generated by $X$. The free group $F(X)$ acts on $X$ from the right by

for $x\in X$ and $g=x_{1}^{\epsilon_{1}}x_{2}^{\epsilon_{2}}\cdots x_{m}^{\epsilon_{m}}\in G$, where $\ast^{1}=\ast$ and $\ast^{-1}=\overline{\ast}$. For example, $x\cdot abc^{-1}d=(((x\ast a)\ast b)\,\overline{\ast}\,c)\ast d$. This action is called the canonical action of $F(X)$ on $X$. The free group $F(X)$ acts on $X$ from the left by

for $x\in X$ and $g=x_{1}^{\epsilon_{1}}x_{2}^{\epsilon_{2}}\cdots x_{m}^{\epsilon_{m}}\in G$.

Note that

Thus, the tensor product $X\otimes X$ is the orbit space of $X\times X$ by the right action of $F(X)$ with $(x_{1},\,x_{2})\cdot g:=(x_{1}\cdot g,\,x_{2}\cdot g)$. We also refer to this action of $F(X)$ on $X\times X$, $(x_{1},\,x_{2})\cdot g:=(x_{1}\cdot g,\,x_{2}\cdot g)$, as the canonical right action of $F(X)$ on $X\times X$.

In Section 5 we compute the tensor product $R_{n}\otimes R_{n}$ for every dihedral qaundle $R_{n}$.

We introduce a method of computing $D(X)\otimes D(X)$ from $X\otimes X$.

Let $X$ be a quandle and let $D(X)$ be the symmetric double. Note that $D(X)=X^{+}\amalg X^{-}=\{x^{+},x^{-}\mid x\in X\}$. For each element $(x,y)\in X\times X$ and $\epsilon,\delta\in\{+,-\}$, let $(x,y)^{\epsilon,\delta}$ denote the element $(x^{\epsilon},y^{\delta})\in D(X)\times D(X)$. Note that every element of $D(X)\times D(X)$ can be written in this form. For a subset $E$ of $X\times X$ and $\epsilon,\delta\in\{+,-\}$, let $E^{\epsilon,\delta}$ denote the subset of $D(X)\times D(X)$ with

We provide two lemmas.

Let $\alpha^{+}:F(X)\to F(D(X))$ be the homomorphism determined by $\alpha^{+}(x)=x^{+}$ for $x\in X$. (Then $\alpha^{+}(x^{-1})=(x^{+})^{-1}$ for $x\in X$.)

Let $\beta:F(D(X))\to F(X)$ be the homomorphism determined by $\beta(x^{+})=x$ and $\beta(x^{-})=x^{-1}$ for $x\in X$. (Then $\beta((x^{+})^{-1})=x^{-1}$ and $\beta((x^{-})^{-1})=x$ for $x\in X$.)

Proof.  It is directly seen from the definition of the canonical action. ∎

Proof.  Let $(x,y)^{\epsilon,\delta}$ be any element of $E^{\epsilon,\delta}$. By definition, $(x,y)\in E$. There is an element $g\in F(X)$ with $(x,y)=(x_{0},y_{0})\cdot g$. By Lemma 3.5, $(x,y)^{\epsilon,\delta}=(x_{0},y_{0})^{\epsilon,\delta}\cdot\alpha_{+}(g)$. Thus, $(x,y)^{\epsilon,\delta}\in[(x_{0},y_{0})^{\epsilon,\delta}]$. Hence, $E^{\epsilon,\delta}\subset[(x_{0},y_{0})^{\epsilon,\delta}]$.

Conversely, let $(x,y)^{\epsilon,\delta}\in D(X)\times D(X)$ be any element of $[(x_{0},y_{0})^{\epsilon,\delta}]$. There is an element $h\in F(D(X))$ with $(x,y)^{\epsilon,\delta}=(x_{0},y_{0})^{\epsilon,\delta}\cdot h$. By Lemma 3.5, $(x,y)=(x_{0},y_{0})\cdot\beta(h)\in E$. Thus, $(x,y)^{\epsilon,\delta}\in E^{\epsilon,\delta}$. Hence, $[(x_{0},y_{0})^{\epsilon,\delta}]\subset E^{\epsilon,\delta}$. ∎

Proof of Theorem 3.4.  It follows from Lemma 3.6. ∎

The following is seen from Example 3.3 and Theorem 3.4.

In this section we introduce involutions on tensor products.

Let $X\otimes X$ be the tensor product of a quandle $X$. Let $\tau:X\times X\to X\times X$ be the involution of $X\times X$ switching the components, $\tau(x_{1},x_{2})=(x_{2},x_{1})$. Since $\tau$ commutes with the canonical right action of $F(X)$ on $X\times X$, it induces an involution on $X\otimes X$, which we denote also by

Let $\rho:X\to X$ be an involution. It induces an invitation $\rho~{}(=\rho\otimes\rho):X\times X\to X\times X$ with $\rho(x_{1},x_{2})=(\rho(x_{1}),\rho(x_{2}))$. Note that $\rho:X\times X\to X\times X$ commutes with the canonical right action of $F(X)$ on $X\times X$ if and only if $\rho:X\to X$ satisfies the first condition (S1) of a good involution.

Let $\rho:X\to X$ be a good involution. Then the involution $\rho:X\times X\to X\times X$ induces an involution on $X\otimes X$, which we denote also by

We denote by $X\otimes X/\langle\tau\rangle$ ($X\otimes X/\langle\rho\rangle$ or $X\otimes X/\langle\tau,\rho\rangle$, resp.) the set of equivalence classes of $X\otimes X$ by the equivalence relation generated by $\tau$ (by $\rho$ or by $\tau$ and $\rho$, resp.).

In Section 5 we compute $R_{n}\otimes R_{n}/\langle\tau\rangle$ for every dihedral qaundle $R_{n}$.

The following lemma is straightforward and we omit the proof.