Arithmetic on q-deformed rational numbers

\[[n]_q=\frac {1-q^n}{1-q}=q^{n-1}+q^{n-2}+\cdots +q+1,\]

is a very classical subject of mathematics. Recently, Morier-Genoud and Ovsienko [MO20] introduced the \(q\)-deformation \([\alpha ]_q\) of a rational number \(\alpha \) based on some combinatorial properties of rational numbers. They further extended this notion to arbitrary real numbers [MO22] by some number-theoretic properties of irrational numbers. These works are related to many directions including Teichmüller spaces [FC99] , the 2-Calabi-Yau category of type \(A_2\)  [BBL23] , the Markov-Hurwitz approximation theory [K22] , [LL22] , [LMOV21] , [R22(a)] , the modular group and Picard groups [LeM21] , [MOV24] , [Ov21] , Jones polynomials of rational knots [KW19(a)] , [LS19] , [NT20] , [MO20] , [BBL23] , [R22(b)] , and combinatorics on fence posets [MSS21] , [Og22] , [OR23] .

For an irreducible fraction \(\frac {r}s >0\), we have

\[\left [\dfrac {r}{s}\right ]_q =\dfrac {\cR _{\frac {r}s}(q)}{\cS _{\frac {r}s}(q)} \ \ \text {for} \ \ \cR _{\frac {r}s}(q), \cS _{\frac {r}s}(q)\in \ZZ _{>0}[q] \ \ \text {with} \ \ \cR _{\frac {r}s}(1)=r \ \ \text {and} \ \ \cS _{\frac {r}s}(1)=s.\]

There are many ways to compute \([\alpha ]_q\) (see Section 2 for details). For example, we have

\[ \left [\frac {6}{5}\right ]_q=\frac {[6]_q}{[5]_q}=\frac {q^5+q^4+q^3+q^2+q+1}{q^4+q^3+q^2+q+1} , \quad \left [\frac {7}{5}\right ]_q =\frac {{q}^{4}+2\,{q}^{3}+2\,{q}^{2}+q+1}{{q}^{3}+2\,{q}^{2}+q+1}, \]

and observe that the denominators of \(\frac {6}{5}\) and \(\frac {7}{5}\) are the same 5, but the denominator polynomials of their \(q\)-deformation are different. In general, the following problem arises. When dose the equation \(\cS _{\frac {r}s}(q)=\cS _{\frac {r'}s}(q)\) hold for two irreducible fractions \(\frac {r}s\) and \(\frac {r'}s\)? By definition, we have \(\cS _{\alpha +n}(q)=\cS _{\alpha }(q)\) for \(n \in \ZZ \), and hence \(r \equiv r' \pmod {s}\) implies \(\cS _{\frac {r}s}(q)=\cS _{\frac {r'}s}(q)\). However, there are more subtle relations.

From these examples, the third author of the present paper and Takeshi Sakurai, who were supervised by the first author, proposed the following conjecture in their master theses [R21] , [S21] . This is the main motivation of the present paper.

In this section, without the assumption that \(p\) is a prime number, we will show that \(ab \equiv -1 \pmod {p}\) implies \(\cS _{\frac {a}{p}}(q)=\cS _{\frac {b}{p}}(q), \) that is, the sufficiency part of Conjecture 1.2 holds. Recall that \(\cS _{\alpha +n}(q)=\cS _{\alpha }(q)\) for all \(\alpha \in \QQ \) and \(n \in \mathbb {Z}\).

In the rest of the paper, \(p\) means a (not necessarily prime) integer with \(p \ge 2\), unless otherwise specified.

Proof.

Proof.

The following lemma is a variant of “Palindrome Theorem" (for example, see [KL02], Theorem 4 ) for continued fractions. We will give a direct proof here for the reader's convenience.

Proof.

Proof.

The following implies the sufficiency of Conjecture  1.2 .

Proof.

The assertion follows from Propositions  3.2 , 3.4 and the fact that \(f^{\vee \vee }(q)=f(q)\) for general \(f(q)\in \ZZ [q]\).

We note that, for the numerator \(\mathcal {R}_{\frac {r}{s}}(q)\) for \(\frac {r}s >1\), a similar result holds. See Lemma 4.1 below.

Regarding \(Q_\alpha ^\cS \) as a finite poset, Oǧuz and Ravichandran [OR23] intensely studied \(\cS _\alpha (q)\) from purely combinatorial point of view. Among other things, they showed that \(\cS _\alpha (q)\) is always unimodal. Here we apply their another result. A polynomial \(f(q)\) is said to be palindromic if \(f^{\vee }(q)=f(q)\).

Proof.

Let \({\mathbf b}=(b_1, \ldots , b_k)\) be an integer sequence such that \(b_1, b_k \ge 0\), \(b_2, \ldots , b_{k-1}>0\) and \(k\) is odd. [OR23], Theorem 1.3 (c) , which was first conjectured in [MSS21] , states that \(\cl (Q({\mathbf b}))\) is palindromic if and only if \(b_i=b_{k+1-i}\) for all \(1 \le i \le k\).

Set \(\frac {r}s=[a_1, \ldots , a_{2m}]\). By the above mentioned result, the \(q\)-polynomial \(\cS _{\frac {r}{s}}(q)=\cl (Q(a_2-1, a_3, \ldots , a_{2m}-1)^{\vee })\) is palindromic if and only if

\(\seteqnumber{0}{3.}{1}\)

\begin{equation} \label {symmetric} a_i =a_{2m+2-i} \quad \text {for all $2 \le i \le 2m$.} \end{equation}

By Lemma  3.3 , the condition ( 3.2 ) holds if and only if \(a^2 \equiv 1 \pmod {p}\).

Proof.

Combining the above results with Chinese remainder theorem, for a general \(s\), we can easily detect all \(r\) such that \(\cS _{\frac {r}s}\) is palindromic (equivalently, \(r^2 \equiv 1 \pmod {s}\)).

Proof.

In [W22] , the fourth author introduced the \(q\)-deformed integers derived from pairs of positive and coprime integers. In this section, by using them we give the second proof of the sufficiency of Conjecture  1.2 . To do this we need the following interpretation of the conjecture.

Proof.

This follows from Lemmas 2.4 and 2.6 .

The polynomial \((a, b)_q\) is convenient to compute \([\alpha \sharp \beta ]_q\).

Any rational number \(\alpha >0\) is associated with a link \(L(\alpha )\) in the \(3\)-sphere \(\mathbb {S}^3\) which is given by the diagram \(D(\alpha )\) below, and such a link is called a rational link or two-bridge link . If \(\alpha \) belongs to the open interval \((0,1)\), then the diagram \(D(\alpha )\) is given as in Figure  1 after the expression of \(\alpha =[0, a_1, \ldots , a_n]\) with odd \(n\).

where

If \(\alpha >1\), then \(D(\alpha )\) is defined by \(D(\alpha ):=D(\alpha ^{-1})\), and if \(\alpha =1\), then \(D(\alpha )=\) .

For a negative rational number \(\alpha \), a rational link \(L(\alpha )\) and its diagram \(D(\alpha )\) are defined in the same way as the positive case. Then we see that the link \(L(\alpha )\) is the mirror image of \(L(-\alpha )\). However, for any \(\alpha \in \QQ \), there is some \(\beta \in \QQ \cap (1, \infty )\) such that \(L(\alpha )\) and \(L(\beta )\) are isotopy. See, for example, [KL02], Theorem 2 . In this sense, we may assume that \(\alpha >1\).

As a useful isotopy invariant for an oriented link \(L\) in \(\mathbb {S}^3\), the Jones polynomial \(V_{L}(t)\) [J85] , [K87] , which is valued in \(\mathbb {Z}[t^{\pm \frac {1}{2}}]\), is well-studied. Lee and Schiffler  [LS19] introduced the following normalization \(J_{\alpha }(q)\) of the Jones polynomial \(V_{\alpha }(t):=V_{L(\alpha )}(t)\) of a rational link \(L(\alpha )\):

\[ J_1(q)=1,\quad J_{\infty }(q)=q. \]

By Lee and Schiffler  [LS19] , it is known that the Jones polynomial \(V_{\alpha }(t)\) can be recovered from \(J_{\alpha }(q)\). By [MO20], Proposition A.1 and the equation ( 2.18 ), we see that, for a rational number \(\alpha >1\), the normalized Jones polynomial \(J_{\alpha }(q)\) can be computed by

The equation ( 4.4 ) corresponds to the equation \(J_{\frac {r}{s}}(q)=\mathcal {R}^{\prime }-q\mathcal {S}^{\prime }\) in [MO20], p.45 under the setting \(\frac {r}{s}=\frac {p}{a}\). In fact, since \(\mathcal {S}^{\prime }=\mathcal {S}_{\frac {p}{a}}(q)\) and \(\mathcal {R}^{\prime }=q\mathcal {R}_{\frac {p}{a}}(q)+\mathcal {S}_{\frac {p}{a}}(q)\) as shown in [MO20], p.45 , we have \(\mathcal {S}^{\prime }=(a-r, r)_q\) by ( 4.6 ) below and \(\mathcal {R}^{\prime }=q(a, p-a)_q+(a-r, r)_q=(a, p)_q\) by ( 4.1 ) and ( 4.5 ) below.

As an application of Theorem  4.4 we have:

By using the above theorem, one can give another proof of Theorem  3.5 , that is, the sufficiency of Conjecture  1.2 .

Another proof of Theorem  3.5 ..

In the study of Conway-Coxeter friezes of zigzag-type developed by the first and the fourth authors [KW19(b)] , [KW19(a)] crucial three operators \(\mathfrak {i}, \mathfrak {r}, \mathfrak {ir}\) on the positive rational numbers are introduced. In this section we examine effect of the operators \(\mathfrak {i}, \mathfrak {r}, \mathfrak {ir}\) on \(\mathcal {R}_{\alpha }(q), \mathcal {S}_{\alpha }(q)\).

Let \(\alpha =\frac {z}{c} >0\) be an irreducible fraction. In the case where \(\alpha \in (0, 1)\), irreducible fractions \(\mathfrak {i}(\alpha ), \mathfrak {r}(\alpha ), (\mathfrak {ir})(\alpha )\) in the interval \((0,1)\) are defined as follows [KW19(a)] :

\[az=a(x+y)=ax+ay=ax+1+bx=1+(a+b)x=1+cx.\]

Hence as operations on \(\QQ \cap (0,1)\), we have \(\mathfrak {i}^2=\mathfrak {r}^2=\operatorname {id}\) and \(\mathfrak {ir}=\mathfrak {ri}\). By Theorem  3.6 , for \(\alpha \in \QQ \cap (0,1)\), \(\mathfrak {r}(\alpha )=\alpha \) if and only if \(\cS _\alpha (q)\) is palindromic.

In the case where \(\alpha >1\), \(\mathfrak {i}(\alpha ), \mathfrak {r}(\alpha )\), and \(\mathfrak {ir}(\alpha )\) are defined as follows:

By Lemmas 3.1 and 3.3 , and the equation ( 5.2 ), one can show that for a positive rational number \(\alpha =[0, a_2, \ldots , a_n]\),

\begin{align} (\mathfrak {ir})(\alpha )&={\begin{cases} [0, a_n, \ldots , a_3, a_2] & \text {if $n$ is odd}, \\ [0, 1, a_n-1, a_{n-1}, \ldots , a_3, a_2] & \text {if $n$ is even and $a_n \ge 2$}, \\ [0, a_{n-1}+1, a_{n-2}, \ldots , a_3, a_2] & \text {if $n$ is even and $a_n =1$,} \end {cases}} \label {eq3-4}\\[0.1cm] \mathfrak {i}(\alpha )&={\begin{cases}[0, 1, a_2-1, a_3, \ldots , a_n] & \text {if $a_2 \geq 2$}, \\ [0, a_3+1, a_4, \ldots , a_n] & \text {if $a_2=1$}, \end {cases}} \label {eq3-5}\\[0.1cm] \mathfrak {r}(\alpha )&={\begin{cases} [0, 1, a_n-1, a_{n-1}, \ldots , a_3, a_2] & \text {if $n$ is odd and $a_n \ge 2$},\\ [0, a_{n-1}+1, a_{n-2}, \ldots , a_3, a_2] & \text {if $n$ is odd and $a_n =1$,}\\ [0, a_n, \ldots , a_3, a_2]& \text {if $n$ is even}. \end {cases}} \label {eq3-6} \end{align}

Proof.

For a quiver \(Q\) of type \(A\), let denote by \(Q^{\mathsf {rot}}\) the quiver obtained from \(Q\) by \(\pi \)-rotation. Since \(Q\simeq Q^{\mathsf {rot}}\) as quivers, their closure polynomials are same;

For \(\alpha \in \mathbb {Q}\cap (1, \infty )\), the equations ( 5.4 ), ( 5.5 ) and ( 5.6 ) imply that

\begin{align} Q_{\mathfrak {\mathfrak {i}}(\alpha )}^\mathcal {R}&=(Q_{\alpha }^\mathcal {R})^{\vee }, \label {eq3-13}\\ Q_{(\mathfrak {ir})(\alpha )}^\mathcal {R}&=(Q_{\alpha }^\mathcal {R})^{\mathsf {rot}}, \label {eq3-14}\\ Q_{\mathfrak {r}(\alpha )}^\mathcal {R}&=(Q_{\alpha }^\mathcal {R})^{\mathsf {rot}\vee }=(Q_{\alpha }^\mathcal {R}) ^{\vee \mathsf {rot}}. \label {eq3-15} \end{align}

Except for the denominator of \(\mathfrak {i}(\alpha )\), the denominator and numerator polynomials of \(q\)-deformations of \(\mathfrak {i}(\alpha ), \mathfrak {r}(\alpha ), (\mathfrak {ir})(\alpha )\) are computed from that of \(\alpha \) and its Farey parent as follows.

Proof.

For a rational number \(\alpha \) with \(0<\alpha <1\), the \(q\)-deformations of \(\mathfrak {i}(\alpha ), \mathfrak {r}(\alpha )\) and \((\mathfrak {ir})(\alpha )\) behave as follows.

Proof.

As an application of Proposition 3.2 we have:

Proof.

On the one hand, for an irreducible fraction \(\alpha >1\), the denominator and numerator polynomials of \([\alpha ]_q\) are given by closure polynomials of some quivers of type \(A\) (see Theorem 2.10 ). On the other hand, from a representation theoretical viewpoint, the closure polynomial of a type \(A\) quiver \(Q\) counts subrepresentations of “the full interval representation" of \(Q\) in which a field \(\mathsf {k}\) corresponds to each vertex and the identity map corresponds to each arrow. In this section, we give an expression to calculate \(\cl (Q)\) that explicitly gives the number of subrepresentations of the full interval representation.

Let \(Q\) be a quiver of type \(A\), that is, the underlying graph of \(Q\) is \(A_n=1-2-3-\cdots -n\). A representation of \(Q\) over a field \(\mathsf {k}\) is a system \(M=(M_{a},\varphi _{\alpha })_{a\in Q_0,\alpha \in Q_1}\) (\(M=(M_a,\varphi _\alpha )\) for short) consisting of \(\mathsf {k}\)-vector spaces \(M_a\) \((a\in Q_0)\), and \(\mathsf {k}\)-linear maps \(\varphi _\alpha :M_{s(\alpha )}\to M_{t(\alpha )}\) (\(\alpha \in Q_1\)). The dimension of \(M\) is the sum of \(\mathsf {k}\)-dimensions of \(M_{a}\). A representation \(M'=(M'_a,\varphi '_\alpha )\) is said to be a subrepresentation of \(M\) if \(M'_a\) is a subspace of \(M_a\), and \(\varphi _\alpha '=\varphi _\alpha |_{M'_a}\). For two representations \(M=(M_a,\varphi _\alpha )\) and \(N=(N_a,\psi _\alpha )\), a morphism of representations \(f:M\to N\) is a family \(f=(f_a)_{a\in Q_0}\) of \(\mathsf {k}\)-linear maps \(f_a:M_a\to N_a\) such that \(\psi _\alpha f_{s(\alpha )}=f_{t(\alpha )}\varphi _\alpha \) for any arrow \(\alpha \).

The category of finite dimensional representations of \(Q\) is denoted by \(\mathsf {rep}(Q)\). It is well-known that there is an \(\mathsf {k}\)-linear equivalence between \(\mathsf {rep}(Q)\) and the category of finitely generated \(\mathsf {k}Q\)-modules, where \(\mathsf {k}Q\) is the path algebra of \(Q\). For a vertex \(i \in Q_0\), we denote by \(S(i)\) the corresponding simple \(\mathsf {k}Q\)-module. For a \(\mathsf {k}Q\)-module \(M\), we also denote by \(\mathsf {rad}(M)\), \(\mathsf {top}(M)\), and \(\mathsf {soc}(M)\) the Jacobson radical, the top, and the socle of \(M\), respectively. The support of \(M\) is the set of composition factors, which is denoted by \(\mathsf {supp}(M)\). In this subsection, any objects of \(\mathsf {rep}(Q)\) are freely regarded as objects of \(\mathsf {mod}~\mathsf {k}Q\). For representations of quivers, see [ASS06], Chapters II and III for more details.

By the Gabriel theorem (for example, see [ASS06], Chapter VII, Theorem 5.10 ), there is one-to-one corresponding between indecomposable objects of \(\mathsf {rep}(Q)\) and positive roots of \(A_n\), that is, pairs \((i,j)\) with \(1\leq i\leq j \leq n\). In this correspondence, each pair \((i,j)\) is assigned with the interval representation \(\mathbb {I}[i,j]=(M_a,\varphi _\alpha )\), where

\[ M_a=\left \{\begin {array}{ll} \mathsf {k} & \text {if $i\leq a\leq j$,} \\ \{0\} & \text {otherwise,} \end {array}\right . \quad \varphi _\alpha =\left \{\begin {array}{ll} 1 & \text {if $i\leq s(\alpha )$ and $t(\alpha )\leq j$,} \\ 0 & \text {otherwise.} \end {array}\right . \]

Following this notation, \(\mathbb {I}[1,n]\) is called the full interval representation of \(Q\). Then, it follows from the definition of the closure polynomial that the coefficient of \(q^{\ell }\) of \(\mathsf {cl}(Q)\) is equal to the number of \(\ell \)-dimensional subrepresentations of \(\mathbb {I}[1,n]\).

Throughout this section, we fix a type \(A\) quiver \(Q=Q(\mathbf {a})\) for some tuple \(\mathbf {a}=(a_1,a_2,\ldots , a_s)\in \mathbb {Z}^s_{\geq 0}\), which has \(n\) vertices, and denote by \(\mathbb {I}(\mathbf {a})\) the full representation of \(Q(\mathbf {a})\). It is clear that \(\rho _1(Q(\mathbf {a}))=\mathsf {dim}_\mathsf {k}~\mathsf {soc}({\mathbb {I}(\mathbf {a})})\), which equals to the number of sinks of \(Q(\mathbf {a})\). Note that the Jordan-Hölder theorem implies that any coefficients of \(\cl (Q(\mathbf {a}))\) are greater than or equal to \(1\). This yields that, for any irreducible fraction \(\alpha >1\), any coefficients of the polynomials \(\cR _\alpha {(q)}\) and \(\cS _\alpha {(q)}\) are greater than \(1\). We also remark that the top and the socle of \(\mathbb {I}(\mathbf {a})\) is given by

\begin{align*} & \mathsf {top}({\mathbb {I}(\mathbf {a})})=\bigoplus _{k\geq 1}S(1+a_1+a_2+\cdots +a_{2k-1}) \\ & \mathsf {soc}({\mathbb {I}(\mathbf {a})}))=\left \{\begin{array}{ll} S(1)\oplus \displaystyle \bigoplus _{k\geq 1}S(1+a_1+a_2+\cdots +a_{2k}) & \text {if $a_1\neq 0$,} \\ [10pt] \displaystyle \bigoplus _{k\geq 1}S(1+a_1+a_2+\cdots +a_{2k}) & \text {if $a_1= 0$.} \end {array}\right . \end{align*} Now, we choose \(1\leq k_1<k_2<\cdots < k_t\) and \(1\leq \ell _1<\ell _2< \cdots < \ell _{t'}\) to be

\begin{align*} & \mathsf {top}({\mathbb {I}(\mathbf {a})})=S(k_1)\oplus \cdots \oplus S(k_t), \\ & \mathsf {soc}({\mathbb {I}(\mathbf {a})})=S(\ell _1)\oplus \cdots \oplus S(\ell _{t'}). \end{align*} Here, we put

\[ \mathcal {T}_{\mathbf {a}}:=\{(k_{i_1},\ldots , k_{i_s})\in \mathbb {Z}^{s}\mid 1\leq i_1<\cdots <i_s\leq t,~s\in \mathbb {N}\}. \]

A subquiver of \(Q(\mathbf {a})\) of the form

\[ \underbrace {\circ \too \circ \cdots \circ \too \circ }_{p_1\, \text {arrows}}\underbrace { \oot \circ \cdots \circ \oot \circ }_{p_2 \, \text {arrows}} \]

is called a \((p_1,p_2)\)- valley . For \((p_1,p_2)\)-valley, we define a polynomial \(\mathsf {val}_q(p_1,p_2)\) by

\[ \val _q(p_1,p_2):=\cl (Q(0,p_1,p_2)).\]

Observe that the equation \(\val _q(p_1,p_2)=\val _q(p_2,p_1)\) holds by ( 5.7 ) and this can be calculated through the following.

Proof.

This lemma follows from direct computation.

Now, we define a sequence of pairs of integers as follows:

1. (i) Compute \(\mathbf {a}-\mathbf {1}:= \left \{\begin {array}{ll} (a_1-1, a_2-1,\ldots , a_s-1) & \text {if $a_1\neq 0$,} \\ ( a_2-1, a_3-1\ldots , a_s-1) & \text {if $a_1=0$.} \end {array}\right . \)
2. (ii) We put

   \[ (b_1,b_2,\ldots , b_{2m}):=\left \{\begin {array}{ll} (0,\mathbf {a}-\mathbf {1},0) & \text {if $a_1\neq 0$ and $s$ is even,} \\ [8pt] (0,\mathbf {a}-\mathbf {1}) & \text {if $a_1\neq 0$ and $s$ is odd,} \\ [8pt] ({\mathbf {a}-\mathbf {1}}, 0)& \text { if $a_1=0$ and $s$ is even,} \\[8pt] ({\mathbf {a}-\mathbf {1}})& \text { if $a_1=0$ and $s$ is odd.} \\[8pt] \end {array}\right . \]

3. (iii) We set \(\mathcal {J}_{\mathbf {a}}:=\{(b_1,b_2), (b_3,b_4),\ldots , (b_{2m-1},b_{2m})\}\).

Proof.

For each \(k_i\) \((i=1,\ldots , t)\), a polynomial \(\Delta _q(k_i)\) is defined as follows.

• (i) Suppose that \(a_1>0\). In this case, we define \(\Delta _q(k_i)\) by

  \[ \Delta _q(k_i):=\left \{\begin {array}{ll} q^{\ell _{i+1}-\ell _{i}+1}[\ell _{i}-k_{i-1}]_q[k_{i+1}-\ell _{i+1}]_q & \text {if $i\neq 1,t$,} \\ [8pt] q^{\ell _{2}}[k_{2}-\ell _{2}]_q& \text { if $i=1$,} \\[8pt] q^{n-\ell _{t}+1}[\ell _t-k_{t-1}]_q& \text { if $i=t$.} \end {array}\right . \]

• (ii) Suppose that \(a_1=0\). In this case, we define \(\Delta _q(k_i)\) by

  \[ \Delta _q(k_i):=\left \{\begin {array}{ll} q^{\ell _{i}-\ell _{i-1}+1}[\ell _{i-1}-k_{i-1}]_q[k_{i+1}-\ell _{i}]_q & \text {if $i\neq 1,t$,} \\ [8pt] q^{\ell _{1}}[k_{2}-\ell _{1}]_q& \text { if $i=1$,} \\[8pt] q^{n-\ell _{t-1}+1}[\ell _{t-1}-k_{t-1}]_q& \text { if $i=t$.} \end {array}\right . \]

\[ \{(v_{2j-1}^{(k_i)},v_{2j}^{(k_i)})\mid j=1,2,\ldots , r_{k_i}\}\subset {\mathcal {J}_{\mathbf {a}}} \]

such that any \((v_{2j-1}^{(k_i)},v_{2j}^{(k_i)})\)-valley is not adjacent to vertex \(k_i\). Then, we set

\[ \widetilde {\Delta }_q(k_i):=\Delta _q(k_i)\prod _{j=1}^{r_{k_i}}\val (v_{2j-1}^{(k_i)},v_{2j}^{(k_i)}). \]

Proof.

Next, for two \(k_{i_1}<k_{i_2}\), we define

\[ \Delta _q(k_{i_1},k_{i_2}):=\left \{\begin {array}{ll} \dfrac {\Delta _q(k_{i_1})\Delta _q(k_{i_2})}{q[\ell _{i_2}-k_{i_2-1}]_q[k_{i_1+1}-\ell _{i_1+1}]_q} & \text {if $i_2=i_1+1$,}\\[15pt] \Delta _q(k_{i_1})\Delta _q(k_{i_2}) & \text {otherwise.} \end {array}\right . \]

Inductively, for \(k_{i_1}<k_{i_2}<\cdots <k_{i_r}\), we define

\[ \Delta _q(k_{i_1},k_{i_2},\ldots , k_{i_r}):= \left \{\begin {array}{ll} \dfrac {\Delta _q(k_{i_1}, k_{i_2},\ldots , k_{i_r-1})\Delta _q(k_{i_r})}{q[\ell _{i_r}-k_{i_r-1}]_q} & \text {if $i_r=i_{r-1}+1$,}\\[15pt] \Delta _q(k_{i_1}, k_{i_2},\ldots , k_{i_r-1})\Delta _q(k_{i_r}) & \text {otherwise.} \end {array}\right . \]

Take a subset

\[ \{(v_{2j-1}^{(k_{i_1},\ldots , k_{i_r})},v_{2j}^{(k_{i_1},\ldots , k_{i_r})})\mid j=1,2,\ldots , r_{(k_{i_1},\ldots , k_{i_r})}\}\subset {\mathcal {J}_{\mathbf {a}}} \]

such that any \((v_{2j-1}^{(k_{i_1},\ldots , k_{i_r})},v_{2j}^{(k_{i_1},\ldots , k_{i_r})})\)-valley is not adjacent to one of vertices \(k_{i_1},\ldots , k_{i_r}\). Then, for \(r=1,\ldots , t\), we set

\[ \widetilde {\Delta }_q(k_1,\ldots , k_r):=\Delta _q(k_1,\ldots ,k_r)\prod _{j=1}^{r_{(k_1,\ldots ,k_r)}}\val (v_{2j-1}^{(k_1,\ldots ,k_r)},v_{2j}^{(k_1,\ldots ,k_r)}). \]

Proof.

By Proposition 6.2 , the first term of the right-hand side of ( 6.1 ) counts \(\ell \)-dimensional subrepresentations of \(\mathsf {rad}(\mathbb {I}(\mathbf {a}))\). Therefore, counting the cases where each \(S(k_i)\) \((i=1,\ldots , t)\) belongs to the support is sufficient. By the proof of Lemma 6.3 , the number of \(\ell \)-dimensional subrepresentations \(N\) of \(\mathbb {I}(\mathbf {a})\) such that \(S(k_{i_1}), \ldots , S(k_{i_r})\in \mathsf {supp}(N)\) but \(S_j\not \in \mathsf {supp}(N)\) for \(j\neq k_{i_1},\ldots ,k_{i_r}\) is the coefficient of \(q^\ell \) of \(\widetilde {\Delta }_q(k_{i_1},\ldots , k_{i_r})\). Thus, the assertion follows.

\[\omega :=\dfrac {-1+\sqrt {-3}}2.\]

Proof.

First, we assume that \(\alpha >1\), and write \(\alpha =[[c_1, \ldots , c_l]]\). By Proposition  2.2 , we have

\[ \begin {pmatrix}\cR _\alpha (\omega )\\ \cS _\alpha (\omega ) \end {pmatrix}= \left ( M_q^-(c_1) M_q^-(c_2) \cdots M_q^-(c_l)\right )|_{q=\omega } \begin {pmatrix} 1\\0 \end {pmatrix}. \]

It is easy to check that \(M_q^-(c)|_{q=\omega }\) for a positive integer \(c\) is one of the following forms:

\[ M_q^-(c)|_{q=\omega }=\left \{ \begin {array}{ll} X:=\begin {pmatrix}0&-\omega ^2\\1&0 \end {pmatrix}& \text {if $c \equiv 0 \pmod {3}$,} \\[15pt] Y:= \begin {pmatrix}1&-1\\1&0 \end {pmatrix} & \text {if $c \equiv 1 \pmod {3}$,} \\[15pt] Z:=\begin {pmatrix}-\omega ^2&-\omega \\1&0 \end {pmatrix} & \text {if $c \equiv 2 \pmod {3}$.} \end {array} \right . \]

Let \(G\) be the subgroup of \(\mathsf {GL}(2,\CC )\) generated by \(X,Y\) and \(Z\). A direct computation shows that the \(X^{12}=Y^6=Z^3=E_2\). Set

\(\seteqnumber{0}{7.}{0}\)

\begin{equation} \label {set A} A:=\left \{ \zeta \! \begin{pmatrix}1\\0\end {pmatrix}, \zeta \! \begin{pmatrix}1\\-\omega \end {pmatrix}, \zeta \! \begin{pmatrix}1\\1\end {pmatrix}, \zeta \! \begin{pmatrix}0\\1 \end {pmatrix} \, \middle | \, \zeta = \pm 1, \pm \omega , \pm \omega ^2 \right \} \subset \CC ^2. \end{equation}

Then, easy calculation shows that \(A\) is closed under the natural action of \(G\). Thus, for any \(W \in G\), all entries of \(W\), especially \(\cR _\alpha (\omega )\) and \(\cS _\alpha (\omega )\) for \(\alpha >1\), belong to the set \(\{0, \pm 1, \pm \omega , \pm \omega ^2 \}\).

Let us consider the case \(\alpha \le 1\). By ( 2.8 ), we have

\(\seteqnumber{0}{7.}{1}\)

\begin{equation} \label {alpha => alpha+1} \begin{pmatrix}\cR _\alpha (\omega )\\ \cS _\alpha (\omega )\end {pmatrix} = \begin{pmatrix}\omega ^2&-\omega ^2\\0&1 \end {pmatrix} \begin{pmatrix} \cR _{\alpha +1}(\omega )\\ \cS _{\alpha +1}(\omega )\end {pmatrix}. \end{equation}

It is easy to check that the set \(A\) is closed under the multiplication of \(\begin {pmatrix}\omega ^2&-\omega ^2\\0&1 \end {pmatrix}\), so we can show that \(\begin {pmatrix} \cR _{\alpha +1}(\omega )\\ \cS _{\alpha +1}(\omega )\end {pmatrix} \in A\) implies \(\begin {pmatrix} \cR _{\alpha }(\omega )\\ \cS _{\alpha }(\omega )\end {pmatrix}\in A\). Since \(\alpha +n >1\) for sufficiently large \(n\), the desired assertion follows from repeated use of the above implication.

Since the leading coefficient of \([n]_q=1+q+\cdots +q^{n-1}\) is 1, when we divide \(f(q) \in \ZZ [q]\) by \([n]_q\), the quotient and the remainder belong to \(\ZZ [q]\). It is clear that if \(\cS _{\frac {r}s}(q)\) can be divided by \([3]_q=1+q+q^2\), then \(s=\cS _{\frac {r}s}(1)\) is a multiple of 3. The following states that the converse is also true.

Proof.

Proof.

In the rest of this section, \(i\) means \(\sqrt {-1}\).

Proof.

Proof.

The next result can be proved by an argument similar to the corresponding results for \(q=\omega \).

Using the computer program Maple, we checked the conjecture for prime numbers up to 739. The following is another piece of evidence.

Proof.

For a general link \(L\), it is a classical fact that

\[V_L(1)=(-2)^{c(L)-1},\]

where \(c(L)\) is the number of the components of \(L\). Hence we have

\[ V_{\frac {r}s}(1)= \begin {cases} -2 &\text {if $r$ is even},\\ 1 & \text {if $r$ is odd}. \end {cases} \]

We can explain this equation using \(q\)-deformed rationals.

Recall the equation ( 4.3 ), which states that the normalized Jones polynomial \(J_\alpha (q)\) of a rational link \(L(\alpha )\) can be computed by the following formula:

\[ J_\alpha (q)=q \cdot \cR _\alpha (q)+(1-q) \cdot \cS _\alpha (q). \]

By an argument similar to the previous section, we can show that

\[\begin {pmatrix}\cR _{\frac {r}s}(-1)\\\cS _{\frac {r}s}(-1)\end {pmatrix}=\pm \begin {pmatrix}1\\0\end {pmatrix}, \pm \begin {pmatrix} 0 \\1\end {pmatrix}, \pm \begin {pmatrix}1\\1\end {pmatrix}\]

(this is a refinement of [MO20, Proposition 1.8]). Hence we have

Next, we will consider the special values of \(J_\alpha (q)\) at \(q=i, \omega , -\omega \). Many parts of the following results should be well-known, but we are interested in the relation to \(q\)-deformed rationals.

Proof.

The next result can be proved similarly to Theorem  8.1 , but we use Proposition  7.5 this time.

For a general link \(L\), it is known that \(V_L(\omega )=(-1)^{c(L)-1}\). Hence, for a rational link \(L(\frac {r}s)\), we have \(V_{\frac {r}s}(\omega )=(-1)^{r-1}\) and hence

\[ \begin {array}{llll} X_0:=\begin {pmatrix}0&\omega ^2\\1&0 \end {pmatrix} & \text {if $c \equiv 0 \pmod {6}$,} \\[15pt] X_1:=\begin {pmatrix}1&-1\\1&0 \end {pmatrix} & \text {if $c \equiv 1 \pmod {6}$,} \\[15pt] X_2:=\begin {pmatrix}1-\omega &\omega \\1&0 \end {pmatrix} & \text {if $c \equiv 2 \pmod {6}$,} \\[15pt] X_3:=\begin {pmatrix}1-\omega +\omega ^2&-\omega ^2\\1&0 \end {pmatrix} & \text {if $c \equiv 3 \pmod {6}$,} \\[15pt] X_4:=\begin {pmatrix}-\omega +\omega ^2&1 \\1&0 \end {pmatrix} & \text {if $c \equiv 4 \pmod {6}$,} \\[15pt] X_5:=\left ( \begin {array}{cccc}\omega ^2&-\omega \\1&0 \end {array}\right ) & \text {if $c \equiv 5 \pmod {6}$.} \end {array} \]

By ( 4.3 ), for \(\alpha =[[c_1, \ldots , c_l]]\), we have

\[\begin {pmatrix} J_\alpha (-\omega )& * \end {pmatrix}= \begin {pmatrix} -\omega & 1+\omega \end {pmatrix} \cdot \left ( M_q^-(c_1) M_q^-(c_2) \cdots M_q^-(c_l)\right )|_{q=-\omega },\]

where \(\begin {pmatrix} J_\alpha (-\omega )& * \end {pmatrix}\) and \(\begin {pmatrix} -\omega & 1+\omega \end {pmatrix}\) are \(1 \times 2\) matrices, and \(\cdot \) means the product of matrices. Easy calculation shows that \(\begin {pmatrix} -\omega & 1+\omega \end {pmatrix}=-\omega \begin {pmatrix} 1& \omega \end {pmatrix}\) and there exists \(\zeta _i \in \{\pm 1, \pm \omega , \pm \omega ^2 \}\) such that

\[\begin {pmatrix} 1 & \omega \end {pmatrix} \cdot X_i = \zeta _i \begin {pmatrix} 1 & \omega \end {pmatrix} \]

for each \(0 \le i \le 5\). So we can show ( 8.4 ) by induction on \(l\).