2. Listing potential Jones polynomials

The Jones polynomial $V_K(t) \in{\mathbb Z}[t^{\pm 1}]$ of links $K \subset S^3$ admits an elegant recursive definition by a skein relation, which we only paraphrase here, since we will not make explicit use of it:

\begin{equation*}t^{-1}\,V(L_+)-t\,V(L_-)=\big(t^{\frac {1}{2}}-t^{-\frac {1}{2}}\big)\,V(L_0).\end{equation*}

As usual, the notation $L_+,L_-,L_{\infty }$ refers to diagrams of links that look locally as follows:

When restricted to knots $K$ , rather than links, the Jones polynomial satisfies the following restrictions in the roots of unity $1,i,\zeta _3,\zeta _6$ :

1. 1. $V_K(1)=1$ ,

2. 2. $V_K^{\prime}(1)=0$ ,

3. 3. $V_K(\zeta _3)=1$ ,

4. 4. $V_K(i)=\pm 1$ ,

5. 5. $V_K(\zeta _6)=\pm (\sqrt{-3})^m$ .

The first four conditions were already derived by Jones [Reference Jones4]; the fifth one by Przytycki [Reference Przytycki6]. In fact, these restrictions admit generalisations to links, giving rise to similar divisibility results for the Jones polynomial of links; we will not discuss these here. The exponent $m$ in condition (5) coincides with the rank of the first homology of the double branched cover $M_2(K)$ with coefficients in ${\mathbb F}_3={\mathbb Z}/3{\mathbb Z}$ , and can also be interpreted as the dimension of the $3$ -colouring invariant of $K$ , as described in [Reference Przytycki6]. The sign in condition (4) is determined by the Arf invariant of $K$ : $V_K(i)=(\!-\!1)^{\text{Arf}(K)}$ . In terms of Vassiliev invariants, the first two conditions reflect the fact that knots admit no non-constant finite type invariants of order zero and one [Reference Birman and Lin1]. Interestingly, this implies that no monomial other than $1$ is the Jones polynomial of a knot [Reference Ganzell3]. Here is a remarkable consequence of the first three conditions together: $V_K(t)-1$ is divisible by $(t-1)^2(t^2+t+1)=(t^3-1)(t-1)$ .

Even better, suppose $p(t) \in{\mathbb Z}[t^{\pm 1}]$ admits the same values as $V_K(t)$ , for $t=1,i,\zeta _3,\zeta _6$ , and satisfies $p^{\prime}(1)=0$ . Then the difference $V_K(t)-p(t)$ is divisible by $(t-1)^2$ times the product of the minimal polynomials of $i, \zeta _3,\zeta _6$ :

\begin{equation*} \begin {aligned} f(t) &= (t-1)^2(t^2+1)(t^2+t+1)(t^2-t+1) \\ &= t^8-2t^7+3t^6-4t^5+4t^4-4t^3+3t^2-2t+1. \end {aligned} \end{equation*}

Therefore, all we need in order to derive Theorem 2 is finding a suitable set of reference polynomials $p(t)$ , with $p^{\prime}(1)=0$ , covering all the possible values of knot Jones polynomials at $t=1,i,\zeta _3,\zeta _6$ . This is easy enough.

Proof of Theorem 2. First, we observe that all the four families of polynomials listed in Theorem 2 satisfy $p(1)=1$ , $p^{\prime}(1)=0$ and $p(\zeta _3)=1$ . Here, we use the fact that $h(t)=(t^3-1)(t-1)(t^2+1)$ has a double root at $t=1$ and a single root at $t=\zeta _3$ .

Next, we observe that all the polynomials of families (i) and (iv) listed in Theorem 2 satisfy $p(i)=1$ , and all the polynomials of families (ii) and (iii) satisfy $p(i)=-1$ . Here, we use that $h(t)$ also has a single root at $t=i$ .

Last, we take care of the value $p(\zeta _6)$ , which should cover all the complex numbers of the form $\pm (\sqrt{-3})^m$ . The values of

\begin{equation*}p(t)=1,V_{3_1}(t),V_{5_1}(t),V_{8_{21}}(t)\end{equation*}

at $t=\zeta _6$ are $1,\sqrt{3}i,-1,\sqrt{3}i$ , respectively. Furthermore, we have $h(\zeta _6)=2$ and $h(\zeta _6)(2\zeta _6-1)=2\sqrt{3}i$ . This implies that the polynomials of families (i) and (iii) cover all the odd integers at $t=\zeta _6$ , while the polynomials of families (ii) and (iv) cover all the odd multiples of $\sqrt{3}i$ at $t=\zeta _6$ . Altogether, the four families listed in Theorem 2 cover all the possible combinations of values of knot Jones polynomial at $t=1,i,\zeta _3,\zeta _6$ , including the double root at $t=1$ . This finishes the proof of Theorem 2.

3. Jones polynomial modulo primes

The goal of this section is to derive Theorem 1 by reducing Theorem 2 modulo a fixed prime number $p$ . We use the notation $\bar{f}(t) \in{\mathbb F}_p[t^{\pm 1}]$ for the reduction of $f(t) \in{\mathbb Z}[t^{\pm 1}]$ modulo $p$ . Theorem 2 remains valid modulo $p$ , with the additional feature that the parameter $n$ is in ${\mathbb F}_p$ . From this, we deduce that the number of Jones polynomials of knots modulo $p$ in the degree range $[0,7]$ is at most $4p$ . This is in accordance with the ratio $4/p^7$ , since there are exactly $p^8$ polynomials modulo $p$ in the degree range $[0,7]$ . We will refer to these $4p$ potential Jones polynomials as reference polynomials $\bar{f}_1,\bar{f}_2,\ldots,\bar{f}_{4p} \in{\mathbb F}_p[t^{\pm 1}]$ .

Proof of Theorem 1. Suppose we are given a degree range $[a,b]$ with $b-a \geq 7$ and a knot $K$ with Jones polynomial $\bar{V}_K(t)$ in that degree range. By Theorem 2, there exists a reference polynomial $\bar{f}_i$ , so that $\bar{V}_K(t)-\bar{f}_i$ is divisible by

\begin{equation*}\bar {f}(t)=t^8-2t^7+3t^6-4t^5+4t^4-4t^3+3t^2-2t+1 \in {\mathbb F}_p[t^{\pm 1}].\end{equation*}

Denote the minimal and maximal degree of $\bar{V}_K(t)-\bar{f}_i$ by $\alpha$ and $\beta$ , respectively. Then there exist unique coefficients

\begin{equation*}c_\alpha,c_{\alpha +1},\ldots,c_{\beta -8} \in {\mathbb F}_p,\end{equation*}

satisfying the following equation:

\begin{equation*}\bar {V}_K(t)-\bar {f}_i\,=\,\bar {f}(t)\left(c_\alpha t^\alpha +c_{\alpha +1} t^{\alpha +1}+\cdots +c_{\beta -8}t^{\beta -8}\right).\end{equation*}

The polynomial $\bar{V}_K(t)$ is therefore determined by $\beta -\alpha -7$ parameters in ${\mathbb F}_p$ . However, since $\bar{V}_K(t)$ is in the degree range $[a,b]$ , all the coefficients $c_\gamma$ with $\gamma \not \in [a,b-8]$ are determined by $f_i$ alone. In other words, only the coefficients $c_a,c_{a+1},\ldots,c_{b-8}$ change if we vary $\bar{V}_K(t)$ in the given degree range. Since there are $4p$ reference polynomials $\bar{f}_i$ , this allows for a maximum of $4p$ times $p^{b-a-7}$ potential Jones polynomials, out of a total of $p^{b-a+1}$ polynomials with coefficients in ${\mathbb F}_p$ in the degree range $[a,b]$ . The resulting ratio is again $4/p^7$ , as claimed.

For odd primes $p \geq 5$ , the bound $4/p^7$ is never sharp, since the parameter $n$ appearing in Theorem 2, case (i), satisfies $1+2n=\pm 3^l$ . In particular, $2n$ cannot be $-1$ (mod $p$ ), since $3^l$ cannot be zero (mod $p$ ). The knot table at our disposition (knotinfo, up to 12 crossings [Reference Livingston and Moore5]) is too small to draw any conclusion about the sharpness of the bound $4/p^7$ for $p=3$ . This leaves us with the case $p=2$ , which is most interesting and deserves its own section.

4. Jones polynomial modulo two

The list of $4p$ potential Jones polynomials in the degree range $[0,7]$ , called reference polynomials in the previous section, boils down to eight polynomials for $p=2$ . These are in fact realised by the following knots: the trivial knot $O$ , $3_1$ , $5_1$ , $5_2$ , $8_{21}$ , $9_{43}$ , $10_{140}$ , $10_{160}$ . The corresponding Jones polynomials (mod $2$ ) are

\begin{equation*}1, \,\, t+t^3+t^4, \,\, t^2+t^4+t^5+t^6+t^7, \,\, t+t^2+t^4+t^5+t^6,\end{equation*}

\begin{equation*}t^3+t^4+t^7, \,\, 1+t+t^7, \,\, 1+t+t^2+t^3+t^5+t^6+t^7, \,\, 1+t^2+t^3+t^5+t^6.\end{equation*}

In order to prove Corollary 1, we need to find $16$ knot Jones polynomials in the degree range $[a,a+8]$ , for all $a \in{\mathbb Z}$ , which appears rather difficult. Luckily, a single knot comes at our rescue: $12n237$ .

As mentioned above, no monomial other than $1$ is the Jones polynomial of a knot. Indeed, no polynomial of the form $p(t)=at^n$ , except $1$ , satisfies $p(1)=1$ and $p^{\prime}(1)=0$ . In contrast, the Jones polynomial of the knot $12n237$ is a non-trivial monomial modulo $2$ :

\begin{equation*}\bar {V}_{12n237}(t)=t^{12} \, (\text {mod 2}).\end{equation*}

Proof of Corollary 1. Using the knot $12n237$ with Jones polynomial $t^{12}$ modulo 2, we can reduce the realisation problem to finitely many values of $a \in{\mathbb Z}$ . Indeed, suppose we find 16 Jones polynomials in a fixed degree range $[a,a+8]$ , realised by the knots $K_1,K_2,\ldots,K_{16}$ . Then, by adding $k$ copies of the knot $12n237$ to the knots $K_i$ , we obtain 16 Jones polynomials in the degree range $[a+12k,a+12k+8]$ . This also works for negative integers $k$ , by adding $|k|$ copies of the mirror image of the knot $12n237$ to the $K_i$ . Hence, in order to cover all degree ranges, it is sufficient to consider the cases $-9 \leq a \leq 2$ . In fact, it is even enough to consider the cases $-4 \leq a \leq 2$ , by the symmetry $V_K(t)=V_{K^*}(t^{-1})$ between the Jones polynomial of a knot $K$ and its mirror image $K^*$ . Based on Rolfsen’s table [Reference Rolfsen7] and knotinfo [Reference Livingston and Moore5], we found $53$ prime knots, plus the trivial knot $O$ , which provide 16 Jones polynomials in all degree ranges of the form $[a,a+8]$ , $a \in \{-4,-3,-2,-1,0,1,2\}$ . These knots include all knots with crossing number $ \leq 8$ , except the knots $8_9,8_{13},8_{16},8_{18}$ (whose Jones polynomials modulo $2$ coincide with the ones of $4_1 \# 4_1,8_4,8_{10},8_{12}$ , in this order), as well as the following knots:

\begin{equation*}9_{42},9_{43},9_{44},10_{124},10_{126},10_{127},10_{128},10_{133},10_{136},10_{140},10_{143},10_{145},\end{equation*}

\begin{equation*}10_{146},10_{147},10_{160},10_{163},10_{165},11n63,11n71,11n99,11n118,11n173.\end{equation*}

The table below indicates the degree range of their corresponding Jones polynomials modulo $2$ . Our convention here is chosen so that $K$ has higher maximal degree than $K^*$ . By taking suitable connected sums of these knots, together with the knot $12n237$ (making it a total of $54$ prime knots), and all their mirror images, we find 16 Jones polynomials in every degree range of the form $[a,a+8]$ , as stated in Corollary 1.

We invite the reader to answer the following concluding questions.

Question 1. Let $p$ be an odd prime. Is there a knot $K \subset S^3$ with

\begin{equation*}\bar {V}_K(t)=t^{12p} \text { (mod $p$)}?\end{equation*}

Table 1. Knots with Jones polynomials of span ≤8