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On Knots, Complements, and 6j-Symbols

Abstract

This paper investigates the relation between colored HOMFLY-PT and Kauffman homology, \({{\,\mathrm{SO}\,}}(N)\) quantum 6j-symbols, and (at)-deformed \(F_K\). First, we present a simple rule of grading change which allows us to obtain the [r]-colored quadruply graded Kauffman homology from the \([r^2]\)-colored quadruply graded HOMFLY-PT homology for thin knots. This rule stems from the isomorphism of the representations \((\mathfrak {so}_6,[r]) \cong (\mathfrak {sl}_4,[r^2])\). Also, we find the relationship among A-polynomials of \({{\,\mathrm{SO}\,}}\) and \({{\,\mathrm{SU}\,}}\) type coming from a differential on Kauffman homology. Second, we put forward a closed-form expression of \({{\,\mathrm{SO}\,}}(N)(N\ge 4)\) quantum 6j-symbols for symmetric representations and calculate the corresponding \({{\,\mathrm{SO}\,}}(N)\) fusion matrices for the cases when representations . Third, we conjecture closed-form expressions of (at)-deformed \(F_K\) for the complements of double twist knots with positive braids. Using the conjectural expressions, we derive t-deformed ADO polynomials.

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Fig. 1
Fig. 2

Notes

  1. 1.

    In this paper, a representation specified by a Young diagram \((r_1 \ge r_2 \ge \cdots )\) is denoted by \(R=[r_1, r_2, \ldots ]\). In particular, we write \([r^s]\) for an \(r \times s\) rectangular Young diagram.

  2. 2.

    We define the normalized quantum invariant

    In this paper, we focus only on knot invariants normalized by the unknot so that HOMFLY-PT \(P_R(K;a,q)\) and Kauffman \(F_R(K;a,q)\) polynomials are also normalized in a similar manner.

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Acknowledgements

We would like to thank Chen Yang for collaboration at the initial stage of the project. S.N. is indebted to Bruno le Floch for collaboration and discussion on 6j-symbols, and he also thanks Ryo Suzuki for identifying Ref. [36] about \({{\,\mathrm{SO}\,}}(N)\) 6j-symbols. We also would like to thank Sunghyuk Park for identifying the relationship between the t-deformed ADO polynomials of \(3_1\) and \(^*3_1\). This work was supported by the National Science Foundation of China under Grant No. NSFC PHY-1748958.

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Communicated by Ruben Minasian.

Appendix A: Derivation of the ADO polynomials

Appendix A: Derivation of the ADO polynomials

In this Appendix, we derive the t-deformed p-th ADO polynomial (4.14) and its higher rank generalization (4.19) from \(F_K(x,a,q,t)\) (4.1) and (4.5). We use \(F_K\) as a short for,

$$\begin{aligned} \lim _{q \rightarrow \zeta _p} F_K(x, a = -t^{-1} q^N, q, t), \end{aligned}$$

and similarly for other functions discussed in this section. We use q and \(\zeta _p\) interchangeably. When an integer k is displayed as \(k = k_1 p + k_0\), it is implied that \(k_1, k_0 \in \mathbb {N}\), and \(0 \le k_0 \le p - 1\).

Our closed formulae are built from q-binomials and q-Pochhammer symbols and we will first discuss their behavior when q goes to roots of unity. The q-Lucas theorem [72] states that,

$$\begin{aligned} {l \brack k}_{q = \zeta _p} = {l_1 \atopwithdelims ()k_1} {l_0 \brack k_0}_{q = \zeta _p}, \end{aligned}$$
(A.1)

where \(l = l_1 p+ l_0,~k = k_1p + k_0\). For q-Pochhammer, we have

$$\begin{aligned} (a;q)_k = (1 - a^p)^{k_1} (a;q)_{k_0}, \end{aligned}$$
(A.2)

where \(q = \zeta _p\), \(k = k_1 p + k_0\). In the following discussion, We will break \(F_K(x,a,q,t)\) into parts. As we will see later, these components enjoy similar properties. It turns out that \(F_K\) can be written as a product of an infinite summation over \(k_1\) and a finite one over \(k_0\), where \(k = k_1p +k_0\) and \(q = \zeta _p\), and the infinite summation over \(k_1\) can be repackaged by the generalized binomial theorem,

$$\begin{aligned} \frac{1}{(1-z)^n} = \sum _{i = 0}^\infty {n + i -1 \atopwithdelims ()i} z^i. \end{aligned}$$
(A.3)

As a result, an ADO polynomial can be expressed as a finite summation of over \(k_0\).

ADO Polynomials of Double Twist Knot \(K_{m,n}~(m,n\in \mathbb {Z}_+)\)

For double twist knots \(K_{m,n}~(m,n \in \mathbb {Z}_+)\), we first analyze (at)-deformed \(F_K\) at radial limit \(q = \zeta _p\). Each component of (4.1) behaves as follows.

  • The twist factor \({\text {tw}}^{(l)}_m(a,q,t)\) (4.4) now becomes

    $$\begin{aligned} {\text {tw}}_m^{(l)} = \sum _{0\le b_1 \le \cdots \le b_{m-1} \le b_m = l} \prod _{i=1}^{m-1}(-q^N)^{b_i} q^{b_i (b_i-1)} t^{ b_i} {b_{i+1} \brack b_{i}}_q. \end{aligned}$$
    (A.4)

    We decompose the summation variables as, \(b_{i} = \alpha _i p + \beta _i\), and \(l = l_1 p + l_0\). Using the q-Lucas theorem, (A.4) can be written as a product of two summations, one over \(\alpha _i\)’s and another one over \(\beta _i\)’s. The summation over \(\beta _i\)’s would give \(\mathrm {tw}_m^{(l_0)}\). Performing the summation over \(\alpha _i\)’s, we obtain

    $$\begin{aligned} \mathrm {tw}_m^{(l)} = S_m^{l_1}((-t)^p)\,\mathrm {tw}_m^{(l_0)}, \end{aligned}$$
    (A.5)

    where \(S_m^{l_1}(x) := \left( S_m(x)\right) ^{l_1} := \left( \sum \limits _{i = 0}^{m-1}x^i\right) ^{l_1}\).

  • The twist factor \({\text {Tw}}_{K_{m,n}}^{(j)}(a,q,t)\) (4.3) becomes

    $$\begin{aligned} {\text {Tw}}_{K_{m,n}}^{(j)} = \sum _{l = 0}^j (-1)^l q^{\frac{1}{2} l (l + 1) - j l} {\text {tw}}_m^{(l)} {\text {tw}}_n^{(l)} {j \brack l}_q. \end{aligned}$$
    (A.6)

    We write the summation variables as \(j = j_1 p + j_0\), and \(l = l_1 p + l_0\). Because of the fact that \(q^p = 1\), the q-Lucas theorem and (A.5), we obtain,

    $$\begin{aligned} {\text {Tw}}_{K_{m,n}}^{(j)} = \left( 1-S_m\left( \left( -t\right) ^p\right) S_n\left( \left( -t\right) ^p\right) \right) ^{j_1} {\text {Tw}}_{K_{m,n}}^{(j_0)}. \end{aligned}$$
    (A.7)
  • Now \(g^{(k)}_{K_{m,n}}(x, a, q, t)\) in (4.2) becomes

    $$\begin{aligned} g^{(k)}_{K_{m,n}} = \sum _{j = 0}^k (x;q^{-1})_k (x t^2 q^N; q)_j (x t^2)^{k-j} q^{k-Nj} {N-2+k \brack k}_q {k \brack j}_q {\text {Tw}}_{K_{m,n}}^{(j)},\nonumber \\ \end{aligned}$$
    (A.8)

    where we have used the fact that

    $$\begin{aligned} \frac{(q^{N-1};q)_k}{(q;q)_k} = {N-2+k \brack k}_q. \end{aligned}$$
    (A.9)

    We decompose the variables as \(j = j_1 p + j_0\), \(k = k_1 p + k_0\) and \(N-2 = A_1 p + A_0\). Plugging (A.1), (A.2) and (A.7) into (A.8), we have

    $$\begin{aligned} g_{K_{m,n}}^{(k)} ={A_1 + k_1 \atopwithdelims ()k_1} \left( 1 - x^p\right) ^{k_1} \left[ 1- \left( 1 - x^p t^{2p} \right) S_m\left( (-t)^p\right) S_n\left( (-t)^p\right) \right] ^{k_1} g^{(k_0)}_{K_{m,n}}.\nonumber \\ \end{aligned}$$
    (A.10)
  • \(F_{K_{m,n}} (x, a, q, t)\) becomes

    $$\begin{aligned} F_{K_{m,n}} = (- t x)^{N-1} \sum _{k = 0}^\infty g_{K_{m,n}}^{(k)}. \end{aligned}$$
    (A.11)

    Given \(k = k_1 p + k_0\), \(N - 2 = A_1 p + A_0\), we have

    $$\begin{aligned} F_{K_{m,n}}= & {} ~ (- t x)^{N - 1} \sum _{k_0 = 0}^{p-1} g_{K_{m,n}}^{(k_0)} \nonumber \\&\times \sum _{k_1 = 0}^\infty {A_1 + k_1 \atopwithdelims ()k_1} \left( 1 - x^p\right) ^{k_1} \left[ 1- \left( 1 - x^p t^{2p}\right) S_m\left( (-t)^p\right) S_n\left( (-t)^p\right) \right] ^{k_1}.\nonumber \\ \end{aligned}$$
    (A.12)

    Using the generalized binomial theorem, we obtain

    $$\begin{aligned} F_{K_{m,n}} = \frac{(- t x)^{N-1} \sum \nolimits _{k = 0}^{p-1}g^{(k)}_{K_{m,n}}}{\left[ x^p + \left( 1 - x^p\right) \left( 1- x^pt^{2p}\right) S_m\left( \left( -t\right) ^p\right) S_n\left( \left( -t\right) ^p\right) \right] ^{A_1 + 1}}. \end{aligned}$$
    (A.13)

    Recall the closed formulae of t-deformed Alexander polynomials (4.15), we can write

    $$\begin{aligned} F_{K_{m,n}} = \frac{(-t x)^{A_0 + 1 - p}}{\Delta _{K_{m,n}}(x^p, -(-t)^p)^{A_1 + 1}} \sum _{k = 0}^{p-1} g^{(k)}_{K_{m,n}}. \end{aligned}$$
    (A.14)

Before we jump to the ADO polynomials, let us first examine the properties of (at)-deformed \(F_K\). When \(p=1\) which leads to \(A_1 = N-2\), \(A_0 = 0\), we have

$$\begin{aligned} \lim _{q \rightarrow 1} F_{K_{m,n}}(x, -t^{-1}q^N, q, t) = \frac{g^{(0)}_{K_{m,n}}}{\Delta _{K_{m,n}}(x,t)^{N-1}} = \frac{1}{\Delta _{K_{m,n}}(x,t)^{N-1}},\quad \quad \end{aligned}$$
(A.15)

in accordance with (4.12). Finally, for the t-deformed \({\text {ADO}}\) polynomials, we have

$$\begin{aligned} {\text {ADO}}_{K_{m,n}}^{{{\,\mathrm{SU}\,}}(N)}(p; x, t)= & {} \Delta _{K_{m,n}}(x^p, -(-t)^p)^{N-1} F_{K_{m,n}}\nonumber \\= & {} (-tx)^{A_0 + 1 - p} \Delta _{K_{m,n}}(x^p, - (-t)^p)^{A_1(p-1) + A_0} \sum _{k=0}^{p-1} g_{K_{m,n}}^{(k)},\nonumber \\ \end{aligned}$$
(A.16)

where \(N-2 = A_1 p + A_0\).

Now let us consider some simple cases. If \(p=1\), then \(A_0 = 0\), we have

$$\begin{aligned} {\text {ADO}}_{K_{m,n}}^{{{\,\mathrm{SU}\,}}(N)}(p=1; x,t) = g^{(0)}_{K_{m,n}} = 1. \end{aligned}$$
(A.17)

For \(N = 2\) and \(p = 2\), then \(A_0 = A_1 = 0\), we have

$$\begin{aligned} {\text {ADO}}_{K_{m,n}}^{{{\,\mathrm{SU}\,}}(2)}(p=2; x,t) = (-t x)^{-1} \left( g^{(0)}_{K_{m,n}} + g^{(1)}_{K_{m,n}}\right) = \Delta _{K_{m,n}}(x,t).\nonumber \\ \end{aligned}$$
(A.18)

A.2 ADO polynomials of double twist knots \(K_{m + \frac{1}{2}, -n}~(m,n \in \mathbb {Z}_+)\)

Following the same procedure in the last subsection, we decompose the closed-form expression of \(F_{K_{m+\frac{1}{2},-n}} (x,a,q,t)\) (4.5) into parts:

  • The twist factor \(\mathbb {t}\!\mathbb {w}_n^{(k)}(x, q, t)\) (4.7) now becomes

    $$\begin{aligned} \!\mathbb {t}\!\mathbb {w}_n^{(k)} (x, q, t) = \sum _{0 = b_0 \le b_1 \le \cdots \le b_{n-1} \le b_n = k} \prod _{i=1}^{n-1} (x^{2 b_i} q^{- b_i (b_{i+1}-1)}t^{2b_i}) {b_{i + 1} \brack b_i}_q.\nonumber \\ \end{aligned}$$
    (A.19)

    We write \(b_i = \alpha _i p + \beta _i\), \(k = k_1 p + k_0\), and we can obtain

    $$\begin{aligned} \!\mathbb {t}\!\mathbb {w}_n^{(k)} = S^{k_1}_n(x^{2p} t^{2p}) \,\!\mathbb {t}\!\mathbb {w}_n^{(k_0)}. \end{aligned}$$
    (A.20)
  • Now \(g^{(k)}_{K_{m+\frac{1}{2},-n}}\) in (4.6) becomes

    $$\begin{aligned} g^{(k)}_{K_{m+\frac{1}{2},-n}} = \sum _{j = 0}^k (x; q^{-1})_k (x t^2 q^N; q)_j x^{k-j} {N-2+k \brack k}_q {k \brack j}_q q^{\frac{1}{2}j(j-1)+k} t^{-j + 2k} {\text {tw}}_m^{(j)} \!\mathbb {t}\!\mathbb {w}_n^{(k)}.\nonumber \\ \end{aligned}$$
    (A.21)

    We write \(k = k_1 p + k_0\), \(j = j_1 p + j_0\), and \(N - 2 = A_1 p + A_0\), and obtain

    $$\begin{aligned} g^{(k)}_{K_{m+\frac{1}{2}, -n}}= & {} ~ \left\{ \left( 1- x^p\right) \left( -t\right) ^p \left[ \left( -x t\right) ^p + \left( x^p t^{2p} - 1\right) S_m \left( \left( - t\right) ^p\right) \right] S_n\left( x^{2p} t^{2p}\right) \right\} ^{k_1}\nonumber \\&\times {A_1 + k_1 \atopwithdelims ()k_1} g_{K_{m+\frac{1}{2},-n}}^{(k_0)}. \end{aligned}$$
    (A.22)
  • \(F_{K_{m+\frac{1}{2},-n}}(x,a,q,t)\) now becomes

    $$\begin{aligned} F_{K_{m+\frac{1}{2},-n}} = (-t x^n)^{N-1} \sum _{k=0}^{\infty } g_{K_{m+\frac{1}{2},-n}}^{(k)}. \end{aligned}$$
    (A.23)

    We write \(k = k_1 p + k_0\), \(N-2 = A_1 p + A_0\). Again, with the help of the generalized binomial theorem, we can get rid of the infinite summation over \(k_1\) and obtain

    $$\begin{aligned} F_{K_{m+\frac{1}{2},-n}} = \frac{(-t x^n)^{A_0 + 1 - p}}{\Delta _{K_{m+\frac{1}{2},-n}}(x^p, -(-t)^p)^{A_1 + 1}} \sum _{k_0 = 0}^{p - 1} g^{(k_0)}_{K_{m+\frac{1}{2},-n}}, \end{aligned}$$
    (A.24)

    where the t-deformed Alexander polynomial is given in (4.15).

When \(p = 1\), then \(A_1 = N - 2\), \(A_0 = 0\), we have

$$\begin{aligned} \lim _{q\rightarrow 1}F_{K_{m+\frac{1}{2}, -n}} (x,-t^{-1}q^N,q,t) = \frac{g^{(0)}_{K_{m+\frac{1}{2},-n}}}{\Delta _{K_{m+\frac{1}{2},-n}}(x,t)^{N-1}} =\frac{1}{\Delta _{K_{m+\frac{1}{2},-n}}(x,t)^{N-1}},\nonumber \\ \end{aligned}$$
(A.25)

in accordance with (4.12). Finally, for the t-deformed \({\text {ADO}}\) polynomials, we have

$$\begin{aligned} {\text {ADO}}_{K_{m+\frac{1}{2},-n}}^{{{\,\mathrm{SU}\,}}(N)}(p; x, t)= & {} \Delta _{K_{m+\frac{1}{2},-n}}(x^p, -(-t)^p)^{N-1} F_{K_{m + \frac{1}{2},-n}} \nonumber \\= & {} (-tx^n)^{A_0 + 1 - p} \Delta _{K_{m + \frac{1}{2},-n}}(x^p, - (-t)^p)^{A_1(p-1) + A_0} \sum _{k=0}^{p-1} g_{K_{m+\frac{1}{2},-n}}^{(k)},\nonumber \\ \end{aligned}$$
(A.26)

where \(N-2 = A_1 p + A_0\).

Now we consider some simple cases. When \(p = 1\), \(A_0 = 0\), we have

$$\begin{aligned} {\text {ADO}}_{K_{m+\frac{1}{2}, -n}}^{{{\,\mathrm{SU}\,}}(N)}(p = 1; x, t) = g^{(0)}_{K_{m+\frac{1}{2},-n}} =1. \end{aligned}$$
(A.27)

For \(N = 2\), \(p = 2\), we have

$$\begin{aligned} {\text {ADO}}_{K_{m+\frac{1}{2},-n}}(p = 2; x, t)= & {} (-t x^n)^{-1} (g^{(0)}_{K_{m+\frac{1}{2},-n}} + g^{(1)}_{K_{m+\frac{1}{2},-n}})\nonumber \\= & {} \Delta _{K_{m+\frac{1}{2}, -n}} (x, t). \end{aligned}$$
(A.28)

A.3 Final formulae

In conclusion, for gauge group \({{\,\mathrm{SU}\,}}(N)\), the t-deformed p-th ADO polynomials are given by

$$\begin{aligned}&{\text {ADO}}_{K_{m,n}}^{{{\,\mathrm{SU}\,}}(N)} (p; x, t) = (-t x)^{A_0 + 1 - p} \Delta _{K_{m,n}}(x^p, -(-t)^p)^{A_1 (p - 1) + A_0} \sum _{k=0}^{p - 1} g^{(k)}_{K_{m,n}}, \\&{\text {ADO}}_{K_{m+\frac{1}{2},-n}}^{{{\,\mathrm{SU}\,}}(N)} (p; x, t) \nonumber \\&\quad = (-t x^n)^{A_0 + 1 - p} \Delta _{K_{m+\frac{1}{2},-n}}(x^p, -(-t)^p)^{A_1 (p - 1)+ A_0} \sum _{k=0}^{p - 1} g^{(k)}_{K_{m + \frac{1}{2},-n}}, \end{aligned}$$

where \(N-2 = A_1 p + A_0\), \(A_1, A_0 \in \mathbb {N}\) and \(0\le A_0 \le p-1\). It is easily seen that the recursion relation conjectured in [50] holds,

$$\begin{aligned} {\text {ADO}}^{{{\,\mathrm{SU}\,}}(N+p)}_{K}(p; x, t) = \Delta _K(x^p, -(-t)^p)^{p-1} {\text {ADO}}^{{{\,\mathrm{SU}\,}}(N)}_{K}(p; x, t). \end{aligned}$$
(A.29)

Note that \(\frac{1}{p}S_m(\zeta _p^n) = 1\), only when m|n. Otherwise, it is zero. Therefore, the t-deformed p-th ADO polynomials can also be written as,

$$\begin{aligned} {\text {ADO}}_{K_{m,n}}^{{{\,\mathrm{SU}\,}}(N)} (p; x, t)= & {} ~ \frac{1}{p}\sum _{l = 0}^{p - 1}(-t x)^{l + 1 - p} \Delta _{K_{m,n}}(x^p, -(-t)^p)^{\frac{(N-2)(p - 1) + l}{p}} S_p(\zeta _p^{N-l-2}) \sum _{k=0}^{p - 1} g^{(k)}_{K_{m,n}} \nonumber \\ {\text {ADO}}_{K_{m+\frac{1}{2},-n}}^{{{\,\mathrm{SU}\,}}(N)} (p; x, t)= & {} \frac{1}{p} \sum _{l=0}^{p - 1}(-t x^n)^{l + 1 - p} \Delta _{K_{m+\frac{1}{2},-n}}(x^p, -(-t)^p)^{\frac{(N-2)(p - 1)+l}{p}}\nonumber \\&\times S_p(\zeta _p^{N - l -2}) \sum _{k=0}^{p - 1} g^{(k)}_{K_{m+\frac{1}{2},-n}}. \end{aligned}$$
(A.30)

Although we write them as summations over l, there is only one non-vanishing term, which corresponds to that l is the remainder of \((N-2)/p\).

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Wang, H.E., Yang, Y.J., Zhang, H.D. et al. On Knots, Complements, and 6j-Symbols. Ann. Henri Poincaré 22, 2691–2720 (2021). https://doi.org/10.1007/s00023-021-01033-4

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