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 (a, t)-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 (a, t)-deformed $F_K$ for the complements of double twist knots with positive braids. Using the conjectural expressions, we derive t-deformed 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 $k_0$.

1.1 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 (a, t)-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 (a, t)-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)

1.2 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)

1.3 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 $(N-2)/p$.