Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 16, 2004, 71 – 78

© F. Nagasato

ABSTRACT. In this paper, we introduce a formula for the homogeneous linear recursive relations of the colored Kauffman brackets, which is more efficient than the formula in [G2].

1. Motivation

In this paper, we discuss the “reducibility” of recursive relations. Assume that for a sequence ${\left\{F_i\right\}}_{i\in \mathbb{\mathbb{Z}}}$ in an integral domain $R$ there exists a non-empty finite subset $S$ in $\mathbb{\mathbb{Z}}$ such that

Then the above relation, called a homogeneous linear recursive relation of ${\left\{F_i\right\}}_{i\in \mathbb{\mathbb{Z}}}$, is said to be reducible if there exist a non-empty proper subset $S^{\prime }\subset S$ such that

If there does not exist such a proper subset $S^{\prime }$, then the recursive relation is said to be irreducible.

Let us focus on the following homogeneous linear recursive relation of the colored Kauffman brackets ${\left\{{\kappa }_n\right\}}_{n\in \mathbb{\mathbb{Z}}}\subset \mathbb{ℂ}\left[t,t^{-1}\right]$ without details:

In fact, for a knot satisfying Ker$\left({\pi }_t\right)\ne 0$, all the recursive relations of the colored Kauffman brackets derived from non-zero elements of Ker$\left({\pi }_t\right)$ represent defining polynomials of a “noncommutative” $SL\left(2,\mathbb{ℂ}\right)$-character variety. Moreover the recursive relations include the information of the A-polynomial. (Refer to [GL, N] for details of these topics and related researches.) In these sense, Theorem $1.1$ is very interesting, and so we now focus on Theorem $1.1$. Note that it is still unknown if Ker$\left({\pi }_t\right)\ne 0$ for any knot.

Now, the formula in Theorem $1.1$ is in fact reducible. Namely, all the recursive relations given by the formula in Theorem $1.1$ are reducible. Indeed, we can get a more efficient formula as follows:

We will first review some concepts needed later through Subsections $2.1$ and $2.3$, prove Theorem $1.2$ in Subsections $2.4$ and $2.5$, and show the efficiency of the above formula in Subsection $2.6$.

2. formula in Theorem $1.2$ and its efficiency

2.1. Glossary. In this paper, we will often consider gluings of 3-manifolds with at least one torus boundary. For convenience, we would like to introduce “the canonical gluing” of such 3-manifolds. Let $M_i$, $i\in \left\{1,2\right\}$, be a 3-manifold with at least one torus boundary $T_i^2$. Fix a longitude ${\lambda }_i$ and a meridian ${\mu }_i$ of $T_i^2$. (In the case of the exterior of a knot in a 3-sphere $S^3$, we fix a preferred longitude of the knot as a longitude of the torus boundary.) Then a gluing of $M_1$ to $M_2$ along the tori $T_1^2$, $T_2^2$ is said to be canonical (in terms of ${\lambda }_i$’s and ${\mu }_i$’s) if ${\lambda }_1$ and ${\mu }_1$ are glued to ${\lambda }_2$ and ${\mu }_2$ respectively.

For an arbitrary compact orientable 3-manifold $M$, a framed link in $M$ is an embedding of the disjoint union of some annuli into $M$. The framing of a framed link is presented by the blackboard framing in the case where $M$ is a 3-sphere, a knot complement or a solid torus. The framing is done by the torus framing in the case where $M$ is a cylinder $T^2\times I$. Here by a framed link in $T^2\times I$ with 0-framing in terms of the torus framing, we mean a framed link isotopic to an embedding of the disjoint union of some annuli into the torus $T^2\times \left\{\frac{1}{2}\right\}$.

For convenience, we fix a longitude $\lambda$ and a meridian $\mu$ of a torus $T^2$, and fix a preferred longitude and a meridian of a knot $K$ in $S^3$ throughout this paper. Note that $\lambda$ and $\mu$ naturally induce a longitude $\lambda \left(c\right)$ and a meridian $\mu \left(c\right)$ of $T^2\times \left\{c\right\}$ for any $c\in I$.

2.2. KBSM. We mention the Kauffman bracket skein module (KBSM for short) needed later. (Refer to [B, BL, HP, P1, P2] for details.) For an arbitrary compact orientable 3-manifold $M$, the Kauffman bracket skein module ${\mathcal{K}}_t\left(M\right)$ is defined by the quotient of the $\mathbb{ℂ}\left[t,t^{-1}\right]$-module $\mathbb{ℂ}\left[t,t^{-1}\right]{\mathcal{L}}_M$ generated by all isotopy classes of framed links in $M$ (including the empty link $\phi$) by the $\mathbb{ℂ}\left[t,t^{-1}\right]$-submodule generated by all possible elements as follows:

where the three drawings of the first line in the above depictions express framed links identically embedded in M, except in an open ball Int$\left(B^3\right)$.

For a framed knot $K_f$ in $M$ and a positive integer $n$, let ${\left(K_f\right)}^n$ be the framed link consisting of $n$ parallel copies of $K_f$. Then we define the element $T_n\left(K_f\right)$ of ${\mathcal{K}}_t\left(M\right)$ as follows:

Also we define the element $S_n\left(K_f\right)$ of ${\mathcal{K}}_t\left(M\right)$ as follows:

Then focus on the following theorem in [P1].

Regarding a solid torus $D^2\times S^1$ as a cylinder $\left(S^1\times I\right)\times I$, we see that ${\mathcal{K}}_t\left(D^2\times S^1\right)$ is free as $\mathbb{ℂ}\left[t,t^{-1}\right]$-module with basis (representatives)

where $\alpha$ is an embedded annulus in $\left(S^1\times I\right)\times I$ isotopic to $\left(S^1\times \left[\frac{1}{3},\frac{2}{3}\right]\right)\times \left\{\frac{1}{2}\right\}$. Let $\left(p,q\right)$ for coprime integers $p$, $q$ be a framed knot in $T^2\times I$ with 0-framing whose core is isotopic to the simple closed curve of slope $p∕q$ on $T^2\times \left\{\frac{1}{2}\right\}$. (Note that the curve of slope $p∕q$ means one homologous to $p\left[\lambda \left(\frac{1}{2}\right)\right]+q\left[\mu \left(\frac{1}{2}\right)\right]$ in $H_1\left(T^2\times \left\{\frac{1}{2}\right\}\right)$.) Then it also follows from Theorem $2.1$ that ${\mathcal{K}}_t\left(T^2\times I\right)$ is free as $\mathbb{ℂ}\left[t,t^{-1}\right]$-module with basis (representatives)

2.3. Colored Kauffman bracket. The colored Kauffman bracket is an invariant of framed knots in $S^3$ defined as follows. For a framed knot $K_f$ in $S^3$ and a non-negative integer $n$, consider an element $S_n\left(K_f\right)$ in ${\mathcal{K}}_t\left(S^3\right)=\mathbb{ℂ}\left[t,t^{-1}\right]$. Then the $n$-th colored Kauffman bracket ${\kappa }_n\left(K_f\right)$ of a framed knot $K_f$ is defined as the element $S_n\left(K_f\right)$. Namely, ${\kappa }_n\left(K_f\right):=S_n\left(K_f\right)$. Note that the equation $S_{-n}\left(K_f\right)=-S_{n-2}\left(K_f\right)$ naturally induces ${\kappa }_{-n}\left(K_f\right)=-{\kappa }_{n-2}\left(K_f\right)$. Here $S_n$ corresponds to the Jones-Wenzl idempotent or “the magic element”. (Refer to [FG, L].)

2.4. Efficient formula. As stated in the first section, the formula in Theorem $1.1$ is reducible, which fact will be observed in Subsection $2.6$. Indeed, we can polish it as seen in Theorem $1.2$. (It is still unknown if the formula in Theorem $1.2$ is irreducible.) In this subsection, we give a proof of Theorem $1.2$.

We first review some propositions and concepts needed later. For a knot $K$ in a 3-sphere $S^3$ let $N\left(K\right)$ be an open tubular neighborhood of $K$ in $S^3$, and let $E_K$ be the exterior $S^3-N\left(K\right)$ of $K$. In [G2] a method is introduced to get a homogeneous linear recursive relation of the colored Kauffman brackets ${\left\{{\kappa }_n\left(K_0\right)\right\}}_{n\in \mathbb{\mathbb{Z}}}$. The method is based on the kernel of the homomorphism as $\mathbb{ℂ}\left[t,t^{-1}\right]$-module

induced by the canonical gluing (see Subsection $2.1$) of a cylinder $T^2\times I$ to the exterior $E_K$ along $T^2\times \left\{1\right\}$ and $\partial E_K$. Indeed, the gluing induces a bihomomorphism

We simply denote by $a\ast b$ the image $C_E\left(a,b\right)$ of $\left(a,b\right)\in {\mathcal{K}}_t\left(T^2\times I\right)\times {\mathcal{K}}_t\left(E_K\right)$. Then the homomorphism ${\pi }_t:{\mathcal{K}}_t\left(T^2\times I\right)\to {\mathcal{K}}_t\left(E_K\right)$ is defined by ${\pi }_t\left({\left(p,q\right)}_T\right)={\left(p,q\right)}_T\ast \phi$.

Now, consider the bihomomorphism

induced by the canonical gluing (see Subsection $2.1$) of $T^2\times I$ to $D^2\times S^1$ along $T^2\times \left\{0\right\}$ and $\partial \left(D^2\times S^1\right)$. We also simply denote by $c\ast b$ the image $C_S\left(c,b\right)$ of $\left(c,b\right)\in {\mathcal{K}}_t\left(D^2\times S^1\right)\times {\mathcal{K}}_t\left(T^2\times I\right)$. Then we get the following formula.

In fact, the above equation can be simplified as follows:

By Proposition $2.2$, we can easily prove Theorem $1.2$. We review Gelca’s construction given in [G2] to prove the theorem. For any knot $K$ in $S^3$, consider the pairing

naturally induced by the $1∕0$-Dehn filling on $E_K$. By the above pairing we can represent the $n$-th colored Kauffman bracket ${\kappa }_n\left(K_0\right)$ of $K_0$ with 0-framing as follows:

(Refer to Subsection $2.3$.) Here we see immediately that for any elements $u\in {\mathcal{K}}_t\left(D^2\times S^1\right)$, $v\in {\mathcal{K}}_t\left(E_K\right)$ and $w\in {\mathcal{K}}_t\left(T^2\times I\right)$,

Let us consider the case where $u=S_n\left(\alpha \right)$, $v=\phi$ and $w={\sum }_{i=1}^k{c}_i{\left(p_i,q_i\right)}_T\in Ker\left({\pi }_t\right)$ in the above equation. Then by Proposition $2.2$ we get the following recursive relation of ${\left\{{\kappa }_n\left(K_0\right)=\langle S_n\left(\alpha \right),\phi \rangle \right\}}_{n\in \mathbb{\mathbb{Z}}}$:

This completes the proof of Theorem $1.2$.

2.5. Proof of Proposition $2.2$. In this subsection, we give a proof of Proposition $2.2$. We first see that $T_n\left(\alpha \right)=S_n\left(\alpha \right)-S_{n-2}\left(\alpha \right)$ by induction. Therefore the following holds by Proposition $2.1$:

Recall $S_n\left(\alpha \right)=-S_{-n-2}\left(\alpha \right)$. Hence the above equation is transformed as follows:

Here let us put $V_n:={\left(-t^{2n+p+2}\right)}^q{S}_{n+p}\left(\alpha \right)+{\left(-t^{2n-p+2}\right)}^{-q}{S}_{n-p}\left(\alpha \right)$. Then the above transformation immediately derives the following recursive relation:

Therefore it suffices to show the equation $S_n\left(\alpha \right)\ast {\left(p,q\right)}_T=V_n$ in the case where $n$ is $-1$ and $0$ for proving Proposition $2.2$. If $n$ is $-1$, then we have

If $n$ is 0, then

By Proposition $2.1$, we have

This shows that $S_0\left(\alpha \right)\ast {\left(p,q\right)}_T-V_0=0$ and completes the proof of Proposition $2.2$.

2.6. Efficiency of the formula in Theorem $1.2$. In this subsection, we show the efficiency of the formula $K_n^s\left(K_0\right)=0$ in Theorem $1.2$ comparing $\widetilde{K_n^s}\left(K_0\right)=0$ in Theorem $1.1$.

According to [G1], for the left-handed trefoil,

is in the kernel of ${\pi }_t$. We pick up this element to show the efficiency. Let $K_0$ be the left-handed trefoil with 0-framing. Then the element $s$ gives rise to the following recursive relation of ${\left\{{\kappa }_n\left(K_0\right)\right\}}_{n\in \mathbb{\mathbb{Z}}}$ via the formula $\widetilde{K_n^{s^{\prime }}}\left(K_0\right)$=0 in Theorem $1.1$:

On the other hand, $s$ gives rise to the following recursive relation of ${\left\{{\kappa }_n\left(K_0\right)\right\}}_{n\in \mathbb{\mathbb{Z}}}$ via the formula $K_n^{s^{\prime }}\left(K_0\right)$=0 in Theorem $1.2$:

As seen in these examples, the formula in Theorem $1.2$ gives us a simpler recursive relation than that in Theorem $1.1$ gives us. In fact, this phenomenon always holds. More concretely, $\widetilde{K_n^s}\left(K_0\right)=K_n^s\left(K_0\right)+K_{n-2}^s\left(K_0\right)$ always holds for any knot $K$ in $S^3$ and any element $s$ in Ker$\left({\pi }_t\right)$ for $K$. (Hence the recursive relations given by the formula in Theorem $1.1$ are reducible.) In this sense, the formula in Theorem $1.2$ is more efficient than that in Theorem $1.1$.

Acknowledgments

I would like to thank Professor Răzvan Gelca for his useful comments. I also grateful to my advisor, Professor Mitsuyoshi Kato, for his encouragement.

References

GRADUATE SCHOOL OF MATHEMATICS, KYUSHU UNIVERSITY, FUKUOKA, 812-8581, JAPAN

E-mail address: fukky@math.kyushu-u.ac.jp

Received February 10,2004; Revised version September 15, 2004