Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 16, 2004, 71 – 78
© F. Nagasato
EFFICIENT FORMULA OF THE COLORED KAUFFMAN BRACKETS
(submitted by M. Malakhaltsev)
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].
2000 Mathematical Subject Classification. Primary 57M27; Secondary 57M25.
Key words and phrases. Colored Kauffman bracket, Kauffman bracket skein module.
The author has been supported by JSPS Research Fellowships for Young Scientists.
In this paper, we discuss the “reducibility” of recursive relations. Assume that for a sequence in an integral domain there exists a non-empty finite subset in such that
Then the above relation, called a homogeneous linear recursive relation of , is said to be reducible if there exist a non-empty proper subset such that
If there does not exist such a proper subset , then the recursive relation is said to be irreducible.
Let us focus on the following homogeneous linear recursive relation of the colored Kauffman brackets without details:
Theorem 1.1 (Gelca [G2]). If for a knot in has a non-zero element , then has the following homogeneous linear recursive relation:
where is a framed knot in with 0-framing such that the core of is isotopic to . ( is uniquely determined up to isotopy.)
In fact, for a knot satisfying Ker, all the recursive relations of the colored Kauffman brackets derived from non-zero elements of Ker represent defining polynomials of a “noncommutative” -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 is very interesting, and so we now focus on Theorem . Note that it is still unknown if Ker for any knot.
Now, the formula in Theorem is in fact reducible. Namely, all the recursive relations given by the formula in Theorem are reducible. Indeed, we can get a more efficient formula as follows:
We will first review some concepts needed later through Subsections and , prove Theorem in Subsections and , and show the efficiency of the above formula in Subsection .
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 , , be a 3-manifold with at least one torus boundary . Fix a longitude and a meridian of . (In the case of the exterior of a knot in a 3-sphere , we fix a preferred longitude of the knot as a longitude of the torus boundary.) Then a gluing of to along the tori , is said to be canonical (in terms of ’s and ’s) if and are glued to and respectively.
For an arbitrary compact orientable 3-manifold , a framed link in is an embedding of the disjoint union of some annuli into . The framing of a framed link is presented by the blackboard framing in the case where is a 3-sphere, a knot complement or a solid torus. The framing is done by the torus framing in the case where is a cylinder . Here by a framed link in 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 .
For convenience, we fix a longitude and a meridian of a torus , and fix a preferred longitude and a meridian of a knot in throughout this paper. Note that and naturally induce a longitude and a meridian of for any .
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 , the Kauffman bracket skein module is defined by the quotient of the -module generated by all isotopy classes of framed links in (including the empty link ) by the -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.
For a framed knot in and a positive integer , let be the framed link consisting of parallel copies of . Then we define the element of as follows:
Also we define the element of as follows:
Then focus on the following theorem in [P1].
Theorem 2.1 (Przytycki [P1]). Let be an orientable surface, and let be an interval . Then the KBSM is the free -module generated by all the isotopy classes of framed links in (including the empty link) isotopic to embeddings of the disjoint union of some annuli into with no trivial component.
Regarding a solid torus as a cylinder , we see that is free as -module with basis (representatives)
where is an embedded annulus in isotopic to . Let for coprime integers , be a framed knot in with 0-framing whose core is isotopic to the simple closed curve of slope on . (Note that the curve of slope means one homologous to in .) Then it also follows from Theorem that is free as -module with basis (representatives)
2.3. Colored Kauffman bracket. The colored Kauffman bracket is an invariant of framed knots in defined as follows. For a framed knot in and a non-negative integer , consider an element in . Then the -th colored Kauffman bracket of a framed knot is defined as the element . Namely, . Note that the equation naturally induces . Here 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 is reducible, which fact will be observed in Subsection . Indeed, we can polish it as seen in Theorem . (It is still unknown if the formula in Theorem is irreducible.) In this subsection, we give a proof of Theorem .
We first review some propositions and concepts needed later. For a knot in a 3-sphere let be an open tubular neighborhood of in , and let be the exterior of . In [G2] a method is introduced to get a homogeneous linear recursive relation of the colored Kauffman brackets . The method is based on the kernel of the homomorphism as -module
induced by the canonical gluing (see Subsection ) of a cylinder to the exterior along and . Indeed, the gluing induces a bihomomorphism
We simply denote by the image of . Then the homomorphism is defined by .
Now, consider the bihomomorphism
induced by the canonical gluing (see Subsection ) of to along and . We also simply denote by the image of . Then we get the following formula.
Proposition 2.1 (Gelca [G2]). For elements and , the following holds:
In fact, the above equation can be simplified as follows:
By Proposition , we can easily prove Theorem . We review Gelca’s construction given in [G2] to prove the theorem. For any knot in , consider the pairing
naturally induced by the -Dehn filling on . By the above pairing we can represent the -th colored Kauffman bracket of with 0-framing as follows:
(Refer to Subsection .) Here we see immediately that for any elements , and ,
Let us consider the case where , and in the above equation. Then by Proposition we get the following recursive relation of :
This completes the proof of Theorem .
2.5. Proof of Proposition . In this subsection, we give a proof of Proposition . We first see that by induction. Therefore the following holds by Proposition :
Recall . Hence the above equation is transformed as follows:
Here let us put . Then the above transformation immediately derives the following recursive relation:
Therefore it suffices to show the equation in the case where is and for proving Proposition . If is , then we have
If is 0, then
By Proposition , we have
This shows that and completes the proof of Proposition .
2.6. Efficiency of the formula in Theorem . In this subsection, we show the efficiency of the formula in Theorem comparing in Theorem .
According to [G1], for the left-handed trefoil,
is in the kernel of . We pick up this element to show the efficiency. Let be the left-handed trefoil with 0-framing. Then the element gives rise to the following recursive relation of via the formula =0 in Theorem :
On the other hand, gives rise to the following recursive relation of via the formula =0 in Theorem :
As seen in these examples, the formula in Theorem gives us a simpler recursive relation than that in Theorem gives us. In fact, this phenomenon always holds. More concretely, always holds for any knot in and any element in Ker for . (Hence the recursive relations given by the formula in Theorem are reducible.) In this sense, the formula in Theorem is more efficient than that in Theorem .
I would like to thank Professor Răzvan Gelca for his useful comments. I also grateful to my advisor, Professor Mitsuyoshi Kato, for his encouragement.
GRADUATE SCHOOL OF MATHEMATICS, KYUSHU UNIVERSITY, FUKUOKA, 812-8581, JAPAN
E-mail address: firstname.lastname@example.org
Received February 10,2004; Revised version September 15, 2004