Fumikazu Nagasato EFFICIENT FORMULA OF THE COLORED KAUFFMANBRACKETS
(submitted by M. Malakhaltsev)
ABSTRACT. In this paper, we introduce a formula for the homogeneouslinear recursive relations of the colored Kauffman brackets, which is moreefficient than the formula in [G2].
Key words and phrases. Colored Kauffman bracket, Kauffman bracket skein module.
The author has been supported by JSPS Research Fellowships for Young Scientists.
1. Motivation
In this paper, we discuss the “reducibility” of recursive relations. Assume that for a sequence
{Fi}i∈ℤ in an integral
domain R there exists a
non-empty finite subset S
in ℤ
such that
∑i∈SciFi=0,whereci(≠0)∈Rfor anyi∈S.
Then the above relation, called a homogeneous linear recursive relation of
{Fi}i∈ℤ,
is said to be reducible if there exist a non-empty proper subset
S′⊂S such
that
∑i∈S′diFi=0,wheredi(≠0)∈Rfor anyi∈S′.
If there does not exist such a proper subset
S′, then
the recursive relation is said to be irreducible.
Let us focus on the following homogeneous linear recursive relation of the colored Kauffman
brackets {κn}n∈ℤ⊂ℂ[t,t−1]
without details:
Theorem 1.1 (Gelca [G2]).If Ker(πt)for a knot KinS3has a non-zeroelement s:= ∑i=1kci⋅(pi,qi)T,then {κn(K0)}n∈ℤhas the following homogeneous linear recursive relation:
where K0is a framedknot in S3with 0-framingsuch that the core of K0is isotopic to K.(K0isuniquely determined up to isotopy.)
In fact, for a knot satisfying Ker(πt)≠0,
all the recursive relations of the colored Kauffman brackets derived from non-zero elements of
Ker(πt)
represent defining polynomials of a “noncommutative”
SL(2,ℂ)-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(πt)≠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:
Theorem 1.2.Under the same notations and the conditions as in Theorem1.1, thefollowing homogeneous linear recursive relation holds:
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 Mi,
i∈{1,2}, be a 3-manifold with at
least one torus boundary Ti2.
Fix a longitude λi
and a meridian μi of
Ti2. (In the case of the exterior
of a knot in a 3-sphere S3,
we fix a preferred longitude of the knot as a longitude of the torus boundary.) Then a gluing
of M1 to
M2 along the
tori T12,
T22 is said to be canonical
(in terms of λi’s
and μi’s)
if λ1 and
μ1 are glued
to λ2
and μ2
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
T2×I. Here by a
framed link in T2×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
T2×{12}.
For convenience, we fix a longitude λ
and a meridian μ
of a torus T2,
and fix a preferred longitude and a meridian of a knot
K in
S3 throughout this
paper. Note that λ
and μ naturally
induce a longitude λ(c)
and a meridian μ(c)
of T2×{c} for
any c∈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 Kt(M) is defined by
the quotient of the ℂ[t,t−1]-module
ℂ[t,t−1]LM generated by all isotopy
classes of framed links in M
(including the empty link φ)
by the ℂ[t,t−1]-submodule
generated by all possible elements as follows:
−t
−t−1
,
L⊔
−(−t2−t−2)L,for any framed link L in M,
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(B3).
For a framed knot Kf
in M and a positive
integer n, let
(Kf)n be the framed link
consisting of n parallel
copies of Kf. Then we
define the element Tn(Kf)
of Kt(M) as
follows:
Tn(Kf)=Kf⋅Tn−1(Kf)−Tn−2(Kf),
T1(Kf)=Kf,T0(Kf)=2⋅φ,T−n(Kf)=Tn(Kf).
Also we define the element Sn(Kf)
of Kt(M) as
follows:
Theorem 2.1 (Przytycki [P1]).Let Fbe an orientable surface, and let Ibe an interval [0,1].Then the KBSM Kt(F×I)is the free ℂ[t,t−1]-modulegenerated by all the isotopy classes of framed links in F×I(including the empty link) isotopic to embeddings of the disjoint union of some annuliinto Fwith no trivial component.
Regarding a solid torus D2×S1
as a cylinder (S1×I)×I,
we see that Kt(D2×S1) is
free as ℂ[t,t−1]-module
with basis (representatives)
{Tn(α)∣n∈ℤ≥0},
where α is an
embedded annulus in (S1×I)×I
isotopic to (S1×[13,23])×{12}. Let
(p,q) for coprime
integers p,
q be a framed
knot in T2×I
with 0-framing whose core is isotopic to the simple closed curve of slope
p∕q on
T2×{12}. (Note that the curve
of slope p∕q means one
homologous to p[λ(12)]+q[μ(12)] in
H1(T2×{12}).) Then it also follows
from Theorem 2.1
that Kt(T2×I) is free
as ℂ[t,t−1]-module
with basis (representatives)
2.3. Colored Kauffman bracket.
The colored Kauffman bracket is an invariant of framed knots in
S3 defined as follows.
For a framed knot Kf in
S3 and a non-negative
integer n, consider
an element Sn(Kf)
in Kt(S3)=ℂ[t,t−1]. Then the
n-th colored Kauffman
bracket κn(Kf) of a framed
knot Kf is defined
as the element Sn(Kf).
Namely, κn(Kf):=Sn(Kf). Note
that the equation S−n(Kf)=−Sn−2(Kf)
naturally induces κ−n(Kf)=−κn−2(Kf).
Here Sn
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
S3 let
N(K) be an open tubular
neighborhood of K
in S3, and
let EK be the
exterior S3−N(K) of
K. In [G2] a
method is introduced to get a homogeneous linear recursive relation of the colored Kauffman
brackets {κn(K0)}n∈ℤ.
The method is based on the kernel of the homomorphism as
ℂ[t,t−1]-module
πt:Kt(T2×I)→Kt(EK),
induced by the canonical gluing (see Subsection
2.1) of a
cylinder T2×I to
the exterior EK
along T2×{1}
and ∂EK.
Indeed, the gluing induces a bihomomorphism
CE:Kt(T2×I)×Kt(EK)→Kt(EK).
We simply denote by a∗b
the image CE(a,b) of
(a,b)∈Kt(T2×I)×Kt(EK). Then the
homomorphism πt:Kt(T2×I)→Kt(EK)
is defined by πt((p,q)T)=(p,q)T∗φ.
Figure 1:
The image
πt((1,0)T)
Now, consider the bihomomorphism
CS:Kt(D2×S1)×Kt(T2×I)→Kt(D2×S1)
induced by the canonical gluing (see Subsection
2.1) of
T2×I to
D2×S1 along
T2×{0} and
∂(D2×S1). We also simply
denote by c∗b
the image CS(c,b)
of (c,b)∈Kt(D2×S1)×Kt(T2×I).
Then we get the following formula.
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
S3,
consider the pairing
〈,〉:Kt(D2×S1)×Kt(EK)→Kt(S3)=ℂ[t,t−1],
naturally induced by the 1∕0-Dehn
filling on EK. By the above pairing
we can represent the n-th
colored Kauffman bracket κn(K0)
of K0
with 0-framing as follows:
〈Sn(α),φ〉=κn(K0).
(Refer to Subsection 2.3.)
Here we see immediately that for any elements
u∈Kt(D2×S1),
v∈Kt(EK) and
w∈Kt(T2×I),
〈u∗w,v〉=〈u,w∗v〉.
Let us consider the case where u=Sn(α),
v=φ and
w= ∑i=1kci(pi,qi)T∈Ker(πt) in the above equation. Then
by Proposition 2.2 we get the
following recursive relation of {κn(K0)=〈Sn(α),φ〉}n∈ℤ:
2.5. Proof of Proposition 2.2.
In this subsection, we give a proof of Proposition
2.2. We first
see that Tn(α)=Sn(α)−Sn−2(α)
by induction. Therefore the following holds by Proposition
2.1:
This shows that S0(α)∗(p,q)T−V0=0 and completes
the proof of Proposition 2.2.
2.6. Efficiency of the formula in Theorem1.2.
In this subsection, we show the efficiency of the formula
Kns(K0)=0 in
Theorem 1.2
comparing Kns ˜(K0)=0 in
Theorem 1.1.
is in the kernel of πt.
We pick up this element to show the efficiency. Let
K0
be the left-handed trefoil with 0-framing. Then the element
s gives rise to the following
recursive relation of {κn(K0)}n∈ℤ
via the formula Kns′˜(K0)=0
in Theorem 1.1:
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,
Kns ˜(K0)=Kns(K0)+Kn−2s(K0) always holds
for any knot K
in S3 and any
element s
in Ker(πt)
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.
Acknowledgements
I would like to thank Professor Răzvan Gelca for his useful comments. I also grateful to
my advisor, Professor Mitsuyoshi Kato, for his encouragement.
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