# The Kauffman bracket skein module of $S^{1}\times S^{2}$ via braids

Ioannis Diamantis  Department of Data Analytics and Digitalisation, School of
Business and Economics, Maastricht University, P.O.Box 616, 6200 MD,
Maastricht, The Netherlands<EMAIL_ADDRESS>
###### Abstract.

In this paper we present two different ways for computing the Kauffman bracket
skein module of $S^{1}\times S^{2}$, ${\rm KBSM}\left(S^{1}\times
S^{2}\right)$, via braids. We first extend the universal Kauffman bracket type
invariant $V$ for knots and links in the Solid Torus ST, which is obtained via
a unique Markov trace constructed on the generalized Temperley-Lieb algebra of
type B, to an invariant for knots and links in $S^{1}\times S^{2}$. We do that
by imposing on $V$ relations coming from the braid band moves. These moves
reflect isotopy in $S^{1}\times S^{2}$ and they are similar to the second
Kirby move. We obtain an infinite system of equations, a solution of which, is
equivalent to computing ${\rm KBSM}\left(S^{1}\times S^{2}\right)$. We show
that ${\rm KBSM}\left(S^{1}\times S^{2}\right)$ is not torsion free and that
its free part is generated by the unknot (or the empty knot). We then present
a diagrammatic method for computing ${\rm KBSM}\left(S^{1}\times S^{2}\right)$
via braids. Using this diagrammatic method we also obtain a closed formula for
the torsion part of ${\rm KBSM}\left(S^{1}\times S^{2}\right)$.

2020 Mathematics Subject Classification. 57K31, 57K14, 20F36, 20F38, 57K10,
57K12, 57K45, 57K35, 57K99, 20C08.

###### Key words and phrases:

skein module, Kauffman bracket, braids, solid torus, $S^{1}\times S^{2}$, lens
spaces, mixed links, mixed braids, braid groups of type B, generalized Hecke
algebra of type B, generalized Temperley-Lieb algebra of type B, torsion

###### Contents

  1. 0 Introduction and overview
  2. 1 Topological Set Up
    1. 1.1 Models of $S^{1}\times S^{2}$
    2. 1.2 Knots and braids in $S^{1}\times S^{2}$
  3. 2 Algebraic Tools
    1. 2.1 Knot algebras related to ST & $S^{1}\times S^{2}$
    2. 2.2 The Kauffman bracket skein module of ST via braids
  4. 3 The Kauffman bracket skein module of $S^{1}\times S^{2}$ via $TL_{1,n}$
    1. 3.1 Useful Lemmata
    2. 3.2 The infinite system
  5. 4 A diagrammatic approach via braids
  6. 5 Conclusions

##  0\. Introduction and overview

Skein modules of $3$-manifolds are generalizations of link invariants in
$S^{3}$ to link invariants in arbitrary $3$-manifolds. They were introduced by
Turaev [29] and Przytycki [28] and they have become very important algebraic
tools in the study of 3-manifolds, since their properties reflect geometric
and topological information about them. Skein modules of 3-manifolds are
modules composed of linear combinations of links in the 3-manifolds, modulo
some properly chosen (local) skein relations. In this paper we compute the
Kauffman bracket skein module of $S^{1}\times S^{2}$, KBSM($S^{1}\times
S^{2}$), namely, the skein module based on the Kauffman bracket skein relation

$L_{+}-AL_{0}-A^{-1}L_{\infty}$

where $L_{\infty}$ and $L_{0}$ are represented schematically by the
illustrations in Figure 1.

The computation of KBSM($S^{1}\times S^{2}$) is equivalent to constructing all
possible analogues of the Jones polynomial for knots and links in $S^{1}\times
S^{2}$, since the linear dimension of KBSM($S^{1}\times S^{2}$) means the
number of independent Kauffman bracket-type invariants defined on knots and
links in $S^{1}\times S^{2}$. In other words, KBSM($S^{1}\times S^{2}$) yields
all possible isotopy invariants of knots which satisfy the Kauffman bracket
skein relation.

The precise definition of KBSM is as follows:

######  Definition 1.

Let $M$ be an oriented $3$-manifold and $\mathcal{L}_{{\rm fr}}$ be the set of
isotopy classes of unoriented framed links in $M$. Let $R=\mathbb{Z}[A^{\pm
1}]$ be the Laurent polynomials in $A$ and let $R\mathcal{L}_{{\rm fr}}$ be
the free $R$-module generated by $\mathcal{L}_{{\rm fr}}$. Let $\mathcal{S}$
be the ideal generated by the skein expressions
$L_{+}-AL_{0}-A^{-1}L_{\infty}$ and $L\bigsqcup{\rm O}-(-A^{2}-A^{-2})L$. Note
that blackboard framing is assumed and that $L\bigsqcup{\rm O}$ stands for the
union of a link $L$ and the trivially framed unknot in a ball disjoint from
$L$.

Figure 1. The links $L$, $L_{0}$ and $L_{\infty}$ locally.

Then the Kauffman bracket skein module of $M$, KBSM$(M)$, is defined to be:

${\rm KBSM}\left(M\right)={\raise 3.01385pt\hbox{$R\mathcal{L}_{{\rm
fr}}$}\\!\mathord{\left/{\vphantom{R\mathcal{L_{{\rm
fr}}}{\mathcal{S}}}}\right.\kern-1.2pt}\\!\lower 3.01385pt\hbox{$S$}}.$

The importance of the Kauffman bracket skein module of a 3-manifold $M$ lies
in the fact that it detects the presence of non-separating 2-spheres and tori
embedded in $M$. On the level of Kauffman bracket skein modules, a non-
separating 2-sphere in $M$ is detected by the existence of torsion in
KBSM($M$).

We consider $S^{1}\times S^{2}$ as being obtained from $S^{3}$ by integral
surgery along the unknot with coefficient zero, that is, $S^{1}\times S^{2}$
is a special type of lens spaces $L(p,q)$ for $p=0$ and $q=1$
($L(0,1)\,\cong\,S^{1}\times S^{2}$). In [1], the Kauffman bracket skein
module of the lens spaces $L(p,q),\,p>0$ is computed via the ‘braid
technique’, and a basis of KBSM($L(p,q)$) is obtained that is different from
the basis presented in [19]. In this paper we apply the ‘braid technique’ to
exhaust the computation of the Kauffman bracket skein module of the family of
lens spaces. More precisely, the integral surgery model of $S^{1}\times S^{2}$
allows us to ‘push’ every knot and link in $S^{1}\times S^{2}$ to the solid
torus, and thus, a basis of KBSM(ST) spans KBSM($S^{1}\times S^{2}$).
Following the pioneering work of V.F.R. Jones ([21, 22]), in [1] we consider
the generalized Temperley-Lieb algebra of type B, $TL_{1,n}$, together with a
Markov trace constructed on $TL_{1,n}$ and the universal invariant $V$ of the
Kauffman bracket type for knots and links in ST defined in [1]. This invariant
gives distinct values to distinct elements of any basis of KBSM(ST), and thus,
it recovers KBSM(ST). Hence, in order to compute KBSM($S^{1}\times S^{2}$),
one needs to extend the invariant $V$ for knots and links in ST to an
invariant of knots and links in $S^{1}\times S^{2}$, by imposing relations
coming from the extra isotopy moves of $S^{1}\times S^{2}$, that we call
(braid) band moves, abbreviated $bm$ (or $bbm$). In other words, the following
infinite system of equations is obtained:

(1) $V_{a}\ =\ V_{bm(a)},\quad{\rm for\ every}\ a\ {\rm in\ a\ basis\ of}\
{\rm KBSM(ST)}.$

By construction, a solution to the infinite system of Equations (1)
corresponds to the computation of the Kauffman bracket skein module of
$S^{1}\times S^{2}$. These equations are simplified with the use of a new
basis of KBSM(ST), $B_{{\rm ST}}$, presented in [1, 2, 17], that arises
naturally on the level of braids. Using the basis $B_{{\rm ST}}$ of KBSM(ST),
we show that the free part of the Kauffman bracket skein module of
$S^{1}\times S^{2}$ is generated by the unknot (or the empty knot) and we also
show how torsion is captured in the solution of the infinite system. It is
worth mentioning that $S^{1}\times S^{2}$ is the first $3$-manifold, where
torsion on its Kauffman bracket skein module is detected via the ‘braid
technique’. Finally, we compute KBSM($S^{1}\times S^{2}$) via a different
diagrammatic method based on braids following [5, 3]. In this way we obtain a
closed formula for the torsion part of the module and our result agrees with
that of Hoste and Przytycki [20]. The diagrammatic method via braids has been
successfully applied in [5] for the case of the Kauffman bracket skein module
of the complement of $(2,2p+1)$-torus knots and in [1] for the case of the
lens spaces $L(p,q)$, $p\neq 0$. The importance of the braid approach lies in
the fact that it can shed light to the problem of computing (various) skein
modules of arbitrary c.c.o. $3$-manifolds (see also [13] for the case of the
HOMFLYPT skein module of the lens spaces $L(p,1)$).

The paper is organized as follows: In §1 we recall the topological setting and
the essential techniques and results from [26, 11]. We first present different
models for $S^{1}\times S^{2}$, which demonstrate the difference between the
knot theory of this $3$-manifold to the knot theory of $S^{3}$. We then
explain how knots and links in $S^{1}\times S^{2}$ can be viewed as mixed
links in $S^{3}$. Mixed links are close related to Kirby diagrams of links in
$S^{3}$, that is, link diagrams in $S^{3}$ that involve the surgery
description of $S^{1}\times S^{2}$. We present isotopy moves for mixed links
in $S^{3}$ that correspond to isotopy moves for knots and links in
$S^{1}\times S^{2}$, and using the analogue of the Alexander theorem for knots
and links in ST, we pass on the level of mixed braids. We then translate
isotopy in $S^{1}\times S^{2}$ to braid equivalence (the analogue of the
Markov theorem for braid equivalence in $S^{1}\times S^{2}$). In §2 we present
results from [23, 24] and [1, 2]. More precisely, we start by recalling
results on the generalized Hecke algebra of type B, $H_{1,n}$, and we present
the universal invariant of the HOMFLYPT type for knots in ST via a unique
Markov trace, $tr$, defined on $H_{1,n}$. This invariant captures the HOMFLYPT
skein module of ST. We then pass to the generalized Temperley-Lieb algebra of
type B, defined as a quotient algebra of $H_{1,n}$ over an appropriate chosen
ideal (see relations (3)), and we obtain the universal invariant $V$ for knots
and links in ST of the Kauffman bracket type, that captures KBSM(ST). We also
present the basis $B_{\rm ST}$ of KBSM(ST) that describes naturally the
isotopy moves for knots and links in $S^{1}\times S^{2}$, and that simplifies
the infinite system of equations (1). In § 3, we solve the infinite system of
equations (1) and we prove that the free part of KBSM($S^{1}\times S^{2}$) is
generated by the unknot (or the empty knot). We also show how torsion is
detected in the solution of this system. Finally, in § 4 we demonstrate the
diagrammatic approach based on braids for computing KBSM($S^{1}\times S^{2}$).

##  1\. Topological Set Up

###  1.1. Models of $S^{1}\times S^{2}$

$S^{1}\times S^{2}$ can be realized as a thickened torus in $4$-dimensions,
such that the cross-sections are two copies of the $2$-sphere $S^{2}$. To
better perceive this, we consider the analogue construction of $S^{1}\times
S^{1}$ in dimension three: we consider copies of $S^{1}$ and we ‘glue’ them
together to obtain a cylinder. Finally, we glue the first and the last $S^{1}$
to obtain a torus (see Figure 2(a) and (b)). This construction is equivalent
to ‘gluing’ the inner and outer $S^{1}$’s from an infinitum concentric
$S^{1}$’s (see Figure 2(c)). Analogously, $S^{1}\times S^{2}$ may be
visualized as a three ball $B^{3}$ with a concentric and of smaller radius
$S^{2}$ being hollowed out, and where the boundary of $B^{3}$ is glued to the
inner $S^{2}$ as illustrated in Figure 2(d).

Figure 2. Models of $S^{1}\times S^{1}$ (a, b, c) and $S^{1}\times S^{2}$ (d).

The model of $S^{1}\times S^{2}$ as illustrated in Figure 2(d) can be used to
highlight the difference between the knot theory of $S^{1}\times S^{2}$ and
the knot theory of $S^{3}$. Consider for example the knot in Figure 3 and
notice that using isotopy, we may unknot this knot with the use of the non-
separating $2$-sphere. This method is known as the Dirac trick.

Figure 3. The Dirac trick for unknotting a knot in $S^{1}\times S^{2}$.

A different way to visualize $S^{1}\times S^{2}$ is by ‘gluing’ two solid tori
via some homeomorphism $h$ on their boundaries, such that $h$ takes a meridian
curve $m_{1}$ on the first solid torus ${\rm ST}_{1}$, to the corresponding
meridian curve $m_{2}$ of the second solid torus ${\rm ST}_{2}$. We may choose
$h$ to be the identity $i:\partial{\rm ST}_{1}\,\rightarrow\,\partial{\rm
ST}_{2}$. Since now $ST\,=\,S^{1}\times D^{2}$, where $D^{2}$ denotes a disc,
and since the gluing of two discs results in a 2-sphere $S^{2}$, the resulting
$3$-manifold can be considered as a family of $2$-spheres parametrized by a
circle (for an illustration see Figure 4).

Figure 4. Gluing two solid tori to obtain $S^{1}\times S^{2}$.

The above construction of $S^{1}\times S^{2}$ coincides with the Dehn surgery
method for obtaining $S^{1}\times S^{2}$. This is a method for obtaining any
$3$-manifold by removing finitely many disjoint solid tori from $S^{3}$ and
then sewing them back in a different way. Recall now that by Alexander’s
trick, the $3$-sphere $S^{3}$ is the union of two solid tori via a
homeomorphism $h$ on their boundaries that maps a meridian curve on the first
solid torus to a longitude curve on the second. Hence, $S^{1}\times S^{2}$ may
be obtained by first removing a solid torus ${\rm ST}_{1}$ from $S^{3}$, and
since $S^{3}\,=\,{\rm ST}_{1}\,\underset{h}{\bigcup}\,{\rm ST}_{2}$, what
remains is another solid torus ${\rm ST}_{2}$. We finally glue ${\rm ST}_{1}$
to ${\rm ST}_{2}$ via $h:m_{1}\mapsto m_{2}$, where $m_{1}$ is a meridian on
${\rm ST}_{1}$ and $m_{2}$ is a meridian on ${\rm ST}_{2}$. Let us present the
above results in a more rigorous format. Consider the tubular neighborhood of
the unknot in $S^{3}$. This is obviously a solid torus, say ${\rm ST}_{1}$.
Then, this solid torus is to be “drilled away” from $S^{3}$ and then sewed
back via $h$. $h$ can be fully determined by a coefficient that instructs how
${\rm ST}_{1}$ is to be sewed back. In the case of $S^{1}\times S^{2}$, the
meridian of ${\rm ST}_{1}$ is to be glued to a meridian of ${\rm ST}_{2}$,
that is a $(0,1)$ curve on the boundary of ${\rm ST}_{2}$. Thus, we say that
$S^{1}\times S^{2}$ is obtained from $S^{3}$ by surgery along the unknot with
coefficient zero, i.e. $S^{1}\times S^{2}\cong{\rm O}^{0}$. It is worth
mentioning that $S^{1}\times S^{2}$ may be considered as a special type of
lens spaces $L(p,q)$, where $p=0$ and $q=1$.

###  1.2. Knots and braids in $S^{1}\times S^{2}$

In this subsection we recall results from [26, 27, 11]. As mentioned before,
we shall consider ST to be the complement of another solid torus in $S^{3}$
and an oriented link $L$ in ST can be represented by an oriented mixed link in
$S^{3}$. A mixed link is a link in $S^{3}$ consisting of the unknotted fixed
part $\widehat{I}$ representing the complementary solid torus in $S^{3}$, and
the moving part $L$ that links with $\widehat{I}$. A mixed link diagram is a
diagram $\widehat{I}\cup\widetilde{L}$ of $\widehat{I}\cup L$ on the plane of
$\widehat{I}$, where this plane is equipped with the top-to-bottom direction
of $I$ (for an illustration see Figure 5).

Figure 5.  A mixed link in $S^{3}$ representing a link in ST.

Isotopy of oriented links in ST consists in isotopy in $S^{3}$ for the moving
part of the mixed link, together with the mixed Reidemeister moves, that
involve both the fixed and the moving part of the mixed link (see Figure 6).

Figure 6.  The mixed Reidemeister moves.

As mentioned above, $S^{1}\times S^{2}$ can be obtained from $S^{3}$ by
surgery on the unknot with surgery coefficient zero. In other words,
$S^{1}\times S^{2}$ can be obtained by “killing” the meridian $m$, that is, by
mirroring $D\times S^{1}$ along its boundary. Knots and links in $S^{1}\times
S^{2}$ may be then visualized in $S^{3}$ as mixed links, together with the
surgery coefficient on their fixed component (“Kirby” like diagrams). It
follows that isotopy in $S^{1}\times S^{2}$ can be then viewed as isotopy in
ST together with the band moves, which reflect the surgery description of the
manifold and that are similar to the second Kirby move. It is worth mentioning
that there are two types of band moves, according to the orientation of the
component of the knot and that of the surgery curve (see Figure 7 for the two
types of band moves for the case of $S^{1}\times S^{2}$). In the $\beta$-type
moves the orientation of the arc is initially opposite to the orientation of
the surgery curve, but after the performance of the move their orientations
agree. In the $\alpha$-type the orientations initially agree, but disagree
after the performance of the move.

Figure 7. The two types of band moves.

In [11] it is shown that in order to describe isotopy for knots and links in a
c.c.o. $3$-manifold, it suffices to consider only one type of band moves (cf.
[11, Theorem 6]) and thus, isotopy between oriented links in $S^{1}\times
S^{2}$ is reflected in $S^{3}$ by means of the following theorem:

######  Theorem 1 (The analogue of Reidemeister’s theorem).

Two oriented links in $S^{1}\times S^{2}$ are isotopic if and only if two
corresponding mixed link diagrams of theirs differ by isotopy in ST together
with a finite sequence of one type of band moves.

As shown in [23], every mixed link is isotopic to the closure of a mixed
braid. A mixed braid in $S^{3}$ is a standard braid where, without loss of
generality, its first strand represents $\widehat{I}$, the fixed part, and the
other strands, $\beta$, represent the moving part. We shall call the subbraid
$\beta$ the moving part of $I\cup\beta$ (see bottom left illustration of
Figure 8).

We now recall the analogue of the Alexander theorem for knots and links in ST
(cf. [24, Theorem 1]):

######  Theorem 2 (The analogue of Alexander’s theorem).

A mixed link diagram $\widehat{I}\cup\widetilde{L}$ of $\widehat{I}\cup L$ may
be turned into a mixed braid $I\cup\beta$ with isotopic closure.

Hence, a link in $S^{1}\times S^{2}$, that is visualized as a mixed link in
$S^{3}$, may be turned to a mixed braid in $S^{3}$. We now translate isotopy
in $S^{1}\times S^{2}$ on the level of mixed braids. We define a braid band
move (abbreviated to bbm) to be a move between mixed braids, which is a
$\beta$-type band move between their closures. It starts with a little band
oriented downward, which, before sliding along a surgery strand, gets one
twist positive or negative (see middle part of Figure 8). Note that in [27] it
was shown that the choice of the position of connecting the two components
after the performance of a bbm is arbitrary.

Figure 8.  The two types of braid band moves.

The sets of braids related to ST form groups, which are in fact the Artin
braid groups of type B, denoted $B_{1,n}$, with presentation:

$B_{1,n}=\left<\begin{array}[]{ll}\begin{array}[]{l}t,\sigma_{1},\ldots,\sigma_{n-1}\\\
\end{array}&\left|\begin{array}[]{l}\sigma_{1}t\sigma_{1}t=t\sigma_{1}t\sigma_{1}\\\
t\sigma_{i}=\sigma_{i}t,\quad{i>1}\\\
{\sigma_{i}}\sigma_{i+1}{\sigma_{i}}=\sigma_{i+1}{\sigma_{i}}\sigma_{i+1},\quad{1\leq
i\leq n-2}\\\
{\sigma_{i}}{\sigma_{j}}={\sigma_{j}}{\sigma_{i}},\quad{|i-j|>1}\\\
\end{array}\right.\end{array}\right>,$

where the generators $\sigma_{i}$ and $t$ are illustrated in Figure 9(i).
Notice that the mixed braid group $B_{1,n}$ can be considered as a subgroup of
the classical braid group on $(n+1)$ strands, $B_{n+1}$, involving elements
whose first strand is fixed.

Let now $\mathcal{L}$ denote the set of oriented knots and links in
$S^{1}\times S^{2}$. Isotopy in $S^{1}\times S^{2}$ is then translated on the
level of mixed braids by means of the following theorem (see also [27, Theorem
5] and [11, Theorem 9]):

######  Theorem 3.

Let $L_{1},L_{2}$ be two oriented links in $S^{1}\times S^{2}$ and let
$I\cup\beta_{1},{\rm\;}I\cup\beta_{2}$ be two corresponding mixed braids in
$S^{3}$. Then $L_{1}$ is isotopic to $L_{2}$ in $S^{1}\times S^{2}$ if and
only if $I\cup\beta_{1}$ is equivalent to $I\cup\beta_{2}$ in $\mathcal{B}$ by
the following moves:

$\begin{array}[]{clll}(i)&Conjugation:&\alpha\sim\beta^{-1}\alpha\beta,&{\rm
if}\ \alpha,\beta\in B_{1,n}.\\\ (ii)&Stabilization\
moves:&\alpha\sim\alpha\sigma_{n}^{\pm 1}\in B_{1,n+1},&{\rm if}\ \alpha\in
B_{1,n}.\\\ (iii)&Loop\ conjugation:&\alpha\sim t^{\pm 1}\alpha t^{\mp
1},&{\rm if}\ \alpha\in B_{1,n}.\\\ (iv)&Braid\ band\
moves:&\alpha\sim\alpha_{+}\sigma_{1}^{\pm 1},&a_{+}\in B_{1,n+1},\end{array}$

where $\alpha_{+}$ is the word $\alpha$ with all indices shifted by +1. Note
that moves (i), (ii) and (iii) correspond to link isotopy in ST.

######  Notation 1.

We denote a braid band move by bbm and, specifically, the result of a positive
(cor. negative) braid band move performed on a mixed braid $\beta$ by
$bbm_{+}(\beta)$ (cor. $bbm_{-}(\beta)$).

##  2\. Algebraic Tools

In this section we recall results from [1] that are crucial for the
computation of KBSM($S^{1}\times S^{2}$). More precisely, we first recall
results on the generalized Temperley-Lieb algebra of type B, $TL_{1,n}$, that
is related to the knot theory of ST, and through $TL_{1,n}$ we obtain the
universal invariant for knots and links in ST of the Kauffman bracket type. We
also present different bases of KBSM(ST) via braids.

###  2.1. Knot algebras related to ST & $S^{1}\times S^{2}$

It is well understood that the generalized Hecke algebra of type B is related
to the knot theory of the solid torus [24]. This algebra is defined as a
quotient of the mixed braid group algebra ${\mathbb{Z}}\left[q^{\pm
1}\right]B_{1,n}$ over the ideal generated by the quadratic relations
${\sigma_{i}^{2}=(q-1)\sigma_{i}+q}$. Namely:

$H_{1,n}(q)=\frac{{\mathbb{Z}}\left[q^{\pm
1}\right]B_{1,n}}{\langle\sigma_{i}^{2}-\left(q-1\right)\sigma_{i}-q\rangle}.$

As shown in [24], $H_{1,n}$ admits a unique Markov trace, $tr$. This Markov
trace gives rise to the most generic analogue of the HOMFLYPT polynomial, $X$,
for links in the solid torus ST. The construction of $tr$ was possible with
the use of appropriate inductive bases on
$\mathcal{H}:=\bigcup_{n=1}^{\infty}\,H_{1,n}$. Namely:

######  Theorem 4 (Proposition 1 & Theorem 1 [24]).

The following sets form linear bases for ${\rm H}_{1,n}(q)$:

(2) $\begin{array}[]{llll}(i)&\Sigma_{n}&=&\\{t_{i_{1}}^{k_{1}}\ldots
t_{i_{r}}^{k_{r}}\cdot\sigma\\},\ {\rm where}\ 0\leq i_{1}<\ldots<i_{r}\leq
n-1,\\\
(ii)&\Sigma^{\prime}_{n}&=&\\{{t^{\prime}_{i_{1}}}^{k_{1}}\ldots{t^{\prime}_{i_{r}}}^{k_{r}}\cdot\sigma\\},\
{\rm where}\ 0\leq i_{1}<\ldots<i_{r}\leq n-1,\\\ \end{array}$

where $k_{1},\ldots,k_{r}\in{\mathbb{Z}}$, $t_{0}^{\prime}\ =\ t_{0}\ :=\
t,\quad t_{i}^{\prime}\ =\ g_{i}\ldots g_{1}tg_{1}^{-1}\ldots
g_{i}^{-1}\quad{\rm and}\quad t_{i}\ =\ g_{i}\ldots g_{1}tg_{1}\ldots g_{i}$
are the ‘looping elements’ in ${\rm H}_{1,n}(q)$ (see Figure 9(ii)) and
$\sigma$ a basic element in the Iwahori-Hecke algebra of type A, ${\rm
H}_{n}(q)$.

Note that a basic element in ${\rm H}_{n}(q)$ is of the form of elements in
the following set [21]:

$S_{n}=\left\\{(g_{i_{1}}g_{i_{1}-1}\ldots
g_{i_{1}-k_{1}})(g_{i_{2}}g_{i_{2}-1}\ldots
g_{i_{2}-k_{2}})\ldots(g_{i_{p}}g_{i_{p}-1}\ldots g_{i_{p}-k_{p}})\right\\},$

for $1\leq i_{1}<\ldots<i_{p}\leq n-1{\rm\;}$.

Figure 9. The generators of $B_{1,n}$ and the ‘looping’ elements
$t^{\prime}_{i}$ and $t_{i}$.

######  Theorem 5.

[24, Theorem 6 & Definition 1] Given $z,s_{k}$ with $k\in{\mathbb{Z}}$
specified elements in $R={\mathbb{Z}}\left[q^{\pm 1}\right]$, there exists a
unique linear Markov trace function on $\mathcal{H}$:

${\rm tr}:\mathcal{H}\to R\left(z,s_{k}\right),\ k\in{\mathbb{Z}}$

determined by the rules:

$\begin{array}[]{lllll}(1)&{\rm tr}(ab)&=&{\rm tr}(ba)&\quad{\rm for}\
a,b\in{\rm H}_{1,n}(q)\\\ (2)&{\rm tr}(1)&=&1&\quad{\rm for\ all}\ {\rm
H}_{1,n}(q)\\\ (3)&{\rm tr}(ag_{n})&=&z{\rm tr}(a)&\quad{\rm for}\ a\in{\rm
H}_{1,n}(q)\\\ (4)&{\rm tr}(a{t^{\prime}_{n}}^{k})&=&s_{k}{\rm
tr}(a)&\quad{\rm for}\ a\in{\rm H}_{1,n}(q),\ k\in{\mathbb{Z}}\\\ \end{array}$

Then, the function $X:\mathcal{L}$ $\rightarrow R(z,s_{k})$

$X_{\widehat{\alpha}}=\Delta^{n-1}\cdot\left(\sqrt{\lambda}\right)^{e}{\rm
tr}\left(\pi\left(\alpha\right)\right),$

is an invariant of oriented links in ST, where $\Delta:=-\frac{1-\lambda
q}{\sqrt{\lambda}\left(1-q\right)}$, $\lambda:=\frac{z+1-q}{qz}$, $\alpha\in
B_{1,n}$ is a word in the $\sigma_{i}$’s and $t^{\prime}_{i}$’s,
$\widehat{\alpha}$ is the closure of $\alpha$, $e$ is the exponent sum of the
$\sigma_{i}$’s in $\alpha$, $\pi$ the canonical map of $B_{1,n}$ to ${\rm
H}_{1,n}(q)$, such that $t\mapsto t$ and $\sigma_{i}\mapsto g_{i}$.

######  Remark 1.

As shown in [24, 12] the invariant $X$ recovers the HOMFLYPT skein module of
ST (see also [13, 15] for a survey on the HOMFLYPT skein module of the lens
spaces $L(p,1)$ via braids).

The algebraic counterpart construction of the Kauffman bracket is the
Temperley-Lieb algebra together with a unique Markov trace constructed by
V.F.R. Jones (see [21] and references therein). It is therefore natural to
consider the generalized Temperley-Lieb algebra of type B, defined as a
quotient of $H_{1,n}(q)$, in order to obtain a universal invariant of the
Kauffman bracket type for knots and links in ST. We have the following
definition ([1, Definition 2]):

######  Definition 2.

The generalized Temperley-Lieb algebra of type B, $TL_{1,n}$, is defined as
the quotient of the generalized Hecke algebra of type B, $H_{1,n}(q)$, over
the ideal generated by expressions of the form:

(3) $\begin{array}[]{lcl}\sigma_{i,i+1}&:=&1+u\
(\sigma_{i}+\sigma_{i+1})+u^{2}\
(\sigma_{i}\sigma_{i+1}+\sigma_{i+1}\sigma_{i})+u^{3}\
\sigma_{i}\sigma_{i+1}\sigma_{i}.\\\ \end{array}$

######  Remark 2.

For the rest of the paper we will adapt a different presentation for
$H_{1,n}$, that involves the parameter $u$ and the quadratic relations:

(4) ${\sigma_{i}^{2}=(u-u^{-1})\,\sigma_{i}+1}.$

Note also that we will be using the $\sigma_{i}$’s for the trace instead of
the $g_{i}$’s. Finally, note that we will be using the symbol
$\widehat{\cong}$ when both stabilization moves and conjugation are performed
(instead of using tr).

In [1], it is shown that for $z=-\frac{1}{u(1+u^{2})}$ the Markov trace
constructed in $H_{1,n}$ factors through to ${TL}_{1,n}$. This leads to the
universal invariant of the Kauffman bracket type for knots and links in ST.
More precisely, we have the following theorem:

######  Theorem 6.

The invariant

$V_{\widehat{\alpha}}(u)\ :=\
\left(-\frac{1+u^{2}}{u}\right)^{n-1}\left(u\right)^{2e}{\rm
tr}\left(\overline{\pi}\left(\alpha\right)\right),$

where $\alpha\in B_{1,n}$ is a word in the $\sigma_{i}$’s and
$t^{\prime}_{i}$’s, $\widehat{\alpha}$ is the closure of $\alpha$, $e$ is the
exponent sum of the $\sigma_{i}$’s in $\alpha$, $\overline{\pi}$ the canonical
map of $B_{1,n}$ to ${\rm TL}_{1,n}$, such that $t\mapsto t$ and
$\sigma_{i}\mapsto g_{i}$, is the universal invariant for knots and links in
ST of the Kauffman bracket type.

The invariant $V$ recovers the Kauffman bracket skein module of ST and as
shown in [1], it can be extended to an invariant for knots and links in
$L(p,q)$ by considering the effect of the braid band moves on elements in some
basis of KBSM(ST) (see also [28]). Equivalently, and since $S^{1}\times
S^{2}\cong L(0,1)$, in order to compute KBSM($S^{1}\times S^{2}$), it suffices
to consider elements in some basis of KBSM(ST), impose on $V$ relations coming
from the band moves and solve the resulting infinite system of equations.

###  2.2. The Kauffman bracket skein module of ST via braids

From the discussion in § 1, it is clear that the knot theory of ST is of
fundamental importance for studying knot theory in other c.c.o. $3$-manifolds.
In [18], the Kauffman bracket skein module of the solid torus is computed
using diagrammatic methods by means of the following theorem:

######  Theorem 7 ([18]).

The Kauffman bracket skein module of ST, KBSM(ST), is freely generated by an
infinite set of generators $\left\\{x^{n}\right\\}_{n=0}^{\infty}$, where
$x^{n}$ denotes a parallel copy of $n$ longitudes of ST and $x^{0}$ is the
affine unknot (see Figure 10).

Figure 10. The standard basis of KBSM(ST).

In the braid setting, elements in the standard basis of KBSM(ST) correspond
bijectively to the elements of the following set (for an illustration see
Figure 11):

(5) $\mathcal{B}^{\prime}_{{\rm
ST}}=\\{t{t^{\prime}_{1}}\ldots{t^{\prime}_{n}},\ n\in\mathbb{N}\\}.$

In other words, the set $\mathcal{B}^{\prime}_{{\rm ST}}$ forms a basis for
KBSM(ST) in terms of braids.

Figure 11. The standard basis of KBSM(ST) in terms of mixed braids.

######  Remark 3.

  * i.

The basis $\mathcal{B}^{\prime}_{{\rm ST}}$ is a subset of $\mathcal{H}$ and,
in particular, $\mathcal{B}^{\prime}_{{\rm ST}}$ is a subset of
$\Sigma^{\prime}=\bigcup_{n}\Sigma^{\prime}_{n}$. Moreover, and in contrast to
elements in $\Sigma^{\prime}$, the elements in $\mathcal{B}^{\prime}_{{\rm
ST}}$ have no ‘gaps’ in the indices, the exponents are all equal to one and
there are no ‘braiding tails’.

  * ii.

The invariant $V$ defined in Theorem 6 recovers KBSM(ST), since it gives
distinct values to distinct elements of $\mathcal{B}^{\prime}_{{\rm ST}}$.
Indeed, we have that ${\rm
tr}(t{t^{\prime}_{1}}\ldots{t^{\prime}_{n}})=s_{1}^{n+1}$.

It follows from Remark 3(ii), that in order to compute the Kauffman bracket
skein module of $S^{1}\times S^{2}$, it suffices to extend the (most generic)
invariant $V$ for knots and links in ST, following the ideas in [12, 13, 14].
That is, it suffices to solve the infinite system of Equations (1),
$V_{\widehat{\alpha}}\ =\ V_{\widehat{bbm_{\pm}(\alpha)}}$, for all $\alpha$
in a basis of KBSM(ST). Note that the above agree with the relation between
KBSM($S^{1}\times S^{2}$) and KBSM(ST) that is presented in [28]. Thus, we
conclude that:

${\rm KBSM(}S^{1}\times S^{2}{\rm)}\,=\,\frac{{\rm
KBSM(ST)}}{<a-bbm_{\pm}(a)>}\ \Leftrightarrow\ V_{\widehat{a}}\ =\
V_{\widehat{bbm_{\pm}(a)}},\quad\ \forall\ a\ {\rm in\ a\ basis\ of\
KBSM(ST)}.$

These equations have very complicated formulations with the use of the
standard basis $\mathcal{B}^{\prime}_{{\rm ST}}$. In Figure 13(a) for example,
we demonstrate the performance of a bbm on
$t{t_{1}^{\prime}}^{2}\in\mathcal{B}^{\prime}_{{\rm ST}}$. To simplify the
infinite system of equations, in [1] a new basis ${B}_{{\rm ST}}$ for the
Kauffman bracket skein module of ST is presented, using the looping generator
$t$ (for an illustration of elements in this basis see Figure 12). More
precisely, we have the following:

######  Theorem 8.

The following set is a basis for KBSM(ST):

(6) $B_{\rm ST}\ =\ \\{t^{n},\ n\in\mathbb{N}\\}.$

Figure 12. Elements in the new basis of KBSM(ST).

######  Remark 4.

Note that elements in the basis $B_{\rm ST}$ have no crossings on the braid
level, and thus, they are more “natural” and appropriate for our purpose.

From Theorem 3, we observe that the equations $V_{\widehat{a}}\ =\
V_{\widehat{bbm_{\pm}(a)}}$ have particularly simple formulations with the use
of the new basis $B_{\rm ST}$. Indeed, we have that
$t^{n}\,\overset{bbm_{\pm}}{\rightarrow}\,t_{1}^{n}\,\sigma_{1}^{\pm 1}$. In
Figure 13(b) we illustrate a bbm applied on $tt_{1}^{2}\in B_{\rm ST}$.

(a) bbm performed on $t{t_{1}^{\prime}}^{2}\in\mathcal{B}^{\prime}_{{\rm
ST}}$.

(b) bbm performed on $tt_{1}^{2}\in B_{ST}$.

##  3\. The Kauffman bracket skein module of $S^{1}\times S^{2}$ via
$TL_{1,n}$

In this section we solve the infinite system of Equations (1), which is
equivalent to computing the Kauffman bracket skein module of the lens spaces
$S^{1}\times S^{2}$. Recall that this infinite system of equations is obtained
by performing braid band moves on elements in the basis $B_{\rm ST}$ of
KBSM(ST) and by imposing to the generic invariant $V$ for knots and links in
ST relations of the form $V_{\widehat{t^{n}}}\,=\,V_{\widehat{bbm(t^{n})}}$,
for all $n\in\mathbb{N}$, where $bbm(t^{n})\,=\,t_{1}^{n}\,\sigma_{1}^{\pm
1}$. Recall also that the unknowns in the system are the $s_{i}$’s, coming
from the fourth rule of the trace function in Theorem 5, that is,
$tr(t^{n})\,=\,s_{n}$, for all $n\in\mathbb{Z}$.

We first observe that the equations obtained by applying the two types of
bbm’s on an element in $B_{\rm ST}$ other than the unknot, are different, and
thus, both bbm’s are needed in order to compute KBSM($S^{1}\times S^{2}$). In
other words, we have that:

For $n\in\mathbb{N}\backslash\\{0\\}$, the equations
$V_{\widehat{t^{n}}}\,=\,V_{\widehat{t_{1}^{n}\sigma_{1}}}$ and
$V_{\widehat{t^{n}}}\,=\,V_{\widehat{t_{1}^{n}\sigma_{1}^{-1}}}$ are not
equivalent.

Consider $t^{n}\in B_{{\rm ST}}$, where $n\geq 1$. Then, the equations
obtained by performing bbm’s on $t^{n}$ are:

$\begin{array}[]{lclc}V_{\widehat{t^{n}}}\,=\,V_{\widehat{t_{1}^{n}\sigma_{1}^{\pm
1}}}&\overset{(+)-bbm}{\underset{(-)-bbm}{\Leftrightarrow}}&\begin{cases}s_{n}&=\,\left(-\,\frac{1+u^{2}}{u}\right)\,u^{4n+2}\,tr(t_{1}^{n}\sigma_{1})\\\
s_{n}&=\,\left(-\,\frac{1+u^{2}}{u}\right)\,u^{4n-2}\,tr(t_{1}^{n}\sigma_{1}^{-1})\end{cases}\end{array}$

We now deal with the equation obtained from a negative bbm and we underline
expressions which are crucial for the next step.

$\begin{array}[]{lclc}s_{n}&=&\left(-\,\frac{1+u^{2}}{u}\right)\,u^{4n-2}\,tr(t_{1}^{n}\underline{\sigma_{1}^{-1}})&\overset{\sigma_{1}^{-1}=\sigma_{1}-(u-u^{-1})}{\Leftrightarrow}\\\
&&&\\\
s_{n}&=&\left(-\,\frac{1+u^{2}}{u}\right)\,u^{4n-2}\,\left[tr(t_{1}^{n}\sigma_{1})\,-\,(u-u^{-1})\,tr(t_{1}^{n})\right]&\Leftrightarrow\\\
&&&\\\
s_{n}&=&\underline{\left(-\,\frac{1+u^{2}}{u}\right)\,u^{4n+2}\,tr(t_{1}^{n}\sigma_{1})}\,u^{-4}+\,\left(\frac{1+u^{2}}{u}\right)\,u^{4n-2}\,\,(u-u^{-1})\,tr(t_{1}^{n})&\overset{(+)-bbm}{\Leftrightarrow}\\\
&&&\\\
s_{n}&=&u^{-4}\,s_{n}+\,\left(\frac{1+u^{2}}{u}\right)\,u^{4n-2}\,\,(u-u^{-1})\,tr(t_{1}^{n})&\\\
\end{array}$

Hence, for $n\in\mathbb{N}\backslash\\{0\\}$ we have that the equations
obtained from the performance of the two types of bbm’s are equivalent if and
only if:

(7) $\frac{u^{4}-1}{u^{4}}\cdot\left[s_{n}-u^{4n}\,tr(t_{1}^{n})\right]\,=\,0$

We now evaluate $tr(t_{1}^{n})$ for $n=1$ and we have:

$\begin{array}[]{lclcl}tr(t_{1})&=&tr(\underline{\sigma_{1}}t\sigma_{1})\ =\
tr(t\underline{\sigma_{1}^{2}})&=&(u-u^{-1})\,tr(t\sigma_{1})+tr(t)\ =\\\
&&&&\\\ &=&(u-u^{-1})\,\underline{z}\cdot s_{1}\ +\
s_{1}&\overset{z=\frac{-1}{u(1+u^{2})}}{=}&\frac{u^{4}+1}{u^{2}(1+u^{2})}\cdot
s_{1},\\\ \end{array}$

while substituting $n=1$ in Eq. (7) we obtain that
$tr(t_{1})\,=\,\frac{1}{u^{4}}\cdot s_{1}$. Therefore, we have a contradiction
and we conclude that in order to compute KBSM($S^{1}\times S^{2}$) we need to
consider the effect of both types of bbm’s on elements in $B_{{\rm ST}}$.

###  3.1. Useful Lemmata

In this subsection we present a series of results toward the solution of the
infinite system (1), by studying the effect of the braid band moves on the
elements in the basis $B_{\rm ST}$. We recall first an ordering relation
defined in [12]. For that we shall need the notion of the index of a word $w$,
denoted $ind(w)$.

######  Definition 3.

[12, Definition 1] Let $w$ be a word in $\Sigma$ or in $\Sigma^{\prime}$.
Then, the index of $w$, $ind(w)$, is defined to be the highest index of the
$t_{i}$’s ($t_{i}^{\prime}$’s respectively) in $w$, by ignoring possible gaps
in the indices of the looping generators and by ignoring the braiding parts.
Moreover, the index of a monomial in $\sigma_{i}$’s is equal to $0$.

######  Definition 4.

[12, Definition 2] Let
$w={t^{\prime}_{i_{1}}}^{k_{1}}\ldots{t^{\prime}_{i_{\mu}}}^{k_{\mu}}\cdot\beta_{1}$
and
$u={t^{\prime}_{j_{1}}}^{\lambda_{1}}\ldots{t^{\prime}_{j_{\nu}}}^{\lambda_{\nu}}\cdot\beta_{2}$
in $\Sigma^{\prime}$, where $k_{t},\lambda_{s}\in\mathbb{Z}$ for all $t,s$ and
$\beta_{1},\beta_{2}\in H_{n}(q)$. Then, we define the following ordering in
$\Sigma^{\prime}$:

  * (a)

If $\sum_{i=0}^{\mu}k_{i}<\sum_{i=0}^{\nu}\lambda_{i}$, then $w<u$.

  * (b)

If $\sum_{i=0}^{\mu}k_{i}=\sum_{i=0}^{\nu}\lambda_{i}$, then:

(i) if $ind(w)<ind(u)$, then $w<u$,

(ii) if $ind(w)=ind(u)$, then:

($\alpha$) if $i_{1}=j_{1},\ldots,i_{s-1}=j_{s-1},i_{s}<j_{s}$, then $w>u$,

($\beta$) if $i_{t}=j_{t}$ for all $t$ and
$k_{\mu}=\lambda_{\mu},k_{\mu-1}=\lambda_{\mu-1},\ldots,k_{i+1}=\lambda_{i+1},|k_{i}|<|\lambda_{i}|$,
then $w<u$,

($\gamma$) if $i_{t}=j_{t}$ for all $t$ and
$k_{\mu}=\lambda_{\mu},k_{\mu-1}=\lambda_{\mu-1},\ldots,k_{i+1}=\lambda_{i+1},|k_{i}|=|\lambda_{i}|$
and $k_{i}>\lambda_{i}$, then $w<u$,

($\delta$) if $i_{t}=j_{t}\ \forall t$ and $k_{i}=\lambda_{i}$, $\forall i$,
then $w=u$.

The ordering in the set $\Sigma$ is defined as in $\Sigma^{\prime}$, where
$t_{i}^{\prime}$’s are replaced by $t_{i}$’s.

Our goal is to express
$bbm(t^{n})\,=\,t_{1}^{n}\,\sigma_{1}^{-1}\,\widehat{\cong}\,t\,t_{1}^{n-1}\,\sigma_{1}$
for $n\in\mathbb{N}$, to sums of elements in $B_{{\rm ST}}$.

######  Lemma 1.

For $n\in\mathbb{N}\backslash\\{0\\}$, the following holds in $TL_{1,n}$:

$\begin{array}[]{llcl}{\rm
i.}&t_{1}^{n}\,\sigma_{1}&\widehat{\cong}&t^{n}\,\sigma_{1}\ +\
\underset{i=0}{\overset{n-1}{\sum}}\,(u-u^{-1})\,t^{i}\,{t_{1}}^{n-i}\\\
&&&\\\ {\rm ii.}&t_{1}^{n}\,\sigma_{1}^{-1}&\widehat{\cong}&t^{n}\,\sigma_{1}\
+\ \underset{i=1}{\overset{n-1}{\sum}}\,(u-u^{-1})\,t^{i}\,{t_{1}}^{n-i}\\\
\end{array}$

###### Proof.

We prove Lemma 1(i) by induction on $n\in\mathbb{N}$. The base of induction is
$t_{1}\sigma_{1}\,\widehat{\cong}\,(u-u^{-1})\,t_{1}\ +\ t\,\sigma_{1}$, that
holds. Assume now that the relations hold for $n$. Then, for $n+1$ we have
that:

$\begin{array}[]{lcl}t_{1}^{n+1}\,\sigma_{1}&=&t_{1}^{n}\,\underline{t_{1}\,\sigma_{1}}\
\widehat{\cong}\
(u-u^{-1})\,t_{1}^{n+1}\,+\,t\,\underline{t_{1}^{n}\,\sigma_{1}}\,\overset{ind.}{\underset{step}{=}}\\\
&&\\\
&=&(u-u^{-1})\,t_{1}^{n+1}\,+\,t^{n+1}\,\sigma_{1}\,+\,\underset{i=0}{\overset{n-1}{\sum}}\,(u-u^{-1})\,t^{i+1}\,{t_{1}}^{n-i}\,=\\\
&&\\\
&=&t^{n+1}\,\sigma_{1}\,+\,\underset{i=0}{\overset{n}{\sum}}\,(u-u^{-1})\,t^{i}\,{t_{1}}^{n+1-i}\end{array}$

∎

We now express monomials of the form $t^{m}t_{1}^{n}\in\Sigma$, where
$m,n\in\mathbb{N}$ in sums of elements of the form
$t^{m}{t_{1}^{\prime}}^{n}\in\Sigma^{prime}$. In [12], a method for expressing
monomials in $\Sigma$ to sums of monomials in $\Sigma^{\prime}$ (and vice-
versa) is presented on the level of the generalized Hecke algebra of type B,
$H_{1,n}$. This was achieved with the use of the ordering relation defined on
the basic sets of $H_{1,n}$ (Definition 4). More precisely, in [12] it is
shown that a monomial $\tau$ in $\Sigma$ can be written as a sum of elements
in the set
$\\{{t^{\prime}_{0}}^{k_{0}}{t^{\prime}_{1}}^{k_{1}}\ldots{t^{\prime}_{m}}^{k_{m}}\
|\ k_{i}\ \geq\ k_{i+1},\ k_{i}\in\mathbb{Z}\setminus\\{0\\},\ \forall i\\}$,
such that the term $\tau^{\prime}$, which is obtained from $\tau$ by changing
$t_{i}$ into $t_{i}^{\prime}$ for all $i$, is the highest order term in the
sum. Recall now that the generalized Temperley-Lieb algebra of type B,
$TL_{1,n}$, is a quotient of $H_{1,n}$ over the ideal generated by elements in
Eq. (3) that only involves the $\sigma$’s, and since the only braiding
generator in the monomials $bbm(t^{n})$ is $\sigma_{1}^{\pm 1}$, it follows
that the results presented in [12] are also true in $TL_{1,n}$ for the
monomials $bbm(t^{n})=t_{1}^{n}\,\sigma_{1}^{\pm 1}$. In particular we have
that for $n\in\mathbb{N}\backslash\\{0\\}$, the following holds in $TL_{1,n}$:

$\begin{array}[]{lclclcl}t_{1}^{n}\,\sigma_{1}&\widehat{\cong}&\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{i}\,{t_{1}^{\prime}}^{n-i}&{\rm\
\
and}&t_{1}^{n}\,\sigma_{1}^{-1}&\widehat{\cong}&\underset{i=1}{\overset{n}{\sum}}\,a_{i}^{\prime}\,t^{i}\,{t_{1}^{\prime}}^{n-i}.\end{array}$

where $a_{i}$ and $a_{i}^{\prime}$ coefficients for all $i$.

We now deal with elements of the form $t^{k}\,{t^{\prime}_{1}}^{m}$, such that
$k,m\in\mathbb{N}$ and $k\geq m>0$.

######  Lemma 2.

For $k,m\in\mathbb{N}$ and $k\geq m>0$, the following relation hold in
$TL_{1,n}$:

$t^{k}\,{t^{\prime}_{1}}^{m}\ \widehat{\cong}\ -A^{-2}\,t^{k+m}\ -\
A^{3}\,t^{k+m-2}\ -\ A\,t^{k-1}\,{t_{1}^{\prime}}^{m-1}$

###### Proof.

We consider $t^{k}\,{t^{\prime}_{1}}^{m}$ and we apply the Kauffman bracket
skein relation as demonstrated in Figure 14. Applying the skein relations and
conjugation, we may express these elements in terms of elements in the basis
$B_{\rm ST}$, together with elements of the form
$t^{k-i}\,{t^{\prime}_{1}}^{m-i},\ i>0$, of lower order than
$t^{k}\,{t^{\prime}_{1}}^{m}$. The process is repeated until we are left with
elements in $B_{\rm ST}$.

Figure 14. The proof of Lemma 2.

∎

######  Remark 5.

In Lemma 2, we considered elements of the form $t^{k}\,{t^{\prime}_{1}}^{m}$,
where $k\geq m>0$. If $k<m$, then we consider the closure operation on the
mixed braids and we order the exponents, since the $t_{i}^{\prime}$’s are
conjugates, i.e.
$t^{k}\,{t^{\prime}_{1}}^{m}\,\widehat{=}\,t^{m}\,{t^{\prime}_{1}}^{k}$. Then,
we braid the resulting loops again (see Figure 15).

Figure 15. Conjugation on the $t_{i}^{\prime}$’s.

######  Corollary 1.

For $k,m\in\mathbb{N}$ and $k\geq m>0$, the following relation hold in
$TL_{1,n}$:

$t^{k}\,{t^{\prime}_{1}}^{m}\ \widehat{\cong}\ -A^{-2}\,t^{k+m}\ +\
\underset{i=0}{\overset{\lfloor\frac{k+m-2}{2}\rfloor}{\sum}}\,a_{i}\,t^{2i},$

where $a_{i}$ are coefficients.

###  3.2. The infinite system

By general position arguments, for computing the Kauffman bracket skein module
of $S^{1}\times S^{2}$, we may first consider the effect of negative bbm’s on
elements in $B_{\rm ST}$, obtain a spanning set for $B^{S^{1}\times
S^{2}}_{-}\,:=\,\frac{B_{\rm ST}}{<\,a-bbm_{-}(a)\,>}$ and then study the
effect of positive bbm’s on elements in $B^{S^{1}\times S^{2}}_{-}$. With a
slight abuse of notation, we denote the above as follows:

${\rm KBSM}\left(S^{1}\times S^{2}\right)\,=\,\frac{{\rm
B}_{ST}}{<\,a-bbm_{\pm}(a)\,>}\,=\,\frac{{\rm
B}_{ST}/<a-bbm_{-}(a)>}{<a-bbm_{+}(a)>}\,=\,\frac{B^{S^{1}\times
S^{2}}_{-}}{<a-bbm_{+}(a)>}.$

Hence, we consider elements in $B_{ST}$ and perform negative braid band moves,
obtaining an infinite system of equations, the solution of which corresponds
to a basis for $B^{-}_{S^{1}\times S^{2}}$. We have the following:

######  Theorem 9.

$B^{-}_{S^{1}\times S^{2}}$ is generated by the unknot $t^{0}$ and $t$.

###### Proof.

We consider the infinite system of equations
$V_{\widehat{t^{n}}}\,=\,V_{\widehat{t_{1}^{n}\,\sigma_{1}^{-1}}}$, for
$n\in\mathbb{N}$. We have that:

$\begin{array}[]{lrcrcl}{\rm For}\ n=0:&1\ \overset{bbm_{-}}{\rightarrow}\
\sigma_{1}^{-1}&\Rightarrow&V_{\hat{1}}\,=\,V_{\hat{\sigma_{1}^{-1}}}&\Rightarrow&1\
=\
\left(-\,\frac{1+u^{2}}{u}\right)\,u^{-2}\,tr(\sigma_{1}^{-1})\,\Rightarrow\\\
&&&&&\\\ &&&1&=&-\,\frac{1+u^{2}}{u^{3}}\,\left(z-(u-u^{-1})\right)\
\overset{z=-\,\frac{1}{u(1+u^{2})}}{\Rightarrow}\\\ &&&&&\\\ &&&1&=&1\\\
&&&&&\\\ {\rm For}\ n=1:&t\ \overset{bbm_{-}}{\rightarrow}\
t_{1}\,\sigma_{1}^{-1}&\Rightarrow&V_{\hat{t}}\,=\,V_{\widehat{t_{1}\,\sigma_{1}^{-1}}}&\Rightarrow&s_{1}\
=\
\left(-\,\frac{1+u^{2}}{u}\right)\,u^{2}\,tr(t_{1}\,\sigma_{1}^{-1})\,\Rightarrow\\\
&&&&&\\\ &&&s_{1}&=&-\,-\,\frac{1+u^{2}}{u}\,u^{2}\,z\,s_{1}\ \Rightarrow\\\
&&&&&\\\ &&&s_{1}&=&s_{1}\\\ \end{array}$

Hence, $t^{0}$, which corresponds to the unknot, and $t$ are free in
$B^{-}_{S^{1}\times S^{2}}$.

Consider now $n\in\mathbb{N}$ such that $n\geq 2$. Then:

$t^{n}\ \overset{bbm_{-}}{\longrightarrow}\ t_{1}^{n}\,\sigma_{1}^{-1}\
\widehat{\cong}\ t\,t_{1}^{n-1}\,\sigma_{1}\
\widehat{\underset{{\rm[13]}}{\cong}}\
\underset{i=1}{\overset{n}{\sum}}\,a_{i}\,t^{i}\,{t_{1}^{\prime}}^{n-i}\
\underset{{\rm Cor.}~{}\ref{cor1}}{\widehat{{\cong}}}\ a_{n}\,t^{n}\ +\
\underset{i=0}{\overset{\lfloor\frac{n-2}{2}\rfloor}{\sum}}\,a_{i}\,t^{2i}$

where $a_{i}$ coefficients for all $i$.

Hence, $tr(t_{1}^{n}\,\sigma_{1}^{-1})\,=\,a_{n}\,s_{n}\ +\
\underset{i=1}{\overset{\lfloor\frac{n}{2}\rfloor}{\sum}}\,b_{i}\,s_{n-2i}$,
for all $n\in\mathbb{N}$. Thus, by ignoring the coefficients, we have the
following:

$\begin{array}[]{lcc}V_{\hat{t^{n}}}\,=\,V_{\widehat{t_{1}^{n}\sigma_{1}^{-1}}}&\Rightarrow&s_{n}\,=\,\begin{cases}\underset{i=0}{\overset{(n-2)/2}{\sum}}\,s_{2i}&,\
{\rm for}\ n\ {even}\\\ \underset{i=0}{\overset{(n-3)/2}{\sum}}\,s_{2i+1}&,\
{\rm for}\ n\ {odd}\end{cases}\,\Rightarrow\\\ \end{array}$

$\begin{array}[]{cc}{\rm\underline{n-even}}&{\rm{\underline{n-odd}}}\\\ &\\\
s_{2}\,=\,s_{0}&s_{3}\,=\,s_{1}\\\ &\\\
s_{4}\,=\,s_{0}\,+\,s_{2}&s_{5}\,=\,s_{1}\,+\,s_{3}\\\ &\\\ \vdots&\vdots\\\
&\\\
s_{2k}\,=\,\underset{i=0}{\overset{k-1}{\sum}}\,s_{2i}&s_{2k+1}\,=\,\underset{i=0}{\overset{k-1}{\sum}}\,s_{2i+1}\\\
&\\\ \Downarrow&\Downarrow\\\ s_{2k}\,\sim\,a\cdot s_{0}&\
s_{2k+1}\,\sim\,b\cdot s_{1},\end{array}$

where $a,b$ coefficients.

Recall now that $tr(t)=s_{1}$ and $tr(t^{0})=s_{0}$. The result follows. ∎

######  Corollary 2.

The set $\\{t^{0},\,t\\}$ forms a basis for $B^{-}_{S^{1}\times S^{2}}$.

We now consider the effect of positive braid band moves on the elements
$t^{0}$ and $t$. We have the following result:

######  Theorem 10.

The free part of KBSM$\left(S^{1}\times S^{2}\right)$ is generated by the
unknot and the torsion part is generated by $t$.

###### Proof.

We consider the elements in the basis of $B^{-}_{S^{1}\times S^{2}}$ and we
study the effect of a positive braid band move on them. We have that:

$\begin{array}[]{lrcrcl}{\rm For}\ n=0:&1\ \overset{bbm_{+}}{\rightarrow}\
\sigma_{1}&\Rightarrow&V_{\hat{1}}\,=\,V_{\hat{\sigma_{1}}}&\Rightarrow&1\ =\
\left(-\,\frac{1+u^{2}}{u}\right)\,u^{-2}\,tr(\sigma_{1})\,\Rightarrow\\\
&&&&&\\\ &&&1&=&-\,(1+u^{2})\,u\,z\
\overset{z=-\,\frac{1}{u(1+u^{2})}}{\Rightarrow}\\\ &&&&&\\\
&&&1&=&1\qquad{\rm(free\ part)}\\\ &&&&&\\\ {\rm For}\ n=1:&t\
\overset{bbm_{+}}{\rightarrow}\
t_{1}\,\sigma_{1}&\Rightarrow&V_{\hat{t}}\,=\,V_{\widehat{t_{1}\,\sigma_{1}}}&\Rightarrow&s_{1}\
=\ \left(-\,\frac{1+u^{2}}{u}\right)\,u^{6}\,tr(t_{1}\,\sigma_{1})\end{array}$

We evaluate $tr(t_{1}\,\sigma_{1})$:

$\begin{array}[]{lclcl}tr(t_{1}\,\sigma_{1})&=&(u-u^{-1})\,tr(t_{1})\,+\,tr(\sigma_{1}\,t)&=&(u-u^{-1})^{2}\,tr(t\,\sigma_{1})\,+\,(u-u^{-1})\,tr(t)\,+\,z\,s_{1}\
=\\\ &&&&\\\ &&&=&(u^{2}-1+u^{-2})\,z\,s_{1}\ +\ (u-u^{-1})\,s_{1}\end{array}$

Substituting $tr(t_{1}\,\sigma_{1})$ in the equation above we obtain the
following:

(8) $V_{\hat{t}}\,=\,V_{\widehat{t_{1}\,\sigma_{1}}}\ \Leftrightarrow\
(1-u^{6})\,(1-u^{2})\,s_{1}\ =\ 0.$

From Equation 8 and Theorem 9 we have that $s_{1}=tr(t)$ generates the torsion
part of KBSM($S^{1}\times S^{2}$). The free part of KBSM($S^{1}\times S^{2}$)
is generated by the unknot (or the empty knot). ∎

Although we don’t obtain a closed formula for the torsion part of
KBSM($S^{1}\times S^{2}$), Theorem 10 implies the following:

######  Corollary 3.

If $\pi$ denotes the natural projection

$\pi:{\rm KBSM}\left(S^{1}\times S^{2}\right)\,\rightarrow\,{\rm
KBSM}\left(S^{1}\times S^{2}\right)/Tor,$

then:

$\pi(t^{2n+1})\,=\,0,\,\forall\,n\in\mathbb{N}\quad{\rm and}\quad{\rm
KBSM}\left(S^{1}\times S^{2}\right)/Tor\,=\,\mathbb{Z}[A^{\pm 1}].$

##  4\. A diagrammatic approach via braids

In this section we present the diagrammatic approach for computing
KBSM($S^{1}\times S^{2}$) via braids. Note that during the process that we
will describe below, the braids “lose” their natural top-to-bottom
orientation, giving rise to what we call unoriented braids. Unoriented braids
were defined in [1] as standard braids by ignoring the natural top-to-bottom
orientation (for an illustration see Figure 16). These tools seem promising in
computing Kauffman bracket skein modules of arbitrary $3$-manifolds (see for
example [3, 5]).

Figure 16. Unoriented braids.

In [11] it is shown that in order to describe isotopy for knots and links in a
c.c.o. $3$-manifold, it suffices to consider only one type of band moves
(recall the discussion in § 2). In the algebraic approach we only considered
the type $\beta$-band moves, that correspond to the braid band moves, and in
the diagrammatic approach we will be using the $\alpha$-type band moves
(recall Figure 7). Our starting point again is the following:

(9) ${\rm KBSM}(S^{1}\times S^{2})\,=\,{\rm KBSM(ST)}\ /\,<\,\alpha-{\rm type\
band\ moves}\,>\ =\ B_{\rm ST}\ /\,<\,\alpha-{\rm type\ band\ moves}\,>$

, that is, we study the effect of the $\alpha$-type band moves on elements in
the $B_{\rm ST}$ basis of KBSM(ST). We denote the effect of $\alpha$-type band
moves on $t^{n}\in B_{\rm ST}$ as $bm(t^{n})=A^{6}x_{n}$, where $x_{n}$ is
illustrated in Figure 17. Obviously, $x_{1}\,=\,\hat{t}$.

Figure 17. Performing band moves on elements in $B_{{\rm ST}}$.

We now return to the infinite system of equations (9) and we have the
following:

(10) $t\ \overset{{\rm band}}{\underset{{\rm move}}{\longrightarrow}}\
A^{6}\,x_{1}\ \Leftrightarrow\ (1-A^{6})\,t\ =\ 0.$

Hence, $t$ produces torsion in KBSM($S^{1}\times S^{2}$).

######  Notation 2.

We denote by $\hat{t}$ the closure of the looping generator $t$ and by
$\widehat{t^{m}}$ the closure of $m$-looping generators in ST. Moreover,
$x_{n}\,\widehat{t^{m}}$, $n,m\in\mathbb{N}$, denotes $x_{n}$ followed by $m$
copies of $\hat{t}$. For an illustration see Figure 18.

Figure 18. The elements $x_{n}\,\widehat{t^{m}}$ for $n,m\in\mathbb{N}$.

In order to clarify the steps needed toward the computation of KBSM(ST) using
the diagrammatic method via braids, we also demonstrate how we obtain the
equation for $t^{2}$. After the performance of the band move on $t^{2}$, we
apply the Kauffman bracket skein relation on the resulting $A^{6}x_{2}$ and we
write $x_{2}$ in terms of $\widehat{t}$’s. Applying the skein relation then
again to the $\widehat{t}$’s, we express them in terms of the $t_{i}$’s,
obtaining the equation of the infinite system. For an illustration see Figure
19. Indeed, we have that:

$\begin{array}[]{rcccl}t^{2}&\rightarrow&bm(t^{2})\,=\,A^{6}\,x_{2}&\widehat{\cong}&-A^{10}\,\widehat{t}^{2}\,-A^{8}\\\
&&&&\\\ {\widehat{t}}^{2}&\overset{\rm
braiding}{\rightarrow}&tt_{1}^{\prime}&\widehat{\cong}&-A^{-2}\,t^{2}\,-\,A^{2}\end{array}$

Hence, we obtain the following equation:

$(1-A^{8})\,t^{2}\ =\ -A^{8}\,(1-A^{4}).$

Figure 19. Expressing $x_{2}$ in terms of elements in $B_{{\rm ST}}$.

We observe that in order to obtain the equations of the infinite system, a
recursive formula for $x_{n}$ is needed. We have the following:

######  Lemma 3.

For $n\in\mathbb{N}$ such that $n\geq 2$, the following relations hold in
KBSM($S^{1}\times S^{2}$):

$x_{n}\ \sim\ -A^{8}\,x_{n-2}\ -\ A^{4}\,x_{n-1}\,\widehat{t}.$

Figure 20. The proof of Lemma 3.

Note now that $\widehat{t}$ won’t affect the process illustrated in Figure 20
for $x_{n-1}\,\widehat{t}$. Indeed, we have that

$x_{n-1}\,\widehat{t}\,\sim\,-A^{8}\,x_{n-3}\widehat{t}\ -\
A^{4}\,x_{n-2}\,\left(\,\widehat{t}\,\right)^{2}.$

Moreover, for $n=2$ we have that
$x_{2}\,\sim\,-A^{4}\,\widehat{t}^{2}\,-A^{2}$.

We now present an ordering relation for elements in the set
$D\,:=\,\\{x_{n}\,\widehat{t}\,^{m}\\}_{n,m\in\mathbb{N}}$.

######  Definition 5.

Let $\alpha\,=\,x_{n}\,\widehat{t}\,^{m}$ and
$\beta\,=\,x_{k}\,\widehat{t}\,^{l}$. Then:

  * i.

If $n+m<k+l$, then $\alpha<\beta$.

  * ii.

If $n+m=k+l$, then:

    * a.

If $n<k$, then $\alpha<\beta$.

    * b.

If $n=k$, then $\alpha=\beta$.

######  Proposition 1.

The set $D\,:=\,\\{x_{n}\,\widehat{t}\,^{m}\\}_{n,m\in\mathbb{N}}$ equipped
with the ordering relation given in Definition 5 is a totally ordered and a
well-ordered set.

###### Proof.

In order to show that the set $D$ is a totally ordered set when equipped with
the ordering given in Definition 5, we need to show that the ordering relation
is antisymmetric, transitive and total. We only show that the ordering
relation is transitive. Antisymmetric property follows similarly and totality
follows from Definition 5, since all possible cases have been considered.

Let $\alpha\,=\,x_{n}\,\widehat{t}\,^{m},\
\beta\,=\,x_{k}\,\widehat{t}\,^{l},\ \gamma\,=\,x_{p}\,\widehat{t}\,^{q}$,
such that $\alpha<\beta$ and $\beta<\gamma$. We prove that $\alpha<\gamma$.
Since $\beta<\gamma$, we have that:

  * (a.)

either $n+m<k+l$, and since $\beta<\gamma$, we have that $k+l\leq
p+q\,\Rightarrow\,n+m<p+q\,\Rightarrow\,\alpha<\gamma$,

  * (b.)

or $n+m=k+l$ such that $n<k$. Then, since $\beta<\gamma$, we have that:

    * (i.)

either $k+l<p+q$, same as in case (a.),

    * (ii.)

or $k+l=p+q$ such that $k<p$. Then, $n<k<p\,\Rightarrow\,\alpha<\gamma$.

We conclude that the ordering relation is transitive. The minimum element in
$D$ is $x_{0}\widehat{t}\,^{0}$, i.e. the unknot, and hence $D$ is a well-
ordered set. ∎

The next step is to convert the $x_{n}$’s to elements in the $B_{\rm ST}$
basis of KBSM(ST) for all $n\in\mathbb{N}$. For that, we need the following
lemmas:

######  Lemma 4.

For $n\in\mathbb{N}$ such that $n\geq 2$, the following relations hold in
KBSM($S^{1}\times S^{2}$):

(11)
$x_{n}\,=\,(-1)^{n-1}\,A^{4n-4}\,\widehat{t}\,^{n}\,+\,\begin{cases}\underset{i=0}{\overset{\frac{n-2}{2}}{\sum}}\,a_{i}\,\widehat{t}\,^{2i}&,\
{\rm for}\ n\ {\rm even}\\\ &\\\
\underset{i=0}{\overset{\lfloor\frac{n-2}{2}\rfloor}{\sum}}\,b_{i}\,\widehat{t}\,^{2i+1}&,\
{\rm for}\ n\ {\rm odd}\end{cases}$

###### Proof.

We prove Lemma 4 by strong induction on $n$. The case $n=2$ is the base of
induction and we have that
$x_{2}\,\widehat{\cong}\,-A^{4}\,\widehat{t}\,^{2}\,-A^{2}$. Hence, relation
(11) holds for $n=2$. Assume that relations (11) hold for all
$n\in\mathbb{N},\,n>2$ up to $k\in\mathbb{N}$. Then, for $k+1$ we have:

$\begin{array}[]{lclc}x_{k+1}&\overset{L.~{}\ref{l3}}{=}&-A^{8}\,\underline{x_{k-1}}\,-\,A^{4}\,\underline{x_{k}}\,\widehat{t}&\overset{ind.}{\underset{step}{=}}\\\
&&&\\\
&=&-A^{8}\,\left((-1)^{k-1}\,A^{4(k-1)-4}\,\widehat{t}\,^{k-1}\,+\,\sum\,{\rm
l.o.t.}\right)&+\\\ &&&\\\
&+&-A^{4}\,\left((-1)^{k-1}\,A^{-4k+4}\,\widehat{t}\,^{k+1}\,+\,\sum\,{\rm
l.o.t.}\right)&=\\\ &&&\\\
&=&(-1)^{k}\,A^{-4k}\,\widehat{t}\,^{k-1}\,+\,(-1)^{k+2}\,A^{-4k}\,\widehat{t}\,^{k+1}\,+\,\sum\,{\rm
l.o.t.}&=\\\ &&&\\\
&=&(-1)^{k+2}\,A^{-4(k+1)+4}\,\widehat{t}\,^{k+1}\,+\,\sum\,{\rm l.o.t.}\\\
\end{array}$

∎

Note that $\widehat{t^{k}}$ corresponds to the monomial $tt_{1}^{\prime}\ldots
t_{k-1}^{\prime}\,=\,\tau^{\prime}_{0,k-1}$, namely, a monomial in the
$\mathcal{B}^{\prime}_{{\rm ST}}$ basis of KBSM(ST). We now convert these
monomials to elements in the $B_{{\rm ST}}$ basis. We also note that this is
done in [1, § 3] via the ordering relation of Definition 4.

######  Lemma 5.

The following relations hold in KBSM(ST), and hence in KBSM($S^{1}\times
S^{2}$), for $k,m\in\mathbb{N}$ such that $k,m\geq 2$:

$\beta\,{t_{i}^{\prime}}^{k}\,{t_{i+1}^{\prime}}^{m}\,\widehat{\cong}\,-A^{-2}\,\beta\,{t_{i}^{\prime}}^{k+m}\,-\,A^{3}\,\beta\,{t_{i}^{\prime}}^{k+m-2}\,-\,A\,\beta\,{t_{i}^{\prime}}^{k-1}\,{t_{i+1}^{\prime}}^{m-1}$

where $\beta$ is a monomial of $t_{i}^{\prime}$’s with index at most $i-1$.

###### Proof.

The proof of Lemma 5 is an immediate result of Lemma 2 (recall also Figure
14). ∎

Lemma 5 implies the following:

######  Proposition 2.

For $n\in\mathbb{N}$, such that $n>0$, the following relations hold:

$\widehat{t}\,^{n}\ \widehat{\underset{{\rm skein}}{\cong}}\
\left(-A^{-2}\right)^{n}\,t^{n+1}\,+\,\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{i},$

where the $a_{i}$’s are coefficients.

###### Proof.

Consider the monomial $\widehat{t}\,^{n}\,:=\,tt^{\prime}_{1}\ldots
t^{\prime}_{n-1}$ and apply Lemma 5 on each pair of loop generators with
consecutive indices. Each time Lemma 5 is applied, the coefficient $-A^{-2}$
will appear on the highest order term of the resulting sum. In total, there
are $n-1$ pairs of $t_{i}^{\prime}$’s where Lemma 5 is to applied. The result
follows. ∎

We are now ready to present the main theorem of this section.

######  Theorem 11.

The following relations hold in KBSM($S^{1}\times S^{2}$):

$\left(1-A^{2n+4}\right)\,t^{n}\,=\,\begin{cases}\underset{i=0}{\overset{(n-2)/2}{\sum}}\,a_{i}\,t^{2i}&,\
{\rm for}\ n\ {\rm even}\\\ &\\\
\underset{i=0}{\overset{(n-3)/2}{\sum}}\,b_{i}\,t^{2i+1}&,\ {\rm for}\ n\ {\rm
odd}\end{cases}$

###### Proof.

$\begin{array}[]{cclclc}t^{n}&\overset{bm}{\rightarrow}&A^{6}\,\underline{x_{n}}&\overset{L.~{}\ref{lemxn}}{=}&A^{6}\,\left[(-1)^{n-1}\,A^{4n-4}\,\underline{\widehat{t}\,^{n}}\,+\,\sum\,{\rm
l.o.t.}\right]&{\widehat{\underset{{\rm Prop.~{}\ref{prop2}}}{\cong}}}\\\
&&&&&\\\ &&&\widehat{\underset{{\rm
skein}}{\cong}}&(-1)^{n-1}\,A^{4n+2}\,\left(-A^{-2}\right)^{n-1}\,\left(t^{n}\,+\,\underset{i=0}{\overset{n-2}{\sum}}\,a_{i}\,t^{i}\right)\,+\,\sum\,{\rm
l.o.t.}&=\\\ &&&&&\\\ &&&=&(-1)^{2n-2}\,A^{4n+2-2n+2}\,t^{n}\,+\,+\,\sum\,{\rm
l.o.t.}&=\\\ &&&&&\\\ &&&=&A^{2n+4}\,t^{n}\,+\,+\,\sum\,{\rm l.o.t.}&\\\
\end{array}$

where “l.o.t.” stands for lower ordered terms than $t^{n}$. ∎

Considering now the result of Theorem 11 on the infinite system of equations
(9), a solution of which corresponds to computing KBSM($S^{1}\times S^{2}$),
we have that the elements of the form $t^{2n+1}$ for $n\in\mathbb{N}$,
correspond to the torsion part of KBSM($S^{1}\times S^{2}$), since these
elements can be written as $a\cdot t$ and $t$ produces torsion in $S^{1}\times
S^{2}$ (recall relation (10)). Moreover, we have that the free part of
KBSM($S^{1}\times S^{2}$) is generated by the unknot. In other words, we have
shown that:

######  Theorem 12.

${\rm KBSM}\left(S^{1}\times S^{2}\right)\ =\ \mathbb{Z}[A^{\pm
1}]\,\oplus\,\underset{i=0}{\overset{\infty}{\bigoplus}}\,\frac{\mathbb{Z}[A^{\pm
1]}}{(1-A^{2i+4})}$

Our results agree with that on [20], where KBSM($S^{1}\times S^{2}$) is
computed with the use of an appropriate basis of Chebyshev polynomials
presented in [25].

##  5\. Conclusions

In this paper we compute the Kauffman brakcet skein module of $S^{1}\times
S^{2}$ using two different methods: the algebraic method based on the
generalized Temperley-Lieb algebra of type B, and the diagrammatic method via
braids. The algebraic approach was first implemented for the computation of
HOMFLYPT skein modules of 3-manifolds via the generalized Hecke algebras of
type B (see for example [13, 14, 16, 4, 15]). It was then extended for the
computation of Kauffman bracket skein modules of $3$-manifolds via the
generalized Temperley-Lieb algebra of type B (see [1]). The diagrammatic
method for computing Kauffman bracket skein modules has been successfully
implemented for the case of the handlebody of genus two ([3]), for the
complement of $(2,2p+1)$\- torus knots ([5]) and for the case of the lens
spaces $L(p,q)$ ([1]). It is also worth mentioning that these techniques have
been applied toward the computation of various skein modules of different
families of ‘knotted objects’ in (various) 3-manifolds. In particular, in [8]
the algebraic background toward the computation of the HOMFLYPT skein module
of the solid torus for singular links is presented (see also [10]) and in [9]
the notion of skein modules is extended for knotoids. Finally, in [7] the
algebraic set up toward the computation of skein modules of tied links in
various $3$-manifolds is presented (see also [6]).

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