# The Kauffman bracket skein module of the lens spaces via unoriented braids

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

In this paper we develop a braid theoretic approach for computing the Kauffman
bracket skein module of the lens spaces $L(p,q)$, KBSM($L(p,q)$), for $q\neq
0$. For doing this, we introduce a new concept, that of an unoriented braid.
Unoriented braids are obtained from standard braids by ignoring the natural
top-to-bottom orientation of the strands. We first define the generalized
Temperley-Lieb algebra of type B, $TL_{1,n}$, which is related to the knot
theory of the solid torus ST, and we obtain the universal Kauffman bracket
type invariant, $V$, for knots and links in ST, via a unique Markov trace
constructed on $TL_{1,n}$. The universal invariant $V$ is equivalent to the
KBSM(ST). For passing now to the KBSM($L(p,q)$), we impose on $V$ relations
coming from the band moves (or slide moves), that is, moves that reflect
isotopy in $L(p,q)$ but not in ST, and which reflect the surgery description
of $L(p,q)$, obtaining thus, an infinite system of equations. By construction,
solving this infinite system of equations is equivalent to computing
KBSM($L(p,q)$). We first present the solution for the case $q=1$, which
corresponds to obtaining a new basis, $\mathcal{B}_{p}$, for KBSM($L(p,1)$)
with $(\lfloor p/2\rfloor+1)$ elements. We note that the basis
$\mathcal{B}_{p}$ is different from the one obtained by Hoste & Przytycki. For
dealing with the complexity of the infinite system for the case $q>1$, we
first show how the new basis $\mathcal{B}_{p}$ of KBSM($L(p,1)$) can be
obtained using a diagrammatic approach based on unoriented braids, and we
finally extend our result to the case $q>1$. The advantage of the braid
theoretic approach that we propose for computing skein modules of c.c.o.
3-manifolds, is that the use of braids provides more control on the isotopies
of knots and links in the manifolds, and much of the diagrammatic complexity
is absorbed into the proofs of the algebraic statements.

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

###### Key words and phrases:

skein module, Kauffman bracket, unoriented braids, solid torus, lens spaces,
mixed links, mixed braids, braid group of type B, generalized Hecke algebra of
type B, generalized Temperley-Lieb algebra of type B

###### Contents

  1. 0 Introduction
  2. 1 Preliminaries
  3. 2 Knot algebras and invariants of knots in ST
  4. 3 The Kauffman bracket skein module of ST via braids
  5. 4 A basis for the Kauffman bracket skein module of $L(p,1)$ via braids
  6. 5 The diagrammatic approach for computing KBSM($L(p,q)$)
  7. 6 Conclusions

##  0\. Introduction

Skein modules were introduced independently by Przytycki [31] and Turaev [32]
as generalizations of knot polynomials in $S^{3}$ to knot polynomials in
arbitrary 3-manifolds. They are quotients of free modules over isotopy classes
of links in 3-manifolds by properly chosen local (skein) relations. A skein
module of a 3-manifold yields all possible isotopy invariants of knot which
satisfy a particular skein relation. Skein modules 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, are called Kauffman bracket skein modules (KBSM).

For example, the Kauffman bracket skein module of $S^{3}$, KBSM($S^{3}$), is
freely generated by the unknot and it is equivalent to the Kauffman bracket
for framed links in $S^{3}$ up to regular isotopy. It’s ambient isotopy
counterpart is the well-known Jones polynomial. Recall that 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 [25] and
references therein). It is worth pointing out the importance of the Temperley-
Lieb algebras. Through the pioneering work of V.F.R. Jones ([25, 26]), these
algebras related knot theory to statistical mechanics, topological quantum
field theories and the construction of quantum invariants for 3-manifolds
(works of Witten, Reshetikhin-Turaev, Lickorish, etc).

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$}}.$

Skein modules of $3$-manifolds have become very important algebraic tools in
the study of $3$-manifolds, since their properties renders topological
information about the $3$-manifolds. In this paper we are interested in the
Kauffman bracket skein module of the lens spaces $L(p,q)$, for $p\neq 0$ using
braid theoretical tools. We consider the lens spaces $L(p,q)$ obtained from
$S^{3}$ by rational surgery along the unknot with coefficient $p/q$. Surgery
along the unknot is realized by considering the complementary solid torus and
attaching to it a solid torus according to a $(p,q)$-homeomorphism on the
boundary. Hence, the Kauffman bracket skein module of the solid torus,
KBSM(ST), is our starting point. KBSM(ST) is essential in the study of
Kauffman bracket skein modules of arbitrary c.c.o. $3$-manifolds, since every
c.c.o. $3$-manifold can be obtained by surgery along a framed link in $S^{3}$
with unknotted components. The family of the lens spaces, $L(p,q)$, comprises
the simplest example, since, as mentioned before, they are obtained by
rational surgery on the unknot. For a survey on skein modules see [31].

From the above it is clear that the knot theory of ST is of fundamental
importance for studying knot theory in other c.c.o. 3-manifolds. A basis for
KBSM(ST) is presented in [32] using diagrammatic methods (see also [22]). More
precisely:

######  Theorem 1 ([32]).

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 $n$ parallel copies of a longitude of ST and $x^{0}$ is the
affine unknot (see Figure 2).

Figure 2. The Turaev basis of KBSM(ST).

In this paper we develop a braid theoretic approach to the Kauffman bracket
skein module of $L(p,q)$, for $q\neq 0$. Note that the KBSM($L(p,q)$) has been
computed via a diagrammatic approach in [23], where it is shown that:

######  Theorem 2 ([23], Theorem 4).

The Kauffman bracket skein module of the lens spaces $L(p,q)$ for $p\geq 1$ is
freely generated by $\\{x^{i}\\}_{i=0}^{\lfloor p/2\rfloor}$, where $\lfloor
p/2\rfloor$ denotes the integer part of $p/2$.

We propose a new and more efficient method for computing Kauffman bracket
skein modules via the notion of unoriented braids, which we define along the
way. The algebraic techniques developed in [29, 30, 27, 28, 11, 12, 17, 14, 1,
2, 3, 4] are of great importance. More precisely, in [28] it is shown that the
generalized Hecke algebra of type B, ${H}_{1,n}$, is related to the knot
theory of the solid torus and the Artin group of the Coxeter group of type B,
$B_{1,n}$. Moreover, the most generic analogue of the HOMFLYPT polynomial,
$X$, for links in the solid torus ST has been derived from ${H}_{1,n}$ via a
unique Markov trace constructed on it. The invariant $X$ recovers the HOMFLYPT
skein module of ST (see [28, 12, 13]) and as shown in [17, 14], it can be
extended to an invariant for knots in $L(p,1)$ by considering the effect of
the braid band moves, i.e. the analogue of band moves on the braid level (see
[29, 30, 11]) on elements in some basis of the HOMFLYPT skein module of ST. As
shown in [31], this is also true for the case of the Kauffman bracket skein
module of $L(p,q),\,p\geq 1$. That is, in order to compute KBSM($L(p,q)$), we
consider elements in some basis of KBSM(ST) and impose on the universal
invariant for knots in ST, relations coming from the band moves. That is:

(1) ${\rm KBSM}\left(L(p,q)\right)=\frac{{\rm KBSM}({\rm
ST})}{<a-bbm(a)>},\quad{\rm where}\ a\ {\rm basis\ element\ of\ KBSM(ST)}.$

In this paper we give the complete analogue of KBSM($L(p,q)$) via braids. We
first define the generalized Temperley-Lieb algebra of type B, $TL_{1,n}$ as
the quotient of ${H}_{1,n}$ over the ideal generated by expressions of the
form:

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

That is, $TL_{1,n}\,:=\,\frac{H_{1,n}}{<\sigma_{i,i+1}>}$. We then show that
the trace function constructed on $H_{1,n}$ ([28]) factors through $TL_{1,n}$
and, by normalizing it, we obtain the universal Kauffman bracket, $V$, for
knots and links in ST. We then extend this invariant to knots and links in
$L(p,q)$ for $p\geq 1$, by solving the infinite system of equations resulting
from the band moves. Namely, we force:

(3) $V_{\widehat{a}}\ =\ V_{\widehat{bm(a)}},$

for all $a$ in the basis of KBSM(ST), where $bm(a)$ denotes the result of a
band move on $a$.

Equations (3) have particularly simple formulations with the use of a new
basis, $B_{{\rm ST}}$, for the Kauffman bracket skein module of ST, that is
presented first in Section 2 (see Eq. (8) in this paper), in terms of mixed
braids (that is, classical braids with the first strand identically fixed). We
prove that $B_{{\rm ST}}$ is a basic set of KBSM(ST) by relating the braided
form of the Turaev basis, ${B}_{{\rm ST}}^{\prime}$, to ${B}_{{\rm ST}}$ via a
lower triangular matrix with invertible elements in the diagonal. For an
illustration of elements in the basis ${B}_{{\rm ST}}$ see the bottom of
Figure 8. Note that the same basis is presented in [1] via the Tempreley-Lieb
algebra of type B and in [21] via a diagrammatic approach based on arrow
diagrams. We then solve the infinite system of Eq. (3) for the case of
$L(p,1)$ and we arrive at the main result of this paper. In particular, we
have the following result:

######  Theorem 3.

The Kauffman bracket skein module of the lens spaces $L(p,1)$, $p\geq 1$, is
freely generated by elements in the set $\mathcal{B}_{p}$, consisting of
element in the form $\\{t^{i}\\}_{i=0}^{\lfloor p/2\rfloor}$. For an
illustration of elements in $\mathcal{B}_{p}$ see Figure 8.

Note that the basis $\mathcal{B}_{p}$ is different from the basis presented by
Hoste and Przytycki in [23] for KBSM($L(p,1)$). After we establish the
algebraic method for the case $q=1$, we present a diagrammatic method based on
braids and we use it to derive the basis $\mathcal{B}_{p}$ for KBSM($L(p,1)$).
We then extend this diagrammatic method for the case of $L(p,q)$, $p,q\geq 1$,
and in particular, we show that the set $\mathcal{B}_{p}$ forms a basis for
KBSM($L(p,q)$). We finally discuss the algebraic method for the case of
KBSM($L(p,q)$).

The paper is organized as follows: In §1 we recall the setting and the
essential techniques and results from [27, 28, 29, 30, 11]. More precisely, we
present isotopy moves for knots and links in $L(p,q)$ and we then describe
braid equivalence for knots and links in $L(p,q)$. In §2 we present results
from [28] and [18] and we extend these results in order to construct the
universal invariant for knots in ST of the Kauffman bracket type. 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 the algebra. 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
from $H_{1,n}$ over the ideal generated by relations (2), and we find
necessary and sufficient conditions so that the trace $tr$ factors through
$TL_{1,n}$. Using the trace, we define the universal invariant $V$ for knots
in ST of the Kauffman bracket type. In § 3 we present results on KBSM(ST) via
braids and we present a new basis, $B_{\rm ST}$, for KBSM(ST) via an ordering
relation defined in [12]. This basis is also presented in [1] using the
Temperley-Lieb algebra of type B, which, roughly speaking, is related to an
invariant for knots in ST, but not the most generic one that captures
KBSM(ST). Moreover, the results presented for obtaining $B_{\rm ST}$ are
essential for the computation of the Kauffman bracket skein module of the lens
spaces $L(p,1)$, $p\geq 1$, that we compute in § 4 using algebraic techniques.
This is done with the use of the newly introduced concept of unoriented braids
defined in § 3.1. These new objects seem promising for studying knots in
c.c.o. 3-manifolds (and c.c.o. 3-manifolds in general) via the proposed
technique. Finally, in § 5 we present the basis $\mathcal{B}_{p}$ for $L(p,1)$
using a diagrammatic approach based on unoriented braids and in § 5.2 we show
how these results can be generalized for the case $L(p,q),q>1$.

It is worth mentioning that the importance of our 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 [4] for the case of the Kauffman bracket
skein module of the complement of $(2,2p+1)$-torus knots). The main difficulty
of the problem lies in solving the infinite system of equations (3).

Acknowledgments I would like to acknowledge several discussions with Professor
Sofia Lambropoulou, who provided insight and expertise that greatly assisted
this research.

##  1\. Preliminaries

###  1.1. Mixed links and isotopy in $L(p,q)$

In this section we recall results from [29, 30, 11]. We shall consider ST to
be the complement of a solid torus in $S^{3}$. Then, an oriented link $L$ in
ST can be represented by an oriented mixed link in $S^{3}$, that 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 3).

Figure 3.  A mixed link.

The lens spaces $L(p,q)$ can be obtained from $S^{3}$ by surgery on the unknot
with surgery coefficient $p\in\mathbb{Z}$. Surgery along the unknot can be
realized by considering first the complementary solid torus and then attaching
to it a solid torus according to some homeomorphism $h$ on the boundary. For
$L(p,1)$, the homeomorphism $h$ on the boundaries of the solid tori maps the
meridian, $m_{1}$, of one solid torus to a $(p,1)$-curve on the other, that
is, a $(p\cdot l_{2}+1\cdot m_{2})$-curve, where $l_{2}$ denotes the longitude
and $m_{2}$ denotes the meridian of the second solid torus:

$\begin{matrix}h&:&\partial\,{\rm ST}_{1}&\rightarrow&\partial\,{\rm
ST}_{2}\\\ &&m_{1}&\mapsto&p\cdot l_{2}+1\cdot m_{2}\end{matrix}$

Thus, isotopy in $L(p,q)$ can be viewed as isotopy in ST together with the
band moves in $S^{3}$, which reflect the surgery description of the manifold.
In Figure 4, the two types of band moves for the case of $L(p,1)$ are
illustrated, according to the orientation of the component of the knot and of
the surgery curve. In the $\alpha$-type the orientation of the arc is opposite
to the orientation of the surgery curve, but after the performance of the move
their orientations agree. In the $\beta$-type the orientations initially
agree, but disagree after the performance of the move. For the case of
$L(p,q)$ see Figure 17. 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. Thm. 6 [11]) and thus, isotopy between oriented links
in $L(p,q)$ is reflected in $S^{3}$ by means of the following theorem:

######  Theorem 4.

Two oriented links in $L(p,q)$ 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.

Figure 4.  The two types of band moves for $q=1$.

###  1.2. Mixed braids and braid equivalence for knots and links in $L(p,q)$

In this subsection we introduce the notion of mixed braids, the mixed braid
group $B_{1,n}$ and braid equivalence in $L(p,q)$. Note that a mixed braid in
$S^{3}$ is a braid where, without loss of generality, its first strand
represents $\widehat{I}$, the fixed part, and the other strands, $\beta$,
represent the moving part $L$. We shall call the subbraid $\beta$ the moving
part of $I\cup\beta$ (see bottom left hand side of Figure 4). We now recall
the analogue of the Alexander theorem for knots in ST (cf. Thm. 1 [28]):

######  Theorem 5 (The analogue of the Alexander theorem for ST).

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.

In order to translate isotopy for links in $L(p,q)$ into braid equivalence, we
first perform the technique of standard parting introduced in [30] in order to
separate the moving strands from the fixed strand that represents the lens
spaces $L(p,q)$. This can be realized by pulling each pair of corresponding
moving strands to the right and over or under the fixed strand that lies on
their right. Then, we define a braid band move to be a move between mixed
braids, which is a $\alpha$-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 bottom of Figure 5 for $q=1$
and Figure 6 for $q>1$). For now we shall only consider the $\alpha$-type band
moves, since the result of an $\alpha$-type band move remain braided, while
the result of a $\beta$-band move does not. The $\beta$-band moves will become
important later, when we compute KBSM($L(p,q)$) diagrammatically. Note also
that there are two different types of braid band moves; the positive and the
negative braid band move, depending on the kind of the twist the moving strand
gets before sliding along the fixed strand.

Figure 5.  Isotopy in $L(p,1)$ and the two types of braid band moves on mixed
braids. Figure 6. The braid band move for $L(p,q),\,q>1$.

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 7(i).

Let now $\mathcal{L}$ denote the set of oriented knots and links in ST.
Isotopy in $L(p,q)$ is then translated on the level of mixed braids by means
of the following theorem:

######  Theorem 6 (Theorem 5, [30]).

Let $L_{1},L_{2}$ be two oriented links in $L(p,1)$ 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 $L(p,1)$ if and only if
$I\cup\beta_{1}$ is equivalent to $I\cup\beta_{2}$ in $\mathcal{B}$ by the
following moves:

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

where $\alpha_{+}$ is the word $\alpha$ with all indices shifted by +1 and
$\beta_{+}$ is the word $\beta$ with all indices shifted by $q$. 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
or negative braid band move performed on a mixed braid $\beta$ by
$bbm_{\pm}(\beta)$.

Note also that in [30] it was shown that the choice of the position of
connecting the two components after the performance of a bbm is arbitrary.

##  2\. Knot algebras and invariants of knots in ST

In this section we present the most generic invariant, $V$, for knots and
links in ST that captures the Kauffman bracket skein module of ST. Therefore,
$V$ is the most appropriate invariant to be extended to a Kauffman bracket
invariant for knots and links in $L(p,q)$. This invariant is derived via a
unique Markov trace on the generalized Temperley-Lieb algebra of type B,
$TL_{1,n}$. This algebra is defined in § 2.2 and the Markov trace factors
through the unique Markov trace on the generalized Hecke algebra of type B. We
first recall some results on the generalized Hecke algebra of type B,
$H_{1,n}$, since $TL_{1,n}$ is defined as the quotient of $H_{1,n}$ over an
appropriate ideal.

###  2.1. The generalized Hecke algebra of type B

In [28] the most generic analogue of the HOMFLYPT polynomial, $X$, for links
in the solid torus ST has been derived from the generalized Hecke algebras of
type $\rm B$, $H_{1,n}$, via a unique Markov trace constructed on them. This
algebra was defined by Lambropoulou as the quotient of
${\mathbb{C}}\left[q^{\pm 1}\right]B_{1,n}$ over the quadratic relations
${g_{i}^{2}=(q-1)g_{i}+q}$. Namely:

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

In [28] it is also shown that the following sets form linear bases for ${\rm
H}_{1,n}(q)$ ([28, Proposition 1 & Theorem 1]):

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

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 7(ii)) and
$\sigma$ a basic element in the Iwahori-Hecke algebra of type A, ${\rm
H}_{n}(q)$, for example in the form of the elements in the set [25]:

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

for $1\leq i_{1}<\ldots<i_{p}\leq n-1{\rm\;}$. In [28] the bases
$\Sigma^{\prime}_{n}$ are used for constructing a Markov trace on
$\mathcal{H}:=\bigcup_{n=1}^{\infty}\,H_{1,n}$, and using this trace, a
universal HOMFLYPT-type invariant for oriented links in ST is constructed.

Figure 7. The generators of $B_{1,n}$ and the ‘looping’ elements
$t^{\prime}_{i}$ and $t_{i}$.

######  Theorem 7.

[28, Theorem 6 & Definition 1] Given $z,s_{k}$ with $k\in{\mathbb{Z}}$
specified elements in $R={\mathbb{C}}\left[q^{\pm 1}\right]$, 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 [28, 12] the invariant $X$ recovers the HOMFLYPT skein module of
ST. For a survey on the HOMFLYPT skein module of the lens spaces $L(p,1)$ via
braids, the reader is referred to [13, 16].

###  2.2. The generalized Temperley-Lieb algebra of type B

We now introduce the analogue of the (normalized) Kauffman bracket polynomial,
$V$, for links in the solid torus via the generalized Temperley-Lieb algebra
of type B.

######  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 the elements
$\sigma_{i,i+1},\,i\in\mathbb{N}\backslash\\{0\\}$ (recall Eq. 2).

######  Remark 2.

In [18] the analogue of the normalized Kauffman bracket polynomial, $V$, for
links in the solid torus $\rm ST$ has been derived from the Temperley-Lieb
algebra of type B, ${TL}_{n}^{{\rm B}}$. This algebra defined as a quotient of
the Hecke algebra of type B, ${H}_{1,n}(q,Q)$, over the ideal generated by the
$\sigma_{i,i+1}$ elements and:

$\begin{array}[]{lcl}h_{B}&:=&1+u\ \sigma_{1}+v\ t+uv\
(\sigma_{1}t+t\sigma_{1})+u^{2}v\ \sigma_{1}t\sigma_{1}+uv^{2}\
t\sigma_{1}t+(uv)^{2}\ \sigma_{1}t\sigma_{1}t\end{array}$

It is also worth mentioning that in [18] a different presentation for
$H_{1,n}$ is used, that involves the parameters $u,v$ and the quadratic
relations

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

One can switch from one presentation to the other by a taking
$\sigma_{i}=u\sigma_{i}$, $t=vt$ and $q=u^{2}$. We will adapt this setting
from now on.

Since the generalized Temperley-Lieb algebra of type B is a quotient of the
generalized Hecke algebra of type B, we look for necessary and sufficient
conditions so that the Markov trace defined in $H_{1,n}$ factors through to
${TL}_{1,n}$. We have the following result:

######  Theorem 8.

The trace defined in $H_{n}(1,q)$ factors through to $TL_{1,n}$ if and only if
the trace parameters take the following values:

(6) $z=-\frac{1}{u(1+u^{2})}.$

###### Proof.

In order to evaluate the values of $z$ so that the trace function factors
through ${TL}_{1,n}$, we solve the equation $tr(g_{i,i+1})\,=\,0$ and we have:

$\begin{array}[]{rclc}u^{3}\,tr(g_{i}\,g_{i+1}\,g_{i})&=&-1\,-2u\,z\,-2u^{2}\,z^{2}&\Leftrightarrow\\\
&&&\\\ u^{3}\,z\,tr(g_{i}^{2})&=&-1\,-2u\,z\,-2u^{2}\,z^{2}&\Leftrightarrow\\\
&&&\\\
u^{3}\,z^{2}\,(u-u^{-1})\,+\,u^{3}\,z&=&-1\,-2u\,z\,-2u^{2}\,z^{2}&\Leftrightarrow\\\
&&&\\\ (uz\,+\,1)\,(u^{3}z\,+\,uz\,+\,1)&=&0&\Leftrightarrow\\\ &&&\\\
z\,=\,-\,\frac{1}{u}&{\rm or}&z\,=\,-\,\frac{1}{u\,(1+u^{2})}.&\\\
\end{array}$

As explained in [18], only the values in (6) are of topological interest and
the proof is now concluded. ∎

Note now that for $z=-\,\frac{1}{u(1+u^{2})}$, one deduces $\lambda\ =\
u^{4}$. Following [18], we obtain the following result:

######  Theorem 9.

The following invariant is the most generic invariant for knots and links in
ST:

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

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
$\sigma_{i}\mapsto g_{i}$.

##  3\. The Kauffman bracket skein module of ST via braids

In the braid setting, the elements of the Turaev-basis of KBSM(ST) correspond
bijectively to the elements of the following set:

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

The set ${B}^{\prime}_{{\rm ST}}$ forms a basis of KBSM(ST) in terms of braids
(for an illustration see Figureallbases). Note that ${B}^{\prime}_{{\rm ST}}$
is a subset of $\mathcal{H}$ and, in particular, ${B}^{\prime}_{{\rm ST}}$ is
a subset of $\Sigma^{\prime}=\bigcup_{n}\Sigma^{\prime}_{n}$. Note also that
in contrast to elements in $\Sigma^{\prime}$, the elements in
${B}^{\prime}_{{\rm ST}}$ have no gaps in the indices, the exponents are all
equal to one and there are no ‘braiding tails’.

The invariant $V$ defined in Theorem 9 recovers KBSM(ST). Indeed, it gives
distinct values to distinct elements of ${B}^{\prime}_{{\rm ST}}$, since ${\rm
tr}(t{t^{\prime}_{1}}\ldots{t^{\prime}_{n}})=s_{1}^{n}$. Hence, in order to
compute the Kauffman bracket skein module of the lens spaces $L(p,q)$, it
suffices to extend the (most generic) invariant $V$ for knots in $L(p,q)$
following the ideas in [12, 13, 14]. That is, it suffices to solve the
infinite system of Equations (3), $V_{\widehat{\alpha}}\ =\
V_{\widehat{bbm_{\pm}(\alpha)}}$, for all $\alpha$ in the basis of KBSM(ST).
These equations have particularly simple formulations with the use of the new
basis, ${B}_{{\rm ST}}$, for the Kauffman bracket skein module of ST first
presented in [1].

We now present a different basis $B_{{\rm ST}}$ for the Kauffman bracket skein
module of the solid torus, which is crucial toward the computation of ${\rm
KBSM}\left(L(p,1)\right)$ and which is described in Eq. (8) in open braid
form. For an illustration see Figure 8. In particular we have the following:

######  Theorem 10.

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

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

Figure 8. Elements in the different bases of KBSM(ST).

The importance of the new basis $B_{\rm ST}$ of KBSM(ST) lies in the
simplicity of the algebraic expression of a braid band move, which extends the
link isotopy in ST to link isotopy in $L(p,q)$ (recall Theorem 1(iv)). This
was our motivation for establishing this new basis ${B}_{\rm ST}$. Note also
that in [2] the same basis was derived using the Temperley-Lieb algebra of
type B. In this section we present a new and different proof for Theorem 10.
The results presented here will also be used for the computation of
KBSM($L(p,1)$) in § 4. The method for proving Theorem 10 is the following:

  * $\bullet$

we first recall the total ordering defined in [12] for elements in the sets
$\Sigma^{\prime}$ and $\Sigma$,

  * $\bullet$

we then pass from the set $B^{\prime}_{{\rm ST}}$ consisting of monomials of
the form $tt_{1}^{\prime}t_{2}^{\prime}\ldots t_{n}^{\prime},\
n\in\mathbb{N}$, to an augmented set ${B_{{\rm ST}}}^{aug}$, consisting of
monomials of the form $t^{n}$, where $n\in\mathbb{Z}$.

  * $\bullet$

We express elements in ${B_{{\rm ST}}}^{aug}\backslash B_{{\rm ST}}$ to sums
of elements in $B_{{\rm ST}}$.

  * $\bullet$

The two (ordered) sets $B^{\prime}_{\rm ST}$ and $B_{\rm ST}$ are then related
via a lower triangular infinite matrix with invertible elements on the
diagonal, and

  * $\bullet$

we conclude that the set $B_{\rm ST}$ forms a basis of KBSM(ST).

Before we proceed with the proof of Theorem 10, we introduce the notion of
unoriented braids, which is crucial for working with unoriented knots on the
“braid” level.

###  3.1. Unoriented braids

The Kauffman bracket polynomial is defined for unoriented knots, but when
working on the braid level, there is a natural orientation from top to bottom.
Applying the Kauffman bracket skein relation on a crossing from a standard
braid results in merging strands with opposite orientations (see Figure 9).

Figure 9.  The skein relation on the braid level.

For this reason, we introduce the notion of unoriented braids as follows:

######  Definition 3.

An unoriented braid is defined as a classical braid by ignoring the natural
top to bottom orientation.

######  Remark 3.

The concept of unoriented braids is new and seems promising in serving as a
tool for studying knots in c.c.o. 3-manifolds and fore deriving invariants for
3-manifolds in general. For a thorough study of unoriented braids see [15].

The significance of unoriented braids for the Kauffman bracket skein module of
the solid torus (and for Kauffman bracket skein modules in general), is
illustrated in Figure 10, where the looping generators $t$ and $t^{-1}$ are
shown to be equivalent in KBSM(ST) (compare also with Figure 14).

Figure 10. Unoriented braids and KBSM(ST).

From now on, by braids we shall mean unoriented braids unless otherwise
stated.

###  3.2. An ordering relation

We now present an ordering relation defined on $\Sigma_{n}^{\prime}$,
$B^{\prime}_{{\rm ST}}$ and $\Sigma_{n}$. We first introduce the notion of the
index of a word in $B^{\prime}_{{\rm ST}}$, in $\Sigma_{n}^{\prime}$ and in
$\Sigma_{n}$.

######  Definition 4.

The index of a word $\tau$ in $B^{\prime}_{{\rm ST}}$ (or in
$\Sigma_{n}^{\prime}$ or in $\Sigma_{n}$), denoted $ind(\tau)$, is defined to
be the highest index of the $t_{i}^{\prime}$’s (respectively of the
$t_{i}$’s), in $\tau$. Similarly, the index of an element in
$\Sigma_{n}^{\prime}$ or in $\Sigma_{n}$ is defined in the same way by
ignoring possible gaps in the indices of the looping generators and by
ignoring the braiding part in $\textrm{H}_{n}$. Moreover, the index of a
monomial in $\textrm{H}_{n}$ and in $TL_{1,n}$ is equal to $0$.

For example,
$ind({t^{\prime}}^{k_{0}}{t^{\prime}_{1}}^{k_{1}}\ldots{t^{\prime}_{n}}^{k_{n}})\,=\,ind(t^{u_{0}}\ldots
t_{n}^{u_{n}})\,=\,n$.

We now proceed with presenting an ordering relation in the sets $\Sigma$ and
$\Sigma^{\prime}$, which passes to their respective subsets $B_{\rm ST}$ and
$B_{\rm ST}^{\prime}$.

######  Definition 5.

[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
$\beta_{1},\beta_{2}$ are standard braid words. 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.

######  Definition 6.

We define the augmented sets $B_{{\rm ST}}^{aug}$ and ${B^{\prime}_{{\rm
ST}}}^{aug}$ as follows:

$\begin{array}[]{lcl}{B_{{\rm ST}}}^{aug}&:=&\\{t^{n},\,n\in\mathbb{Z}\\},\\\
&&\\\ {B^{\prime}_{{\rm
ST}}}^{aug}&:=&\\{t^{k_{0}}{t^{\prime}_{1}}^{k_{1}}\ldots{t^{\prime}_{m}}^{k_{m}},\,k_{i}\in\mathbb{Z},\,\forall
i\\}.\end{array}$

######  Proposition 1.

The sets ${B_{{\rm ST}}}$, ${B_{{\rm ST}}}^{aug}$, ${B_{{\rm ST}}^{\prime}}$
and ${B_{{\rm ST}}^{\prime}}^{aug}$ equipped with the ordering relation of
Definition 5, are totally ordered sets. Moreover, the sets ${B_{{\rm ST}}}$
and ${B_{{\rm ST}}^{\prime}}$ are well-ordered sets.

###### Proof.

In [12], Proposition 1, it is shown that the sets ${\Sigma_{n}}^{\prime}$ and
${\Sigma_{n}}$ equipped with the ordering relation of Definition 5, are
totally ordered sets. Since ${B_{{\rm ST}}^{\prime}}$ and ${B_{{\rm
ST}}^{\prime}}^{aug}$ are subsets of ${\Sigma_{n}}^{\prime}$, and since
${B_{{\rm ST}}}$ and ${B_{{\rm ST}}}^{aug}$ are subsets of ${\Sigma_{n}}$, the
sets ${B_{{\rm ST}}}$, ${B_{{\rm ST}}}^{aug}$, ${B_{{\rm ST}}^{\prime}}$ and
${B_{{\rm ST}}^{\prime}}^{aug}$ inherit the property of being totally ordered
sets from ${\Sigma_{n}}$ (cor. $\Sigma_{n}^{\prime}$).

Moreover, $t^{0}$ is the minimum element of both ${B_{{\rm ST}}}$ and
${B_{{\rm ST}}^{\prime}}$ and thus, these sets are well-ordered sets. ∎

###  3.3. Useful Lemmata

In this subsection we prove a series of results in order to convert elements
in $B^{\prime}_{\rm ST}$ to elements in $B_{\rm ST}$. We will use the symbol
$\widehat{\cong}$ when conjugation and stabilization moves are performed, and
$\widehat{\underset{{\rm skein}}{\cong}}$ when both conjugation, stabilization
moves and the Kauffman bracket skein relation is performed. Note that in order
to simplify our results, we will omit the coefficients involved. Finally, it
is worth mentioning that the $t_{i}^{\prime}$’s are conjugates, i.e.
$t_{i}^{\prime}\,t_{j}^{\prime}\,\widehat{\cong}\,t_{j}^{\prime}\,t_{i}^{\prime}$
for all $i\neq j$. A general case is illustrated in Figure 11, for
$w\in{B^{\prime}_{\rm ST}}^{aug}$.

Figure 11. Conjugation on the $t_{i}^{\prime}$’s.

In order to simplify the algebraic expressions obtained throughout this
procedure and throughout the paper in general, we first introduce the
following notation:

######  Notation 2.

We set
${\tau^{\prime}_{i,i+m}}^{k_{i,i+m}}:={t^{\prime}}_{i}^{k_{i}}\ldots{t^{\prime}}^{k_{i+m}}_{i+m}\in{B^{\prime}_{{\rm
ST}}}^{aug}$, for $m\in\mathbb{N}$, $k_{j}\neq 0$ for all $j$.

######  Lemma 1.

Let $w\in{B^{\prime}_{\rm ST}}^{aug}$ such that $ind(w)=n-1$. Then, for
$k,m\in\mathbb{Z}$ the following relations hold in KBSM(ST):

$w\,{t_{n}^{\prime}}^{k}\,{t_{n+1}^{\prime}}^{m}\ \widehat{\underset{{\rm
skein}}{\cong}}\ a\cdot w\,{t_{n}^{\prime}}^{k-m}\,+\,b\cdot
w\,{t_{n}^{\prime}}^{k+m},$

where $a,b$ coefficients.

The proof of Lemma 1 is illustrated in Figure 12.

Figure 12. The proof of Lemma 1.

Note that when applying Lemma 1 on an element in $B^{\prime}_{\rm ST}$, we
obtain an element in $B^{\prime}_{\rm ST}$ and an element in ${B^{\prime}_{\rm
ST}}^{aug}$ of lower index than the initial element in $B^{\prime}_{\rm ST}$.
Indeed we have:

${\tau^{\prime}_{0,n}}\ =\
t_{0}\,t^{\prime}_{1}\,\ldots\,t^{\prime}_{n-1}\,t^{\prime}_{n}\,\overset{Lemma~{}\ref{lemst}}{\cong}\
t_{0}\,t^{\prime}_{1}\,\ldots\,t^{\prime}_{n-2}\ +\
t_{0}\,t^{\prime}_{1}\,\ldots\,{t^{\prime}_{n-1}}^{2}\ =\
{\tau^{\prime}_{0,n-2}}\ +\ {\tau^{\prime}_{0,n-2}}\,{t^{\prime}_{n-1}}^{2}.$

We now convert elements in ${B^{\prime}_{\rm ST}}^{aug}$ to sums of elements
in $B_{\rm ST}^{aug}$. We have the following result:

######  Lemma 2.

The following relations hold in KBSM(ST):

${\tau^{\prime}}^{k_{0,n}}_{0,n}\ \widehat{\underset{{\rm skein}}{\cong}}\
\underset{i\leq k}{\sum}\,a_{i}\,t^{i},\\\ $

where $k\,=\,\underset{i=0}{\overset{n}{\sum}}\,k_{i}$, $k_{i}\in\mathbb{N}$
and $a_{i}$ coefficients for all $i$.

###### Proof.

We prove Lemma 2 by strong induction on the order of
${\tau^{\prime}}^{k_{0,n}}_{0,n}$.

The base of the induction is the monomial $tt_{1}^{\prime}\in B^{\prime}_{\rm
ST}$ of index $1$ and it is illustrated in Figure 13.

Figure 13. The base of induction.

Assume now that Lemma 2 holds for all monomials of lower order than
${\tau^{\prime}}^{k_{0,n}}_{0,n}$. Then, we have:

${\tau^{\prime}}^{k_{0,n}}_{0,n}\ =\
{\tau^{\prime}}^{k_{0,n-2}}_{0,n-2}\,\underline{{t^{\prime}_{n-1}}^{k_{n-1}}\,{t_{n}^{\prime}}^{k_{n}}}\
\overset{{\rm Lemma}~{}\ref{lemst}}{\cong}\
{\tau^{\prime}}^{k_{0,n-2}}_{0,n-2}\,{t_{n-1}^{\prime}}^{k_{n-1}-k_{n}}\,+\,{\tau^{\prime}}^{k_{0,n-2}}_{0,n-2}\,{t_{n-1}^{\prime}}^{k_{n-1}+k_{n}}.$

According to the ordering relation, on the right hand side of this equation we
have two monomials of index $n-1<n=ind({\tau^{\prime}}^{k_{0,n}}_{0,n})$.
Thus, these monomials are of lower order than
${\tau^{\prime}}^{k_{0,n}}_{0,n}$. By the induction hypothesis the proof is
concluded. ∎

######  Remark 4.

Figure 12 suggests that Lemma 2 is true for $k_{i}\in\mathbb{Z}$, that is:

(9) ${\tau^{\prime}}^{k_{0,n}}_{0,n}\ \widehat{\underset{{\rm skein}}{\cong}}\
\underset{i\leq k}{\sum}\,a_{i}\,t^{i},$

where $k\,=\,\underset{i=0}{\overset{n}{\sum}}\,|\,k_{i}\,|$,
$k_{i}\in\mathbb{Z}$ and $a_{i}$ coefficients for all $i$.

Applying now Lemma 2 (together with Equations 9 when needed) on elements in
$B^{\prime}_{{\rm ST}}$ leads to the following result:

######  Corollary 1.

Elements in $B^{\prime}_{{\rm ST}}$ can be written as sums of elements in
${B_{{\rm ST}}}^{aug}$. In particular:

(10) $\tau^{\prime}_{0,n}\ \widehat{\underset{{\rm skein}}{\cong}}\
\underset{i\leq n+1}{\sum}\,a_{i}\,t^{i},$

where $a_{i}$ coefficients for all $i$. Equivalently, the set ${B_{{\rm
ST}}}^{aug}$ spans KBSM(ST).

In the following example, we demonstrate in detail all calculations involved
when applying Lemma 2 and Equations 9 to elements in $B^{\prime}_{{\rm ST}}$.

######  Example 1.

$\begin{array}[]{rclclcl}t\,t_{1}^{\prime}&\widehat{\underset{{\rm
skein}}{\cong}}&1\,+\,t^{2}&&&&\\\ &&&&&&\\\
t\,t_{1}^{\prime}\,t_{2}^{\prime}&\widehat{\underset{{\rm
skein}}{\cong}}&t\,{t^{\prime}_{1}}^{2}\,+\,t&\widehat{\underset{{\rm
skein}}{\cong}}&t^{-1}\,+\,t\,+\,t^{3}&&\\\ &&&&&&\\\
t\,t_{1}^{\prime}\,t_{2}^{\prime}\,t_{3}^{\prime}&\widehat{\underset{{\rm
skein}}{\cong}}&t\,t^{\prime}_{1}\,{t^{\prime}_{2}}^{2}\,+\,t\,t_{1}^{\prime}&\widehat{\underset{{\rm
skein}}{\cong}}&t\,{t_{1}^{\prime}}^{3}\,+\,t\,{t_{1}^{\prime}}^{-1}\,+\,t\,t_{1}^{\prime}&\widehat{\underset{{\rm
skein}}{\cong}}&t^{-2}\,+\,1\,+\,t^{2}\,+\,t^{4}\\\ &&&&&&\\\
t\,t_{1}^{\prime}\,t_{2}^{\prime}\,t_{3}^{\prime}\,t_{4}^{\prime}&\widehat{\underset{{\rm
skein}}{\cong}}&t\,t^{\prime}_{1}\,{t^{\prime}_{2}}\,{t^{\prime}_{3}}^{2}\,+\,t\,t_{1}^{\prime}\,{t^{\prime}_{2}}&\widehat{\underset{{\rm
skein}}{\cong}}&t\,t_{1}^{\prime}\,{t_{2}^{\prime}}^{3}\,+\,t\,{t_{1}^{\prime}}\,{t_{2}^{\prime}}^{-1}&+&t\,t_{1}^{\prime}\,t_{2}^{\prime}\\\
&&&&&&\\\ &\widehat{\underset{{\rm
skein}}{\cong}}&t\,{t_{1}^{\prime}}^{4}\,+\,t\,{t_{1}^{\prime}}^{-2}&+&t\,+\,t\,{t_{1}^{\prime}}^{2}\,+\,t\,t_{1}^{\prime}\,t_{2}^{\prime}&\\\
&&&&&&\\\ &\widehat{\underset{{\rm
skein}}{\cong}}&t^{-3}\,+\,t^{-1}&+&t\,+\,t^{3}&+&t^{5}\\\ \end{array}$

Following the calculations in Example 1, one may notice that there are closed
formulas for Equations (10). More precisely, simple calculations lead to the
following result:

######  Lemma 3.

For $n\in\mathbb{N}$, the following relations hold in KBSM(ST):

$\tau_{0,n}^{\prime}\ \widehat{\underset{{\rm
skein}}{\cong}}\begin{cases}\underset{i=0}{\overset{(n+1)/2}{\sum}}\,a_{i}\,t^{2i}\,+\,\underset{i=1}{\overset{(n-1)/2}{\sum}}\,b_{i}\,t^{-2i}&,{\rm
for}\ n\ {\rm odd}\\\ &\\\
\underset{i=0}{\overset{n/2}{\sum}}\,c_{i}\,t^{2i+1}\,+\,\underset{i=1}{\overset{n/2}{\sum}}\,d_{i}\,t^{-2i+1}&,{\rm
for}\ n\ {\rm even}\end{cases},$

where $a_{i},b_{i},c_{i},d_{i}$ are coefficients.

###  3.4. Dealing with negative exponents

We now deal with monomials in ${B_{{\rm ST}}}^{aug}$ with negative exponents,
i.e. elements in ${B_{{\rm ST}}}^{aug}\,\backslash\,B_{{\rm ST}}$, and express
them to sums of elements in $B_{{\rm ST}}$. As mentioned before, due to the
fact that the Kauffman bracket doesn’t take orientation into consideration, we
use unoriented braids. In the figures that follow, we start with standard
braids and we ignore the orientation when a Kauffman bracket skein relation is
performed, resulting in unoriented braids. Note that we will be using the same
notation for standard braids and unoriented braids. In Figure 14 the cases of
$t^{-1}$ and $t^{-2}$ are illustrated and it is important to note that the
resulting monomial $t\,t_{1}^{\prime}$ is an unoriented braid. Finally, it is
worth mentioning that in order to demonstrate the importance of the new
proposed method for computing Kauffman bracket skein modules, we will follow a
more algebraic procedure for converting elements in ${B_{{\rm
ST}}}^{aug}\,\backslash\,B_{{\rm ST}}$ to elements in $B_{{\rm ST}}$ using the
Kauffman bracket skein relation, instead of straightforward using the fact
that $t^{-n}\,\sim\,t^{n}$, for all $n\in\mathbb{N}$ on the unoriented braid
level.

Figure 14. Converting $t^{-1}$ and $t^{-2}$ to elements in $B^{\prime}_{{\rm
ST}}$.

Note that in the case of $t^{-2}$, when we apply the skein relation for the
first time, a closed looping $\hat{t}$ is “isolated”. We then apply again the
skein relation by ignoring $\hat{t}$ (readers may find useful to think of it
as being shrunk). Recall also that $tt_{1}^{\prime}\,\widehat{\underset{{\rm
skein}}{\cong}}\,1+t^{2}$ (recall Figure 13). Thus,
$t^{-2}\,\widehat{\underset{{\rm skein}}{\cong}}\,1\,+\,t^{2}$ in KBSM(ST).

Let us consider now the general case for $n\in\mathbb{N}$. We first perform an
RI move on the top of the looping generator creating a kink. We then apply the
Kauffman bracket skein relation on the resulting crossing and the result is
$t^{-n+2}\,+\,t^{-n+1}\,t_{1}^{\prime}$. We apply the same procedure on the
resulting monomials. Note that in the monomial $t^{-n+1}\,t_{1}^{\prime}$, the
looping $t_{1}^{\prime}$ will not interfere with the above mentioned process
on the closure $\widehat{t^{-n+1}\,t_{1}^{\prime}}$. We continue until all
negative exponents become either zero or one. Obviously, this depends on $n$
being odd or even. Hence, we have proved the following result:

######  Lemma 4.

For $n\in\mathbb{N}$, the following relations hold in KBSM(ST):

$t^{-n}\ \widehat{\underset{{\rm skein}}{\cong}}\
\begin{cases}\underset{i=0}{\overset{\frac{n-1}{2}}{\sum}}\,a_{i}\,\tau_{0,2i}^{\prime}&,\
{\rm for}\ n\ {\rm odd}\\\
\underset{i=0}{\overset{\frac{n}{2}}{\sum}}\,a_{i}^{\prime}\,\tau_{0,2i}^{\prime}&,\
{\rm for}\ n\ {\rm even}\end{cases},$

where $a_{i},a^{\prime}_{i}$ coefficients.

It is crucial now to note that when we apply Lemma 3 on a monomial
$\tau^{\prime}$ in $B_{\rm ST}^{\prime}$, we obtain monomials in $B_{\rm ST}$
and in ${B_{\rm ST}}^{aug}\backslash B_{\rm ST}$. We then apply Lemma 4 on the
resulting monomials in ${B_{\rm ST}}^{aug}\backslash B_{\rm ST}$ and obtain
elements in $B^{\prime}_{\rm ST}$ again, but of lower order than the initial
monomial $\tau^{\prime}$. Thus, if we continue this way, we will eventually
reach at elements in $B_{\rm ST}$. Indeed we have the following result:

######  Proposition 2.

For $n\in\mathbb{N}$, the following hold in KBSM(ST):

$t^{-n}\ \widehat{\underset{{\rm skein}}{\cong}}\
\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{i},$

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

###### Proof.

Let $n$ be an odd natural number. The case that $n$ is even follows similarly.
We have that:

$\begin{array}[]{lclcl}t^{-n}&\overset{{\rm
Lemma}~{}\ref{fl2}}{\sim}&\underset{i=0}{\overset{\frac{n-1}{2}}{\sum}}\,a_{i}\,\tau_{0,2i}^{\prime}&\overset{{\rm
Lemma}~{}\ref{fl1}}{\sim}&\underset{i=0}{\overset{\frac{n-1}{2}}{\sum}}\,a_{i}\,\left(\underset{j=0}{\overset{i}{\sum}}\,a_{i}^{\prime}\,t^{2j+1}\
+\ \underset{j=1}{\overset{i}{\sum}}\,a_{i}^{\prime\prime}\,t^{-2j+1}\right)\
=\\\ &&&&\\\
&&&=&\underset{A}{\left(\underbrace{\underset{i=0}{\overset{\frac{n-1}{2}}{\sum}}\,\underset{j=0}{\overset{i}{\sum}}\,b_{i}\,t^{2j+1}}\right)}\
+\
\underset{B}{\left(\underbrace{\underset{i=0}{\overset{\frac{n-1}{2}}{\sum}}\,\underset{j=1}{\overset{i}{\sum}}\,b_{i}^{\prime}\,t^{-2j+1}}\right)},\\\
\end{array}$

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

We now deal with terms in $B$ that consist of elements in ${B_{{\rm
ST}}}^{aug}\,\backslash\,B_{{\rm ST}}$, since all terms in $A$ are in $B_{{\rm
ST}}$. The highest exponent in the terms in $B$ is $(-n+3)>-n$ and thus, we
may continue applying Lemmata 3 and 4 until we eliminate all negative
exponents. All resulting monomials will eventually belong to $B_{{\rm ST}}$
and the proof is concluded. ∎

######  Remark 5.

It is possible to find a closed formula for $t^{-n}$, $n\in\mathbb{N}$ in
KBSM(ST). More precisely, simple calculations lead to the following result:

$t^{-n}\ \widehat{\underset{{\rm skein}}{\cong}}\
\begin{cases}\underset{i=0}{\overset{n/2}{\sum}}\,a_{i}\,t^{2i}&,\ {\rm for}\
n\ {\rm even}\\\
\underset{i=0}{\overset{\frac{n-1}{2}}{\sum}}\,a^{\prime}_{i}\,t^{2i+1}&,\
{\rm for}\ n\ {\rm odd}\\\ \end{cases},$

where $a_{i},a^{\prime}_{i}$ coefficients, for all $i$.

###  3.5. The new basis of KBSM(ST)

We are now in position to prove the main result of this section. More
precisely:

######  Theorem 11.

The set $B_{\rm ST}$ forms a basis for the Kauffman bracket skein module of
the solid torus.

###### Proof.

Consider $\tau^{\prime}_{0,n}\in B^{\prime}_{{\rm ST}}$ of index $n$. We apply
Corollary 1 and obtain a sum of elements in ${B_{{\rm ST}}}^{aug}$ with
$(n+1)$ being the highest exponent appearing in the sum. More precisely, for
$n$ odd we have:

$\begin{array}[]{lclcl}B^{\prime}_{{\rm
ST}}\,\ni\,\tau^{\prime}_{0,n}&\widehat{\underset{{\rm
skein}}{\cong}}&\underset{\in B_{\rm ST},\,\forall
i}{\underbrace{\underset{i=0}{\overset{\frac{n+1}{2}}{\sum}}\,a_{i}\,t^{2i}}}&+&\underset{\in{B_{\rm
ST}}^{aug}}{\underbrace{\underset{i=1}{\overset{\frac{n-1}{2}}{\sum}}\,\underline{a_{i}\,t^{-2i}}}}\\\
&&&&\\\ &\widehat{\underset{{\rm skein}}{\cong}}&\underset{\in B_{\rm
ST},\,\forall
i}{\underbrace{\underset{i=0}{\overset{\frac{n+1}{2}}{\sum}}\,a_{i}\,t^{2i}}}&+&\underset{\in
B^{\prime}_{\rm
ST}}{\underbrace{\underset{i=1}{\overset{\frac{n-1}{2}}{\sum}}\,\left(\underset{j=0}{\overset{i}{\sum}}\,a_{i}^{\prime}\,\tau^{\prime}_{0,2j}\right)}}\end{array}$

The monomial of highest order in the sum
$\underset{i=1}{\overset{\frac{n-1}{2}}{\sum}}\,\left(\underset{j=0}{\overset{i}{\sum}}\,a_{i}^{\prime}\,\tau^{\prime}_{0,2j}\right)$
is $\tau_{0,n-1}^{\prime}$, which according to the ordering relation of
Definition 5, is of less order than the initial monomial
$\tau_{0,n}^{\prime}$. Hence, if we continue applying the same procedure on
the resulting monomials, we will eventually reach at a sum of elements in
$B_{{\rm ST}}$, with $(n+1)$ being the highest exponent.

Thus, we have proved that any monomial $\tau^{\prime}$ in $B^{\prime}_{{\rm
ST}}\,:\,ind(\tau^{\prime})=n$, can be written as a sum of elements in
$B_{{\rm ST}}$ and in particular, $\tau^{\prime}\,\widehat{\underset{{\rm
skein}}{\cong}}\,\underset{i=0}{\overset{n+1}{\sum}}\,a_{i}\,t^{i}$, for some
coefficients $a_{i}$. Moreover, $t^{n+1}$ is the monomial with the highest
order appearing when converting the monomial $\tau^{\prime}\in
B^{\prime}_{{\rm ST}}$ of index $n$ to elements in $B_{{\rm ST}}$. Hence, the
sets $B^{\prime}_{{\rm ST}}$ and $B_{{\rm ST}}$ are related via an infinite
lower triangular matrix with invertible elements in the diagonal. Thus, the
set $B_{{\rm ST}}$ forms a basis of KBSM(ST). ∎

##  4\. A basis for the Kauffman bracket skein module of $L(p,1)$ via braids

In this section we solve the infinite system of Equations (3) for $q=1$, which
is equivalent to computing the Kauffman bracket skein module of the lens
spaces $L(p,1)$. Recall that this infinite system of equations is obtained
from elements in the basis of KBSM(ST), $B_{\rm ST}$, by performing braid band
moves and then 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 $bbm(t^{n})\,=\,t^{p}\,t_{1}^{n}\,\sigma_{1}^{\pm
1}$ (recall Theorem 6). Recall also that the unknowns in the system are the
$s_{i}$’s, coming from the fourth rule of the trace function in Theorem 7,
that is, $tr(t^{n})\,=\,s_{n}$, for all $n\in\mathbb{Z}$.

###  4.1. The infinite system

In this subsection we present a series of results toward the solution of the
infinite system. Our main task is the evaluation of
$tr(t^{p}\,t_{1}^{n}\,\sigma_{1}^{\pm 1})$ for $n\in\mathbb{N}$. Recall now
that the trace function is defined using the $t_{i}^{\prime}$’s and not the
$t_{i}$’s that we obtain when performing a bbm on elements in $B_{\rm ST}$
(recall the fourth rule of Theorem 7). Thus, our first task is to express the
monomials $t^{p}\,t_{1}^{n}\,\sigma_{1}^{\pm 1}\in\Sigma$ to sums of monomials
in $\Sigma^{\prime}$ (recall Eq. (4)). In [12], a method for expressing
monomials in $\Sigma^{\prime}$ to sums of monomials in $\Sigma$ (and vice-
versa) is presented on the generalized Hecke algebra of type B, $H_{1,n}$,
level. More precisely, it is shown that a monomial $\tau\in\Sigma$ can be
written as a sum of elements in $\Sigma^{\prime}$, such that the homologous
word of $\tau$ in $\Sigma^{\prime}$, i.e. the monomial $\tau$ where all
$t_{i}$’s are replaced by $t_{i}^{\prime}$’s, is the monomial of the highest
order appearing in the resulting sum. All other monomials appearing in the sum
have smaller index, or equal index but different exponents (recall the
ordering relation of Definition 5). Finally, the braiding “tails”, i.e. the
$\sigma_{i}$’s that come after the $t_{i}$’s in monomials in $\Sigma$, are
eliminated in this process.

Since now 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. (2) that
only involves the $\sigma$’s, and since the only braiding generator in the
monomials $bbm(t^{n})$ is $\sigma_{1}$, it follows that the results presented
in [12] are also true in $TL_{1,n}$ for the monomials
$bbm(t^{n})=t^{p}\,t_{1}^{n}\,\sigma_{1}^{\pm 1}$. Hence, we obtain the
following result:

######  Lemma 5.

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

$t^{p}\,t_{1}^{n}\,\sigma_{1}\ \widehat{\cong}\
\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{p+i}\,{t_{1}^{\prime}}^{n-i},$

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

Note that in the monomial of highest order in
$\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{p+i}\,{t_{1}^{\prime}}^{n-i}$ is
the homologous monomial of $t^{p}\,t_{1}^{n}$, $t^{p}\,{t^{\prime}_{1}}^{n}$.
In order now to evaluate
$tr\left(\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{p+i}\,{t_{1}^{\prime}}^{n-i}\right)$,
we first perform Lemma 1 on the monomials in the sum and we have the following
result:

######  Lemma 6.

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

$t^{p}\,{t_{1}^{\prime}}^{n}\ \widehat{\underset{{\rm skein}}{\cong}}\
a_{i}\,t^{p-n}\,+\,b_{i}\,t^{p+n},$

where $a_{i},b_{i}$ are coefficients for all $i$.

Hence, we have that $tr(t^{p}\,t_{1}^{n}\,\sigma_{1}^{\pm 1})\ =\
\underset{i=0}{\overset{n}{\sum}}\,\left(a_{i}^{\prime}\,s_{p+n}\,+\,b_{i}^{\prime}\,s_{p-n+2i}\right)$,
for some coefficients $a_{i}^{\prime}$ and $b_{i}^{\prime}$, since:

$tr(t^{p}\,t_{1}^{n}\,\sigma_{1}^{\pm 1})\ =\
tr\left(\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{p+i}\,{t_{1}^{\prime}}^{n-i}\right)\
=\
tr\left(\underset{i=0}{\overset{n}{\sum}}\,\left(a_{i}^{\prime}\,t^{p+n}\,+\,b_{i}^{\prime}\,t^{p-n+2i}\right)\right).$

######  Remark 6.

Consider now a negative braid band move applied on elements in $B_{\rm ST}$,
i.e. $t^{n}\,\rightarrow\,t^{p}\,t_{1}^{n}\,\sigma_{1}^{-1}$. Observe that
$t^{p}\,t_{1}^{n}\,\sigma_{1}^{-1}\,=\,t^{p}\,t_{1}^{n-1}\,\sigma_{1}\,t\,\widehat{\cong}\
t^{p+1}\,t_{1}^{n-1}\,\sigma_{1}$. Thus, applying Lemmata 5 and 6 on
$t^{p+1}\,t_{1}^{n-1}\,\sigma_{1}$ we obtain:

$t^{p+1}\,t_{1}^{n-1}\,\sigma_{1}\ \widehat{\cong}\
\underset{i=0}{\overset{n-1}{\sum}}\,a_{i}\,t^{p+1+i}\,{t_{1}^{\prime}}^{n-1-i}\
\widehat{\underset{{\rm skein}}{\cong}}\
\underset{i=0}{\overset{n-1}{\sum}}\,\left(b_{i}\,t^{p+n}\,+\,c_{i}\,t^{p-n+2+2i}\right),$

for some coefficients $a_{i},b_{i}$ and $c_{i}$. Evaluating now the trace of
these elements, we have that

$tr(t^{p}\,t_{1}^{n}\,\sigma_{1}^{-1})\ =\
\underset{i=0}{\overset{n-1}{\sum}}\,\left(b_{i}\,s_{p+n}\,+\,c_{i}\,s_{p-n+2+2i}\right).$

Less number of unknowns of the infinite system appear in the equation
$V_{\widehat{t^{n}}}\ =\ V_{\widehat{bbm_{-}(t^{n})}}$, compared to the
equation $V_{\widehat{t^{n}}}\ =\ V_{\widehat{bbm_{+}(t^{n})}}$, and thus, we
may consider only the positive braid band moves for obtaining the equations of
the infinite system for $n\in\mathbb{N}$.

###  4.2. Dealing with indices greater than $p$

We are now in position to prove the first fundamental result toward the
solution of the infinite system. We have the following:

######  Proposition 3.

For $n\in\mathbb{N}$ he following relations hold in KBSM(L(p,1)) for
$n\in\mathbb{N}$:

$s_{p+n}\ =\
\begin{cases}\underset{i=0}{\overset{n/2}{\sum}}\,(c_{i}\,s_{2i}\,+\,d_{i}\,s_{p-2i})&,\,{\rm
for}\ n\ {\rm even}\\\
\underset{i=0}{\overset{(n-1)/2}{\sum}}\,(c^{\prime}_{i}\,s_{2i-1}\,+\,d^{\prime}_{i}\,s_{p-2i-1})&,\,{\rm
for}\ n\ {\rm odd}\end{cases},$

where $c_{i},d_{i},c^{\prime}_{i},d^{\prime}_{i}$ are coefficients.

###### Proof.

The following diagram describes how the equations of the infinite system are
obtained:

$\begin{array}[]{ccccccc}t^{n}&\overset{bbm}{\longrightarrow}&t^{p}t_{1}^{n}\,\sigma_{1}&\hat{\cong}&\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{p+i}\,{t_{1}^{\prime}}^{p-i}&\hat{\underset{{\rm
skein}}{\cong}}&\underset{i=0}{\overset{n}{\sum}}\,\left(t^{p-n+2i}\,+\,t^{p+n}\right)\\\
|&&&&&&|\\\ tr&&&&&&tr\\\ \downarrow&&&&&&\downarrow\\\
s_{n}&&&&&&\underset{i=0}{\overset{n}{\sum}}\,s_{p-n+2i}\\\ |&&&&&&|\\\
V&&&&&&V\\\ \downarrow&&&&&&\downarrow\\\
s_{n}&&&&=&&\left(-\,\frac{1+u^{2}}{u}\right)\,u^{2e}\cdot\underset{i=0}{\overset{n}{\sum}}\,s_{p-n+2i}\\\
\end{array}$

Hence,
$s_{n}\,=\,\left(-\,\frac{1+u^{2}}{u}\right)\,u^{2e}\cdot\underset{i=0}{\overset{n}{\sum}}\,s_{p-n+2i}$,
for all $n\in\mathbb{N}$.

We now consider the following cases:

  * •

For $n=2k,\,k\in\mathbb{N}$, we have:

$\begin{array}[]{lclcccl}{\rm For}\
n=0&:&s_{0}\,=\,s_{p}&\Leftrightarrow&s_{p}&=&s_{0}\,:=\,1\\\ &&&&&&\\\ {\rm
For}\
n=2&:&s_{2}\,=\,s_{p-2}\,+\,s_{p}\,+\,s_{p+2}&\Leftrightarrow&s_{p+2}&=&s_{0}\,+\,s_{2}\,+\,s_{p-2}\\\
&&&&&&\\\ {\rm For}\
n=4&:&s_{4}\,=\,s_{p-4}\,+\,s_{p-2}\,+\,s_{p}\,+\,s_{p+2}\,+\,s_{p+4}&\Leftrightarrow&s_{p+4}&=&s_{0}\,+\,s_{2}\,+\,s_{4}\,+\\\
&&&&&&+\,s_{p-4}\,+\,s_{p-2}\\\ \vdots&&&&&&\\\ {\rm For}\
n=2k&:&&\Leftrightarrow&s_{p+2k}&=&\underset{i=0}{\overset{k}{\sum}}\,s_{2i}\,+\,\underset{i=1}{\overset{k}{\sum}}\,s_{p-2i}\\\
\end{array}$

  * •

For $n=2k+1,\,k\in\mathbb{N}$, we have:

$\begin{array}[]{cclcccl}{\rm For}\
n=1&:&s_{1}\,=\,s_{p-1}\,+\,s_{p+1}&\Leftrightarrow&s_{p+1}&=&s_{1}\,+\,s_{p-1}\\\
&&&&&&\\\ {\rm For}\
n=3&:&s_{3}\,=\,s_{p-3}\,+\,s_{p-1}\,+\,s_{p+1}\,+\,s_{p+3}&\Leftrightarrow&s_{p+3}&=&s_{1}\,+\,s_{3}\,+\,s_{p-3}\,+\\\
&&&&&&+\,s_{p-1}\\\ \vdots&&&&&&\\\ {\rm For}\
n=2k+1&:&&\Leftrightarrow&s_{p+2k+1}&=&\underset{i=0}{\overset{k}{\sum}}\,(s_{2i+1}+s_{p-2i-1})\\\
\end{array}$

The proof is concluded. ∎

An immediate result of Proposition 3 is that every unknown of the infinite
system with index greater than or equal to $p$, can be written as a sum of the
unknowns with index less than $p$. That is:

######  Corollary 2.

For $k\in\mathbb{N}$, the following hold in KBSM($L(p,1)$):

$s_{p+k}\ =\ \underset{i<p}{\sum}\,q_{i}\,s_{i},\quad{\rm where}\
i\in\mathbb{Z}\ {\rm and}\ q_{i}\ {\rm coefficients}.$

######  Remark 7.

Note that since $tr(t^{n})\,=\,s_{n}$, we have proved that the loop elements
$t^{p+i}\in{\rm KBSM(ST)},\,i\in\mathbb{N}$ can be expressed as a sum of
elements in the form $t^{p-j},\,j\in\mathbb{N}$. Roughly speaking, the
exponents of the $t$’s are mod$p$ in KBSM($L(p,1)$).

It is important to observe now that depending on $p$, the exponents on some
monomials of the resulting sum may be negative. We shall deal with negative
indices in the next subsection.

###  4.3. Dealing with negative indices

We now deal with the unknowns of the infinite system that have negative
indices and express them in terms of $s_{i}$’s where $i\in\mathbb{N}$. Recall
that $s_{-n}=tr(t^{-n})$ and that $t^{-n}\,\widehat{\underset{{\rm
skein}}{\cong}}\,\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{i}$ (Proposition
2 ). Thus, we have that:

$s_{-n}\,=\,tr(t^{-n})\,=\,tr\left(\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,t^{i}\right)\,=\,\underset{i=0}{\overset{n}{\sum}}\,a_{i}\,s_{i}.$

If instead of Proposition 2 we use relations from Remark 5, we obtain the
following:

######  Lemma 7.

The following relations hold in KBSM($L(p,1)$) for all $n\in\mathbb{N}$:

$s_{-n}\ =\begin{cases}\underset{i=0}{\overset{n/2}{\sum}}\,s_{2i}&,{\rm for}\
n\ {even}\\\ \underset{i=0}{\overset{(n-1)/2}{\sum}}\,s_{2i+1}&,{\rm for}\ n\
{odd}\end{cases}$

###### Proof.

We have that:

$\begin{array}[]{cccc}t^{-n}&\widehat{\underset{{\rm
skein}}{\cong}}&\underset{i=0}{\overset{n/2}{\sum}}\,a_{i}\,t^{2i},\,{\rm
for}\,n\ {\rm even\ \ \
or}&\underset{i=0}{\overset{(k-1)/2}{\sum}}\,b_{i}\,t^{2i+1},\,{\rm for}\,k\
{\rm odd}\\\ |&&|&|\\\ {\rm V}&&{\rm V}&{\rm V}\\\
\downarrow&&\downarrow&\downarrow\\\
s_{-n}&&\underset{i=0}{\overset{n/2}{\sum}}\,a_{i}\,s_{2i}&\underset{i=0}{\overset{(n-1)/2}{\sum}}\,b_{i}\,s_{2i+1}\\\
\end{array}$

for some coefficients $a_{i},b_{i}$. Omitting the coefficients we have that:

$s_{-2}\,=\,s_{0}\,+\,s_{2},\ s_{-4}\,=\,s_{0}\,+\,s_{2}\,+\,s_{4},\ \ldots,\
s_{-j}\,=\,s_{0}\,+\,s_{2}\,+\,\ldots\,+\,s_{j},\ {\rm for}\ j\ {\rm even\
and}$ $s_{-1}\,=\,s_{1},\ s_{-3}\,=\,s_{1}\,+\,s_{3},\ \ldots,\
s_{-j}\,=\,s_{1}\,+\,s_{3}\,+\,\ldots\,+\,s_{j},\ {\rm for}\ j\ {\rm odd}.$

The result follows. ∎

###  4.4. The solution of the infinite system

We are now in position to present the solution of the infinite system that is
equivalent to computing the Kauffman bracket skein module of the lens spaces
$L(p,1)$. We first generalize Proposition 3 and in particular we show that the
unknowns with indices greater than or equal to $p$ can be written as sums of
unknowns with non-negative indices and also that these indices are $\mod p$.
Indeed, we have the following:

######  Proposition 4.

The following relations hold in KBSM($L(p,1)$) for all $n\in\mathbb{N}$:

$s_{p+n}\ =\ \underset{i=0}{\overset{p}{\sum}}\,a_{i}\,s_{i}.$

###### Proof.

For $n\leq p$ the results follows from Proposition 3. Let now $n>p$ and
consider the case where $n$ is even. The case $n$ being odd follows similarly.

Let $j\in\mathbb{N}\backslash\\{0\\}$ such that $n=p+j$. Omitting the
coefficients, we have:

$\begin{array}[]{lcl}s_{p+n}&\overset{Prop.~{}\ref{th1}}{=}&\underset{i=0}{\overset{n/2}{\sum}}\,(s_{2i}\,+\,s_{p-2i})\
=\\\ &&\\\ &=&\underset{A}{\left[\underbrace{(s_{0}\,+\,s_{p})\ +\
(s_{2}\,+\,s_{p-2})\ +\ \ldots\ +\ (s_{p}\,+\,s_{0})}\right]}\ +\\\ &&\\\
&=&\underset{B}{\left[\underbrace{(s_{p+2}\,+\,s_{-2})\ +\
(s_{p+4}\,+\,s_{-4})\ +\ \ldots\ +\ (s_{p+j}\,+\,s_{-j})}\right]}\ +\\\
\end{array}$

Note now that all indices in terms in $A$ are in $\\{0,1,\ldots,p\\}$ and
thus, we focus on terms in $B$, i.e. terms of the form $s_{-k}$ and $s_{p+k}$
for $k\in\\{1,2,\ldots,j\\}$.

We distinguish the following cases:

  * Case I:

We deal first with terms of the form $s_{-k}$ for $k\in\\{1,2,\ldots,j\\}$. We
apply Lemma 7 on these terms and we obtain sums of unknowns with all indices
being less than or equal to $j$. Thus, if $j<p$ the result follows. If $j>p$,
we repeat the same procedure until we reduce all indices $\mod p$.

  * Case II:

We now deal with terms of the form $s_{p+k}$ for $k\in\\{1,2,\ldots,j\\}$. We
apply Proposition 3 and obtain elements in $s_{i}$’s such that $i$ is either
in $\mathbb{Z}\backslash\mathbb{N}$ (see Case I), or in $\mathbb{N}$ such that
$i<p$. Continuing that way, we eventually obtain a sum of elements of the form
$\underset{i=0}{\overset{p}{\sum}}\,s_{i}$.

∎

We shall now reduce the indices in the unknowns further. Let

$I=\\{0,1,\ldots,\lfloor p/2\rfloor\\},$

where $\lfloor k\rfloor$ denotes the greatest integer less than or equal to
$k$. We have:

######  Theorem 12.

For all $n\in\mathbb{N}\backslash I$ we have that $s_{n}\,=\,\underset{i\in
I}{\sum}\,s_{i}$.

###### Proof.

From Proposition 4, it follows that it suffices to show that
$s_{n}\,=\,\underset{i\in I}{\sum}\,s_{i}$ for $n\in\\{\lfloor
p/2\rfloor+1,\ldots,p\\}$.

Consider elements in ${B_{\rm ST}}^{aug}\backslash B_{\rm ST}$ of the form
$t^{-n}$ for $n\in I$. We perform bbm’s on these elements in order to obtain
equations for the infinite system. Using Lemma 7 and Proposition 3, we may
convert the unknowns with negative indices to sums of unknowns with indices in
$I$ and thus, we conclude that the unknowns with indices $I$ suffice to
generate all unknowns $s_{k}$, where $k\notin I$. Equivalently, the elements
$t^{k}$, where $k\notin I$, can be expressed as sums of elements of the form
$t^{n}$, where $n\in I$. More precisely, we have:

$\begin{array}[]{clclcl}\bullet&t^{0}\ \overset{bbm}{\rightarrow}\
t^{p}\,\sigma_{1}^{\pm
1}&\overset{V}{\Rightarrow}&1:=s_{0}\,=\,s_{p}&\Rightarrow&s_{p}\ =\ s_{0}\\\
&&&&&\\\ \bullet&t^{-1}\ \overset{bbm}{\rightarrow}\
t^{p}\,t_{1}^{-1}\sigma_{1}^{\pm
1}&\overset{V}{\Rightarrow}&s_{-1}\,=\,s_{p-1}&\overset{Lemma~{}\ref{nein}}{\underset{s_{-1}=s_{1}}{\Rightarrow}}&s_{p-1}\
=\ s_{1}\\\ &&&&&\\\ \bullet&t^{-2}\ \overset{bbm}{\rightarrow}\
t^{p}\,t_{1}^{-2}\sigma_{1}^{\pm
1}&\overset{V}{\Rightarrow}&s_{-2}\,=\,s_{p-2}+s_{p}+s_{p+2}&\overset{Lemma~{}\ref{nein}\,\&\,Prop.~{}\ref{fp1}}{\underset{\left\\{\begin{matrix}s_{-2}&=&s_{0}+s_{2}\\\
s_{p+2}&=&\underset{i=0}{\overset{p}{\sum}}\,s_{i}\end{matrix}\right\\}}{\Rightarrow}}&s_{p-2}\
=\ \underset{i=0}{\overset{p-3}{\sum}}\,s_{i}\\\ &&&&&\\\
\vdots&&\vdots&&\vdots&\\\ \end{array}$

$\begin{array}[]{llcl}\bullet&t^{-\lfloor p/2\rfloor+1}\
\overset{bbm}{\rightarrow}\ t^{p}\,t_{1}^{-\lfloor
p/2\rfloor+1}\sigma_{1}^{\pm 1}&\overset{V}{\Rightarrow}&s_{-\lfloor
p/2\rfloor+1}\,=\,s_{p-\lfloor p/2\rfloor-1}+\ldots+s_{p+\lfloor
p/2\rfloor+1}\\\ &&&\\\ &&\Rightarrow&s_{p-\lfloor p/2\rfloor+1}\ =\
\underset{i=0}{\overset{p-\lfloor p/2\rfloor}{\sum}}\,s_{i}\end{array}$

Note that in all computations above, we omit the coefficients in order to
simplify the algebraic expressions. The proof is now concluded. ∎

######  Corollary 3.

The set $\mathcal{B}_{p}\ :=\\{t^{n},\,n\in I\\}$ spans the Kauffman bracket
skein module of the lens spaces $L(p,1)$.

We now show that the set $\mathcal{B}_{p}$ is linearly independent and thus,
we prove the main theorem of the paper, Theorem 3:

###### Proof.

From the discussion and the results presented above, it follows that it
suffices to show that $t^{\lfloor p/2\rfloor}$ cannot be written as a sum of
elements in $B_{p}\backslash\\{t^{\lfloor p/2\rfloor}\\}$, or equivalently,
that the unknown $s_{\lfloor p/2\rfloor}$ cannot be written as a sum of
unknowns of indices in $I\backslash\\{\lfloor p/2\rfloor\\}$. We consider the
most interesting case that $p$ is even. The case that $p$ is odd follows
immediately from the proof of Theorem 12.

Note that since $t^{p/2}$ is in the basis of KBSM(ST), and since
KBSM($L(p,1)$)$=$KBSM(ST)$/<{\rm band\ moves}>$, it suffices to show that the
braud band moves do not affect $t^{\lfloor p/2\rfloor}$. Recall first that
isotopy in $L(p,1)$ can be viewed as isotopy in ST together with one of the
two different types of band moves (Theorem 6). So far we have been using the
$\alpha$-type band moves that are immediately translated on the level of
braids by the braid band moves. In order to prove that $t^{p/2}$ for $p$ even,
cannot be expressed as a sum of looping generators of exponents in
$I\backslash\lfloor p/2\rfloor$, we use the $\beta$-type band moves. This
simplifies the calculations involved a lot.

Indeed, consider $t^{p/2}\in B_{\rm ST}$ and perform the $\beta$-type band
move as illustrated in Figure 15. Note that the result of the performance of a
type $\beta$ band move on a looping generator is not a mixed braid. But as
shown in Figure 15, isotopy in ST results in the looping generator
$\widehat{t^{p-p/2}}\,=\,\widehat{t^{p/2}}$, that we braid and obtain
$t^{p/2}$ again. Hence, $t^{p/2}$ cannot be written as sum of elements in
$\mathcal{B}_{p}\backslash\\{t^{p/2}\\}$ and the proof is now concluded. ∎

######  Remark 8.

It is possible to prove that $t^{p/2}$ cannot be written as a sum of elements
in $\mathcal{B}_{p}\backslash\\{t^{p/2}\\}$ using the braid band moves, i.e.
the $\alpha$-type band moves. The difficulty lies in keeping track of all
coefficients involved, since following the same procedure as in the proof of
Theorem 12 for
$t^{-p/2}\,\overset{bbm}{\rightarrow}\,t^{p}\,t_{1}^{-p/2}\,\sigma_{1}^{\pm
1}$ (applying Lemma 7 and Proposition 3), results in an equation of the form
$\underset{i=0}{\overset{p/2}{\sum}}\,a_{i}\,s_{i}\ =\
\underset{i=0}{\overset{p/2}{\sum}}\,b_{i}\,s_{i}$, where $a_{i},b_{i}$
coefficients for all $i$. We would then have to prove that $a_{i}=b_{i}$ for
all $i$, which is very complicated and technical.

######  Remark 9.

The case $L(0,1)\,\cong\,S^{1}\times S^{2}$ is of special interest and in
[10], the algebraic approach is applied in order to compute KBSM($S^{1}\times
S^{2}$). As shown in [24], KBSM($S^{1}\times S^{2}$) contains torsion, and in
[10] we show how torsion is detected by solving the corresponding infinite
system of equations (3). This emphasizes on the importance of the algebraic
approach based on unoriented braids that we present in this paper.

##  5\. The diagrammatic approach for computing KBSM($L(p,q)$)

In this section we compute the Kauffman bracket skein module of the lens
spaces $L(p,q)$ using a diagrammatic method that is different to the
diagrammatic method used in [23]. We first demonstrate this method for the
$q=1$ case and we then extend it for $q>1$. It is worth mentioning that in
[4], a new basis for the Kauffman bracket skein module of the complement of
$(2,2p+1)$-torus knots is presented via this new method, which is based on
unoriented braids.

###  5.1. The $L(p,1)$ case

In this subsection we describe how the basis $\mathcal{B}_{p}$ can be obtained
using a diagrammatic approach based on unoriented braids. Instead of the
$\alpha$-band move, we will be using the $\beta$-type band move throughout
this section (recall Figure 4).

Let $t^{n}\in B_{\rm ST}$. Apply a $\beta$-type band move as shown in the
first two illustrations of Figure 15. The red strand corresponds to the strand
that appears after the band move is performed, which wraps around the surgery
curve $I$, $p$ times. The orange strands in the middle illustration indicate
the places where isotopy in ST is performed in order to unwrap the moving
strand from the surgery strand.

Figure 15. The $\beta$-type band move on an element in $B_{\rm ST}$ and its
result in KBSM($L(p,1)$).

As shown in Figure 15, using the $\beta$=band move and isotopy in ST, we may
reduce $t^{n}$ to $t^{n-p}$ for all $n\in\mathbb{Z}$. In particular, for
$n\in\mathbb{N}$, we have that:

$\begin{array}[]{ccccccc}n=0&:&t^{0}\ =\ t^{p}&,&n=1&:&t^{1}\ =\ t^{p-1}\\\
&&&&&&\\\ n=2&:&t^{2}\ =\ t^{p-2}&,&n=3&:&t^{3}\ =\ t^{p-3}\\\
\vdots&&\vdots&&\vdots&&\vdots\\\ n=\lfloor p/2\rfloor-1&:&t^{\lfloor
p/2\rfloor-1}\ =\ t^{p-\lfloor p/2\rfloor+1}&&&&\end{array}$

and we conclude that the $t^{n}$’s for $n\in\\{\lfloor
p/2\rfloor+1,\ldots,p\\}$ are not in the basis of KBSM($L(p,1)$).

Similarly, for $n\in\mathbb{Z}\backslash\mathbb{N}$, we have:

$\begin{array}[]{ccccccc}n=-1&:&t^{-1}\ =\ t^{p+1}&,&n=-2&:&t^{-2}\ =\
t^{p+2}\\\ &&&&&&\\\ n=-3&:&t^{-3}\ =\ t^{p+3}&,&n=-4&:&t^{-4}\ =\ s_{p+4}\\\
\vdots&&\vdots&&\vdots&&\vdots\\\ \end{array}$

and we conclude that $t^{p+k}\,\sim\,t^{-k}$, for $k\in\mathbb{N}$. Recall now
that $t^{-k}\,\widehat{\underset{{\rm
skein}}{\cong}}\,\underset{i=0}{\overset{k}{\sum}}\,t^{i}$ (Proposition 2).
Thus, $t^{p+1}\,\sim\,t^{-1}\,=\ t$, etc. We conclude again that the $t^{n}$’s
for $n>p$ are not in the basis of KBSM($L(p,1)$). Equivalently, we have that
the set $\mathcal{B}_{p}$ spans KBSM($L(p,1)$). Linear independence follows
then from the proof of Theorem 3.

###  5.2. The $L(p,q),\,q>1$ cases

In this subsection we adjust the proof of Theorem 3 for the case of
$L(p,q),\,q>1$ and in particular we show that the diagrammatic approach using
the unoriented braids that we presented in § 5.1, also works for $q>1$. Hence,
we shall conclude that the set $\mathcal{B}_{p}$ forms a basis for
KBSM($L(p,q)$).

We first note that the band moves are more complicated for $q>1$, in the sense
that when a moving strand of a mixed link approaches the surgery curve
$\hat{I}$, $q$ new strands will appear wrapping around $\hat{I}$, $p$-times.
For an illustration see Figure 16, for $p=3$ and $q=2$ and where band move is
shortened to bm.

Figure 16. The $\beta$-type band move for $L(3,2)$.

On the unoriented braid level, a generic $(p,q)$-band move is illustrated in
Figure 17. The shaded area depicts the braided $(p,q)$-torus knots that
appears after the performance of the band move, coming from the fact that this
torus knot bounds a disc in $L(p,q)$. Note also that the standard closure for
mixed braids on the resulted “braid” is a $\beta$-band move on the mixed link
level.

Figure 17. The band move for $L(p,q),q>1$.

Our starting point is the basis $B_{\rm ST}$ of the Kauffman bracket skein
module of the solid torus. The idea is the same as in the $q=1$ case. More
precisely, we consider elements $t^{n}\in\,B_{\rm ST}$ and perform the
$\beta$-type band moves. Then, using isotopy in ST we may reduce $t^{n}$ to
$t^{n-p}$ for all $n$. In Figure 18 we demonstrate this process for $L(3,2)$
and in particular we show how to reduce $t^{2}$ to $t$ in $L(3,2)$.

Figure 18.  $t^{2}\,\sim t$ in $L(3,2)$.

Thus, for $k\notin I$, we have that $t^{k}\,\widehat{\underset{\rm
skein}{\cong}}\,t^{n}$, for some $n\in I$. The same arguments that we used in
the proof of Theorem 3 for the $q=1$ case, in order to prove that the set
$\mathcal{B}_{p}$ is linearly independent, also apply for the $q>1$ case, and
we conclude that the set $\mathcal{B}_{p}$ forms a basis for KBSM($L(p,q)$).

######  Remark 10.

It is possible to use the algebraic method presented in § 4 to derive the
basis $\mathcal{B}_{p}$ for KBSM($L(p,q)$), $q>1$. Only the calculations
involved are more complicated due to the complexity of the $(p,q)$-braid band
move:
$bbm\left(t^{n}\right)\,=\,\left[(\sigma_{q-1}\,\ldots\,\sigma_{1})\,t\right]^{p}\,t_{q}^{n}\,\sigma_{q}^{\pm
1}$. If we consider $bbm(t^{n})$ in KBSM(ST), we may express it as a sum of
elements in $B_{\rm ST}$. Using the Kauffman bracket skein relations together
with Lemma 7 and Proposition 3, one may show that
$bbm\left(t^{n}\right)\,=\,\underset{i=0}{\overset{p+n}{\sum}}\,a_{i}\,t^{i}$,
for some coefficients $a_{i}$ and for $n\in\mathbb{N}$. The result then
follows similarly to $L(p,1)$ case.

##  6\. Conclusions

In this paper we introduce the concept of unoriented braids, that play an
important role for computing Kauffman bracket skein modules of c.c.o.
3-manifolds, and that seems promising in serving as a tool for studying c.c.o.
3-manifold invariants in general. Then, we introduce the generalized
Temperley-Lieb algebra of type B, $TL_{1,n}$, and Jones’ original idea ([25]),
we present the most generic invariant, $V$, for knots and links in the solid
torus ST, of the Kauffman bracket type. We then extend $V$ to an invariant for
knots and links in the lens spaces by imposing on $V$ relations coming from
the (braid) band moves. Our starting point is the Kauffman bracket skein
module of ST, KBSM(ST), since

${\rm KBSM(}L(p,q){\rm)}\,=\,{\rm KBSM(ST)}/<{\rm band\ moves}>.$

We first present a new basis for KBSM(ST), $B_{\rm ST}$, using the concept of
unoriented braids and we then solve the infinite system of equations
$V_{\widehat{a}}\ =\ V_{\widehat{bbm(a)}}$, where $bbm(a)$ denotes the result
of a $(p,1)$-braid band move on an element $a$ in the basis of KBSM(ST). This
is equivalent to computing the Kauffman bracket skein module of $L(p,1)$ and
it is the first time that a skein module is fully computed via this algebraic
approach based on braids.

For the case of the HOMFLYPT skein module of the lens spaces $L(p,1)$, the
reader is referred to [13, 14, 17, 3] for an algebraic approach based on
braids and the generalized Hecke algebra of type B, and [20] for a
diagrammatic approach via arrow diagrams (see also [19, 16]). We then present
a new diagrammatic approach for computing KBSM($L(p,1)$) and we finally extend
this approach for computing KBSM($L(p,q)$), $q>1$. Finally, it is worth
mentioning that in [5, 6, 7, 8, 9], skein modules are also discussed for
different families of “knotted objects” in 3-manifolds, such as tied links,
pseudo and singular links and knotoids.

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