Extensions of Braid Group Representations to the Monoid of Singular Braids

Abstract

Given a representation $\varphi :B_n \rightarrow G_n$ of the braid group $B_n$, $n \ge 2$ into a group $G_n$, we are considering the problem of whether it is possible to extend this representation to a representation $\Phi :SM_n \rightarrow A_n$, where $SM_n$ is the singular braid monoid and $A_n$ is an associative algebra, in which the group of units contains $G_n$. We also investigate the possibility of extending the representation $\Phi :SM_n \rightarrow A_n$ to a representation $\widetilde{\Phi } :SB_n \rightarrow A_n$ of the singular braid group $SB_n$. On the other hand, given two linear representations $\varphi _1, \varphi _2 :H \rightarrow GL_m(\Bbbk )$ of a group H into a general linear group over a field $\Bbbk$, we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of $SB_n$ which is an extension of the Lawrence–Krammer–Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence–Krammer–Bigelow representation.

1 Introduction

The monoid of singular braids or the Baez–Birman monoid, $SM_n$, $n \ge 2$, was introduced independently by J. Baez [3], J. Birman [9] and L. Smolin [33]. This monoid $SM_n$ is generated by the standard generators $\sigma _1^{\pm 1},$ $\sigma _2^{\pm 1}$, $\ldots , \sigma _{n-1}^{\pm 1}$ of the braid group $B_n$ in addition to the singular generators $\tau _1$, $\tau _2$, $\ldots , \tau _{n-1}$ depicted in Fig. 2. It is shown in Ref. [15] that the monoid $SM_n$ embeds into a group $SB_n$ that is said to be the singular braid group. The reader is referred to Refs. [13, 14, 17, 35]for more on the singular braid monoid and the singular braid group.

It is well known that the Artin representation of $B_n$ may be used to calculate the fundamental group of knot complements while the Burau representation can be used to calculate the Alexander polynomial of knots. In Ref. [17], Gemein studied extensions of the Artin representation and the Burau representation to the singular braid monoid and the relation between them which is induced by Fox free calculus.

In Ref. [14], Dasbach and Gemein investigated extensions of the Artin representation $B_n \rightarrow {\text {Aut}}(F_n)$ and the Burau representation $B_n \rightarrow GL_n (\mathbb {Z}[t, t^{-1}])$ to $SM_n$ and found connections between these representations. They also showed that a certain linear representation of $SM_3$ is faithful.

Just as with braids and classical links, closing a singular braid yields a singular link. Thus, the extensions of the Artin representation and the Burau representation give rise to invariants of singular knots. Gemein [17] studied invariants coming from the extended Artin representation. Indeed, he obtained an infinite family of group invariants, all of them in relation with the fundamental group of the knot complement.

Recall that a group G is said to be linear if there exists a faithful representation of G into the general linear group $\textrm{GL}_m(\Bbbk )$ for some integer $m\ge 2$ and a field $\Bbbk$. In Ref. [34], linear representations of the virtual braid groups $VB_n$, and the welded braid groups $WB_n$ into $\text{ GL}_n(\mathbb {Z}[t, t^{-1}])$ were constructed. These representations extend the Burau representation.

The Lawrence–Krammer–Bigelow representation - is one of the most famous linear representations of the braid group $B_n$. Lawrence [26] constructed a family of representations of $B_n$. It was shown in Refs. [8, 24] that one of these representations is faithful for all $n \in \mathbb {N}$. This leads to a natural question regarding the linearity of the singular braid group $SB_n$. It is worth mentioning here that a linear representation of $SM_3$ which is faithful was constructed in Ref. [14]. This representation is an extension of the Burau representation.

It is a natural approach to construct an extension of the Lawrence–Krammer–Bigelow representation to $SB_n$. In the present article, we discuss the construction of such extension. Notice that in Ref. [6], the first author constructed a linear representation $\rho :VB_{n}\mapsto GL(V_{m})$ of the virtual braid group $VB_n$, where $V_m$ is a free module of dimension $m=n(n -1)/2$ with a basis $\left\{ v_{i,j}\right\}$, $1\leqslant i<j\le n$. This representation is not an extension of the Lawrence–Krammer–Bigelow representation of $B_n$.

In his pioneering work [18], V.F.R. Jones constructed the HOMFLY polynomial P(q, z), an isotopy invariant of classical knots and links, using the Iwahori–Hecke algebras $H_n(q)$, the Ocneanu trace, and the natural surjection of the classical braid groups $B_n$ onto the algebras $H_n(q)$. In Ref. [20], the Yokonuma–Hecke algebras have been used for constructing framed knot and link invariants following the method of Jones.

The relation between singular knots and singular braids is just the same as in the classical case. A lot of papers are dedicated to the construction of invariants of singular links. For instance, the HOMFLY and Kauffman polynomials were extended to three-variable polynomials of singular links by Kauffman and Vogel [23]. The extended HOMFLY polynomial was recovered by the construction of traces on singular Hecke algebras [32]. Juyumaya and Lambropoulou [19] used a similar approach to define invariants of singular links.

A generalization of the Alexander polynomial for oriented singular links and pseudo-links was introduced in Ref. [28]. The Alexander polynomials of a cube of resolutions (in Vassiliev’s sense) of a singular knot were categorified in Ref. [1]. Moreover, a one-variable extension of the Alexander polynomial for singular links was categorified in Ref. [29]. The generalized cube of resolutions (containing Vassiliev resolutions as well as those smoothings at double points which preserve the orientation) was categorified in Ref. [30]. On the other hand, Fiedler [16] extended the Kauffman state models of the Jones and Alexander polynomials to the context of singular knots.

A singular link can be regarded as an embedding in $\mathbb {R}^3$ of a four-valent graph with rigid vertices. We can think of such vertices as being rigid disks with four strands connected to it which turn as a whole when we flip the vertex by 180 degrees. It is well-known that polynomial invariants of classical links extend (in various ways) to invariants of rigid-vertex isotopy of knotted four-valent graphs, see Refs. [21] and [23] for instance.

In Ref. [11], a homomorphism of $SM_n$ into the Temperley–Lieb algebra was constructed leading to a polynomial invariant of singular links which is an extended Kauffman bracket. Also, in Ref. [11], it was shown how to define this invariant, by interpreting singular link diagrams as abstract tensor diagrams and employing a solution to the Yang–Baxter equation. For classical links, this was done by Kauffman in Ref. [22].

The theory of singular braids is related to the theory of pseudo-braids. In particular, it was proved in Ref. [7] that the monoid of pseudo-braids is isomorphic to the singular braid monoid. Hence, the group of the singular braids is isomorphic to the group of pseudo-braids. On the other side, the theory of pseudo-links is a quotient of the theory of singular links by the singular first Reidemeister move. A similar approach has been used by Diamantis to study singular and pseudo-knots*** in the solid torus [12].

This paper is organized as follows. In Sect. 2, we recall some basic definitions and facts on braid group, singular braid monoid, and Artin and Burau representations. In Sect. 3, we shall discuss the extension of the LKBR to the singular braid monoid. Extensions of other braid group representations are discussed in Sect.  4. In Sect. 5, we shall study the defect of the extension of the LKBR with respect to the exterior product of two extensions of the Burau representations. Finally, some open questions and directions for further research are given in Sect. 6.

Notations. In this paper, we shall use the following notations and conventions. If $\varphi _*$ is a representation of the braid group, where $*$ is some index, such as A, B, LKB, etc., corresponding to Artin, Burau, Lawrence–Krammer–Bigelow, and so forth, then $\Phi _*$ denotes an extension of this representation to the singular braid monoid $SM_n$. Here, extension means that $\Phi _* |_{B_n} = \varphi _*(B_n)$. If all $\Phi _*(\tau _i)$ are invertible, then we obtain a representation of the singular braid group $SB_n$ that we shall denote by $\widetilde{\Phi }_*$.

2 Basic Definitions

In this section, we recall some basic definitions and results needed in the sequel. More details can be found in Refs. [2, 10, 27].

The braid group $B_n$, $n\ge 2$, on n strands can be defined as the group generated by $\sigma _1,\sigma _2,\ldots ,\sigma _{n-1}$ with the defining relations

$$\begin{aligned} \sigma _i \, \sigma _{i+1} \, \sigma _i= & \sigma _{i+1} \, \sigma _i \, \sigma _{i+1},~~~ i=1,2,\ldots ,n-2, \end{aligned}$$

(1)

$$\begin{aligned} \sigma _i \, \sigma _j= & \sigma _j \, \sigma _i,~~~|i-j|\ge 2. \end{aligned}$$

(2)

The geometric interpretation of $\sigma _i$, its inverse $\sigma _{i}^{-1}$ and the unit e of $B_n$ are depicted in Fig. 1.

Fig. 1

The elementary braids $\sigma _i$, $\sigma _i^{-1}$ and the unit e

The group $B_n$ has a faithful representation into the automorphism group $\textrm{Aut}(F_n)$ of the free group $F_n = \langle x_1, x_2, \ldots , x_n \rangle .$ In this case, the generator $\sigma _i$, $i=1,2,\ldots ,n-1$, is mapped to the automorphism

$$\begin{aligned} \sigma _{i} \mapsto \left\{ \begin{array}{ll} x_{i} \longmapsto x_{i} \, x_{i+1} \, x_i^{-1}, & \\ x_{i+1} \longmapsto x_{i}, & \\ x_{l} \longmapsto x_{l}, & l\ne i,i+1. \end{array} \right. \end{aligned}$$

This representation is known as the Artin representation and is denoted hereafter by $\varphi _A$.

Now, we shall define the Burau representation

$$\begin{aligned} \varphi _B :B_n \longrightarrow GL(W_n) \end{aligned}$$

of $B_n$, where $W_n$ is a free $\mathbb {Z}[t^{\pm 1}]$-module of rank n with the basis $w_1, w_2, \ldots , w_n$. The automorphisms $\varphi _B (\sigma _i)$, $i = 1, 2, \ldots , n-1$, of module $W_n$ act by the rule

$$\begin{aligned} \varphi _B(\sigma _i) :\left\{ \begin{array}{l} w_i \longmapsto (1-t) w_i + t w_{i+1}, \\ w_{i+1} \longmapsto w_i, \\ w_k \longmapsto w_k,~~k \not =i, i+1. \end{array} \right. \end{aligned}$$

The Baez–Birman monoid [3, 9, 33] or the singular braid monoid $SM_n$ is generated (as a monoid) by the elements $\sigma _i,$ $\sigma _i^{-1}$, $\tau _i$, $i = 1, 2, \ldots , n-1$. The elements $\sigma _i,$ $\sigma _i^{-1}$ generate the braid group $B_n$. The generators $\tau _i$ satisfy the defining relations

$$\begin{aligned} \tau _i \, \tau _j = \tau _j \, \tau _i, ~~~|i - j| \ge 2, \end{aligned}$$

(3)

and the mixed relations:

$$\begin{aligned} \tau _{i} \, \sigma _{j}= & \sigma _{j} \, \tau _{i}, ~~~|i - j| \ge 2, \end{aligned}$$

(4)

$$\begin{aligned} \tau _{i} \, \sigma _{i}= & \sigma _{i} \, \tau _{i},~~~ i=1,2,\ldots ,n-1, \end{aligned}$$

(5)

$$\begin{aligned} \sigma _{i} \, \sigma _{i+1} \, \tau _i= & \tau _{i+1} \, \sigma _{i} \, \sigma _{i+1},~~~ i=1,2,\ldots ,n-2, \end{aligned}$$

(6)

$$\begin{aligned} \sigma _{i+1} \, \sigma _{i} \, \tau _{i+1}= & \tau _{i} \, \sigma _{i+1} \, \sigma _{i}, ~~~ i=1,2,\ldots ,n-2. \end{aligned}$$

(7)

For a geometric interpretation of the elementary singular braid $\tau _i$ see Fig. 2.

Fig. 2

The elementary singular braid $\tau _i$

It is proved by R. Fenn, E. Keyman, and C. Rourke [15] that the Baez–Birman monoid $SM_n$ is embedded into a group $SB_n$ which they call the singular braid group.

3 Extension of the Lawrence–Krammer–Bigelow Representation

The primary goal of this section is to find extensions of the Lawrence–Krammer–Bigelow representation of the braid group $B_n$ to a representation of the singular braid monoid $SM_n$. In particular, we will explicitely determine all such extensions in the cases $n=3$ and $n=4$.

Now, let us recall the definition of the Lawrence–Krammer–Bigelow representation (LKBR for short) of the braid group $B_n$, see Refs. [8, 24, 26]. Let $R =\mathbb {Z}[t^{\pm 1}, q^{\pm 1}]$ be the ring of Laurent polynomials on two variables q and t over the ring of integers. Let $V_m$ be a free module over R with basis $\{v_{ij}\},$ $1 \le i < j \le n$. Then the LKBR $\varphi _{LKB} :B_n \longrightarrow GL(V_{m})$ is defined by action of $\sigma _i$, $i=1,2, \ldots , n-1$, on the basis of $V_{m}$ as follows

$$\begin{aligned} \varphi _{LKB} (\sigma _i)(v_{k,l})= {\left\{ \begin{array}{ll} v_{k,l}, & \{k,l\} \cap \{i,i+1\} = \emptyset ,\\ v_{i,l}, & k = i +1,\\ tq(q-1)v_{i,i+1}+(1-q)v_{i,l}+qv_{i+1,l}, & k = i \, \text{ and } \, i+1< l,\\ tq^{2}v_{i,i+1}, & k = i \, \text{ and } \, l= i+1,\\ v_{k,i}, & l = i+1 \, \text{ and } \, k<i,\\ (1-q)v_{k,i}+qv_{k,i+1}+q(q-1)v_{i,i+1}, & l=i.\ \end{array}\right. }\nonumber \\ \end{aligned}$$

(8)

As usual, we can present linear transformations $\varphi _{LKB} (\sigma _i)$ by matrices of size $m \times m$ in the basis $v_{ij}$, $1 \le i < j \le n$, where we use lexicographical order of basis vectors. Notice that we are considering coordinates of vectors as rows and the basis vectors of $V_m$ as columns. We have an isomorphism ${\text {GL}}(V_n) \cong {\text {GL}}_m(R)$; hence, we can consider LKBR as a homomorphism $\varphi _{LKB} :B_n \rightarrow {\text {GL}}_m(R)$.

Example 3.1

(1) Under the representation $\varphi _{LKB} :B_3\longrightarrow GL_3(\mathbb {C})$, the generators of $B_3$ are mapped to the matrices,

$$\begin{aligned} \sigma _{1} \mapsto \begin{pmatrix} tq^2 & \quad 0 & \quad 0 \\ tq(q-1) & \quad 1-q & \quad q \\ 0 & \quad 1 & \quad 0 \end{pmatrix},~~~ \sigma _{2} \mapsto \begin{pmatrix} 1-q & \quad q & \quad q(q-1) \\ 1 & \quad 0 & \quad 0 \\ 0 & \quad 0& \quad tq^2 \end{pmatrix}. \end{aligned}$$

(2) Under the representation $\varphi _{LKB} :B_4 \longrightarrow GL_6(\mathbb {C})$, the generators of $B_4$ are mapped to the matrices,

$$\begin{aligned} & \sigma _{1} \mapsto \begin{pmatrix} tq^2 \quad & \quad 0 & 0 & \quad 0 & \quad 0 & \quad 0 \\ tq(q-1)\quad & \quad 1-q & \quad 0& \quad q & \quad 0& \quad 0 \\ tq(q-1)& \quad 0& \quad 1-q & \quad 0 & \quad q & \quad 0\\ 0 & \quad 1 & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 1 & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0& \quad 1 \end{pmatrix}, \\ & \sigma _{2} \mapsto \begin{pmatrix} 1-q & \quad q & \quad 0& \quad q(q-1)& \quad 0& \quad 0 \\ 1 & \quad 0 & \quad 0 & \quad 0 & \quad 0& \quad 0 \\ 0 & \quad 0 & \quad 1 & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0& \quad tq^2 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad tq(q-1) & \quad 1-q & \quad q \\ 0 & \quad 0 & \quad 0 & \quad 0 & \quad 1 & \quad 0 \end{pmatrix},\\ & \sigma _{3} \mapsto \begin{pmatrix} 1 & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 1-q & \quad q& \quad 0 & \quad 0 & \quad q(q-1) \\ 0 & \quad 1 & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad 1-q& \quad q& \quad q(q-1) \\ 0 & \quad 0 & \quad 0 & \quad 1 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad tq^2 \end{pmatrix}. \end{aligned}$$

To formulate our main result of this section, we will assume that the ring $R =\mathbb {Z}[t^{\pm 1}, q^{\pm 1}]$ is a subring of the complex numbers $\mathbb {C}$, where t and q are transcendental numbers over $\mathbb {Q}$ and $V_m$ is a vector space over $\mathbb {C}$.

Theorem 3.2

Let $\varphi _{LKB} :B_n \longrightarrow GL(V_{m})$ be the Lawrence–Krammer–Bigelow representation and $u, v \in \mathbb {C}$. Then the map

$$\begin{aligned} \Phi ^{u,v}_{LKB} :SM_n \longrightarrow GL(V_{m}), \end{aligned}$$

which is defined on the generators by the formulas

$$\begin{aligned} \Phi ^{u,v}_{LKB}(\sigma _{i})= & \varphi _{LKB} \left( \sigma _{i} \right) ,\\ \Phi ^{u,v}_{LKB}(\tau _{i})= & u\varphi _{LKB} \left( \sigma _{i} \right) +v e, ~~e = \textrm{id}, \end{aligned}$$

defines a representation of $SM_n$ which is an extension of the LKBR of $B_n$. If all $\Phi ^{u,v}_{LKB}(\tau _{i})$ are invertible, then we get a representation of the group $SB_n$. Moreover, for $n=3, 4$ any extension of the LKBR to $SM_n$ has this form.

Proof

It can be easily checked that the transformations $\Phi ^{u,v}_{LKB}(\sigma _{i})$ and $\Phi ^{u,v}_{LKB}(\tau _{i})$, $i = 1, 2, \ldots , n-1$ satisfy all defining relations of $SM_n$. Hence, $\Phi ^{u,v}_{LKB}$ defines a representation of $SM_n$. Obviously, if all transformations $\Phi ^{u,v}_{LKB}(\sigma _{i})$ are invertible, then we get a linear representation of the singular braid group $SB_n$. Now it remains to prove that in the cases $n=3,4$ any extension of the LKBR to $SM_n$ is of the form $\Phi ^{u,v}_{LKB}$.

Let us consider the case $n=3$. We shall proceed as follows. Take as images of $\tau _1$ and $\tau _2$ two matrices of size $3 \times 3$ with 9 unknown entries. Then include these matrices with the images of $\sigma _1$ and $\sigma _2$ under the LKBR (see Example 3.1(1), into the defining relations of $SM_3$. Elementary but tedious calculations show that the images of $\tau _1$ and $\tau _2$ must be the following

$$\begin{aligned} & \tau _{1} \mapsto \begin{pmatrix} uq^2t+v & \quad 0 & \quad 0 \\ utq(q-1) & \quad u(1-q)+v & \quad uq \\ 0 & \quad u & \quad v \\ \end{pmatrix},~~~\\ & \tau _{2} \mapsto \begin{pmatrix} u(1-q)+v & \quad uq & \quad uq(q-1) \\ u & \quad v & \quad 0 \\ 0 & \quad 0 & \quad uq^{2}t+v\\ \end{pmatrix}. \end{aligned}$$

In the case $n=4$, using the same calculations as for the case $n=3$ and the matrices from Example 3.1(2), we should be able to prove that

$$\begin{aligned} & \tau _{1} \mapsto \begin{pmatrix} utq^2+v & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\ utq(q-1)& \quad u(1-q)+v & \quad 0& \quad uq & \quad 0& \quad 0 \\ utq(q-1)& \quad 0& \quad u(1-q)+v & \quad 0 & \quad uq & \quad 0\\ 0 & \quad u & \quad 0 & \quad v & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad u & \quad 0 & \quad v & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0& \quad u+v \end{pmatrix}, \\ & \tau _{2} \mapsto \begin{pmatrix} u(1-q)+v & \quad uq & \quad 0& \quad uq(q-1)& \quad 0& \quad 0 \\ u & \quad v & \quad 0 & \quad 0 & \quad 0& \quad 0 \\ 0 & \quad 0 & \quad u+v & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0& \quad utq^2+v & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad utq(q-1) & \quad u(1-q)+v & \quad uq \\ 0 & \quad 0 & \quad 0 & \quad 0 & \quad u & \quad v \end{pmatrix}, \\ & \tau _{3} \mapsto \begin{pmatrix} u+v & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad u(1-q)+v & \quad uq& \quad 0 & \quad 0 & \quad uq(q-1) \\ 0 & \quad u & \quad v & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad u(1-q)+v& \quad uq& \quad uq(q-1) \\ 0 & \quad 0 & \quad 0 & \quad u & \quad v & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad utq^2+v \end{pmatrix}. \end{aligned}$$

$\square$

Remark 3.3

One may ask whether it is possible to find conditions under which $\det (\Phi ^{u,v}_{LKB}(\tau _{i})) \not = 0$. Indeed, using the relations

$$\begin{aligned} \tau _{i+1} = \sigma _{i} \sigma _{i+1} \tau _{i} \sigma _{i+1}^{-1} \sigma _{i}^{-1}, ~~i = 1, 2, \ldots , n-2, \end{aligned}$$

we see that in $SM_n$, all $\tau _i$ are conjugate with $\tau _1$. Hence,

$$\begin{aligned} \det (\Phi ^{u,v}_{LKB}(\tau _{1})) =\det (\Phi ^{u,v}_{LKB}(\tau _{2})) = \ldots = \det (\Phi ^{u,v}_{LKB}(\tau _{n-1})). \end{aligned}$$

It means that it is enough to find only $\det (\Phi ^{u,v}_{LKB}(\tau _{1}))$ in $B_3$, $B_4$ and so on.

In $B_3$ we have

$$\begin{aligned} \det (\Phi ^{u,v}_{LKB}(\tau _{1})) = (u q^2 t + v) (v^2 + v u (1 - q) - u^2 q). \end{aligned}$$

In $B_4$, we have

$$\begin{aligned} & \det (\Phi ^{u,v}_{LKB}(\tau _{i}))=4tu^6+v^6-2quv^5+q^2tuv^5+3uv^5+q^2u^2v^4-6qu^2v^4\\ & \quad -2q^3tu^2v^4+3q^2tu^2v^4+3u^2v^4+3q^2u^3v^3-6qu^3v^3\\ & \quad +q^4tu^3v^3-6q^3tu^3v^3+3q^2tu^3v^3 +u^3v^3+3q^2u^4v^2-2qu^4v^2\\ & \quad +3q^4tu^4v^2-6q^3tu^4v^2+q^2tu^4v^2+q^2u^5v+3q^4tu^5v-2q^3tu^5v. \end{aligned}$$

Remark 3.4

Theorem 3.2 implies the existence of extensions of the LKBR to the singular braid group $SB_n$. In contrast, it has been proved in Ref. [4] that there are no extensions of the LKBR to the virtual braid group $VB_n$ nor to the welded braid group $WB_n$ for $n \ge 3$.

3.1 Burau Representation

We shall now show that some analogous of Theorem 3.2 holds for the Burau representation. We will assume that the Burau representation is a representation,

$$\begin{aligned} \varphi _B :B_n \rightarrow {\text {GL}}_n(\mathbb {Z}[t^{\pm 1}]) \le {\text {GL}}_n(\mathbb {C}) \end{aligned}$$

into the general linear group over the field $\mathbb {C}$. Here we take as t some transcendental over $\mathbb {Q}$. It was proved in Ref. [17] that any linear local homogeneous representation $\Phi _B :S{M_n} \rightarrow {\text {GL}}_n(\mathbb {C})$ that is an extension of the Burau representation of $B_n$ can be defined on the generators:

where $a \in \mathbb {C}$ is transcendental over $\mathbb {Q}$ such that $a \not = \frac{t}{t+1}$, then we get a representation of $SB_n$.

In Ref. [25], it was proved that the representation $\Phi _B :S{M_n} \rightarrow {\text {GL}}_n(\mathbb {C})$ is reducible. Furthermore, a reduced representation $\Phi _B^r :S{M_n} \rightarrow {\text {GL}}_{n-1}(\mathbb {C})$ was constructed and was proved to be irreducible.

A proof of the following proposition is straightforward.

Proposition 3.5

The images of the generators $\sigma _i$ and $\tau _i$ in the representation $\Phi _B :{SM_n} \rightarrow {\text {GL}}_n(\mathbb {C})$ are related by the formulas

$$\begin{aligned} \Phi _B(\tau _i) = (1-a) \varphi _B(\sigma _{i})+ a \cdot \textrm{id}, ~~i = 1, 2, \ldots , n-1. \end{aligned}$$

4 Extensions of the Braid Group Representations

Suppose that we have a representation $\varphi :B_n \rightarrow G_n$ of the braid group into a group $G_n$. In this section, we discuss whether it is possible to extend this representation to a representation $\Phi :SM_n \rightarrow A_n$, where $A_n$ is an associative algebra such that $G_n$ lies in the group of units $A_n^*$.

Proposition 4.1

Let $\varphi :B_n \rightarrow G_n$ be a representation of the braid group $B_n$, $\Bbbk$ be a field and $a, b, c \in \Bbbk$. Then a map $\Phi _{a,b,c} :SM_n \rightarrow \Bbbk [G_n]$ which acts on the generators by the rule

$$\begin{aligned} \Phi _{a,b,c}(\sigma _i^{\pm 1})= & \varphi (\sigma _i^{\pm 1}), ~~~\Phi _{a,b,c}(\tau _i)=a \varphi (\sigma _i)+b \varphi (\sigma ^{-1}_i)+ce, ~~\\ & \quad i = 1, 2, \ldots , n-1, \end{aligned}$$

defines a representation of $SM_n$ into $\Bbbk [G_n]$. Here e is the unit element of $G_n$.

Proof

We need to verify that the defining relations of $SM_n$ are mapped to the defining relations of $\Bbbk [G_n]$. Since this is true for the defining relations of $B_n$, we have to check the mixed relations and relations which involve only the generators $\tau _i$ (see relations (3)–(7)). At first, let us consider the relation (3),

$$\begin{aligned} \tau _i \, \tau _j = \tau _j \, \tau _i, ~~~|i - j| \ge 2. \end{aligned}$$

Acting by $\Phi _{a,b,c}$, we get the equality

$$\begin{aligned} & (a \varphi (\sigma _{i})+b \varphi (\sigma _i^{-1})+ce) (a \varphi (\sigma _{j})+b \varphi (\sigma _{j}^{-1})+ce) \\ & \quad = (a \varphi (\sigma _{j})+b \varphi (\sigma _{j}^{-1})+ce) (a \varphi (\sigma _{i})+b \varphi (\sigma _i^{-1})+ce). \end{aligned}$$

Since,

$$\begin{aligned} \varphi (\sigma _{i}^{\pm }) \varphi (\sigma _{j}) = \varphi (\sigma _{j}) \varphi (\sigma _{i}^{\pm }),~~\varphi (\sigma _{i}^{\pm }) \varphi (\sigma _{j}^{-1}) = \varphi (\sigma _{j}^{-1}) \varphi (\sigma _{i}^{\pm }), \end{aligned}$$

the needed relation holds. Relations (4)–(5) can be checked in a similar way.

Let us check the long relation (6) (the checking of the last relation (7) is similar),

$$\begin{aligned} \sigma _{i} \sigma _{i+1} \tau _i = \tau _{i+1} \sigma _{i} \sigma _{i+1}. \end{aligned}$$

Taking the images by $\Phi _{a,b,c}$ of both sides, we get

$$\begin{aligned} & \varphi (\sigma _{i}) \varphi (\sigma _{i+1})(a \varphi (\sigma _{i})+b \varphi (\sigma _i^{-1})+ce) \\ & \quad = (a \varphi (\sigma _{i+1})+b \varphi (\sigma _{i+1}^{-1})+ce) \varphi (\sigma _{i} \sigma _{i+1}), \end{aligned}$$

which is equivalent to the relation

$$\begin{aligned} & a \varphi (\sigma _{i}) \varphi (\sigma _{i+1}) \varphi (\sigma _{i})+b \varphi (\sigma _{i}) \varphi (\sigma _{i+1}) \varphi (\sigma _i^{-1})+c \varphi (\sigma _{i}) \varphi (\sigma _{i+1})e \\ & \quad =a \varphi (\sigma _{i+1}) \varphi (\sigma _{i}) \varphi (\sigma _{i+1})+b \varphi (\sigma _{i+1}^{-1}) \varphi (\sigma _{i}) \varphi (\sigma _{i+1}) +c \varphi (\sigma _{i}) \varphi (\sigma _{i+1}) e. \end{aligned}$$

Taking into consideration relations of $B_n$ and the fact that $\varphi$ is a representation, we can easily see that

$$\begin{aligned} \Phi _{a,b,c}(\sigma _{i} \sigma _{i+1} \tau _i) =\Phi _{a,b,c}( \tau _{i+1} \sigma _{i} \sigma _{i+1}). \end{aligned}$$

$\square$

Let us give some examples of representations of this type.

Birman representation. Motivated by the study of invariants of finite type (or Vassiliev invariants) of classical knots, Birman [9] introduced a representation of $SM_n$ into the group algebra $\mathbb {C}[B_n]$ by the expression

$$\begin{aligned} \sigma _{i}^{\pm 1} \mapsto \sigma _{i}^{\pm 1},~~ \tau _i \mapsto \sigma _{i}- \sigma _i^{-1}, ~~i = 1, 2, \ldots , n-1. \end{aligned}$$

It is easy to see that if we put in Proposition 4.1, $\varphi = \textrm{id}$, $a = 1$, $b=-1$, $c=0$, we get $\Phi _{1,-1,0}$ that is the Birman representation. Paris [31] proved that this representation is faithful.

A natural question that arises here is the following:

Question 4.2

For what values of $a, b, c \in \mathbb {C}$ the representation $\Phi _{a,b,c}$ is faithful?

Further, we can formulate a question about the possibility of extending the representation $\Phi _{a,b,c}$ to the singular braid group $SB_n$. To construct a representation of $SB_n$, it is required that the image of $\tau _i$ has an inverse, for all $i \in \{ 1, 2, \ldots , n-1 \}$. Let

$$\begin{aligned} B = \sigma _i(a \sigma _i + c) + b+e. \end{aligned}$$

Using the formula

$$\begin{aligned} (e - A)^{-1} = e + A + A^2 + A^3 + \ldots , \end{aligned}$$

we get

$$\begin{aligned} \Phi _{a,b,c}(\tau _i)^{-1} = (a \sigma _i + b \sigma _i^{-1} + c e)^{-1} = \sigma _i (e - B + B^2 - \ldots ). \end{aligned}$$

Hence, we obtain a representation

$$\begin{aligned} \tilde{\Phi }_{a,b,c} :SB_n \rightarrow \mathbb {C}[[B_n]]. \end{aligned}$$

Question 4.3

For what values of $a, b, c \in \mathbb {C}$ the representation $\tilde{\Phi }_{a,b,c}$ is faithful?

5 Comparing LKBR and the Exterior Square of Burau Representation

Suppose that we have two representations

$$\begin{aligned} \varphi , \psi :G \rightarrow {\text {GL}}_l(\Bbbk ) \end{aligned}$$

of a group G into a general linear group over a field $\Bbbk$. To compare these two representations, we introduce the following definition.

Definition 5.1

The additive defect of an element $g\in G$ is the matrix $d_g = \varphi (g) - \psi (g)$. The multiplicative defect of an element $g \in G$ is the matrix $k_g = \varphi (g)^{-1} \psi (g)$.

5.1 Exterior Square of Burau Representation

Consider the Burau representation

$$\begin{aligned} \varphi _B :B_n \rightarrow {\text {GL}}(W_n), \end{aligned}$$

where $W_n$ is a vector space over $\mathbb {C}$ with a basis $w_1$, $w_2$, $\ldots$, $w_{n-1}$. Let us take the second exterior power $\wedge ^{2}$ $W_n$ that is the quotient of $W_n \otimes W_n$ by the subspace generated by the set $\left\{ w\otimes w\mid \ w \in W_n \right\} .$ The vector space $\wedge ^{2}$ $W_n$ has a basis

$$\begin{aligned} u_{ij} = e_i \wedge e_j,~~~1 \le i < j \le n. \end{aligned}$$

We will denote by $\varphi _{DB} :B_n \rightarrow GL(\wedge ^{2}W_n)$ the homomorphism which is defined on the generators of $B_n$ by the rule

$$\begin{aligned} \varphi _{DB}(\sigma _k)(u_{ij}) = \varphi _B(\sigma _k)(e_i) \wedge \varphi _B(\sigma _k)(e_j),~~~1 \le i < j \le n, \end{aligned}$$

where $\varphi _B$ is the Burau representation of $B_n$.

Using elementary calculations, one can prove the following:

Proposition 5.2

The generators of $B_n$ act on $\wedge ^{2}W_n$ by automorphisms,

$$\begin{aligned} \varphi _{DB} \left( \sigma _{i} \right) :{\left\{ \begin{array}{ll}u_{k, i}\mapsto (1-q)u_{k,i}+qu_{k, i+1},& k<i; \\ u_{k, i+1}\mapsto u_{k, i},& k<i; \\ u_{i, i+1}\mapsto (1-q)u_{i, i+1};& \\ u_{i, l} \mapsto (1-q) u_{i, l}+q u_{i+1, l},& i+1< l;\\ u_{i+1, l} \mapsto u_{i, l}, & i+1< l; \\ u_{k, l} \mapsto u_{k, l}, & \left\{ k,l \right\} \cap \left\{ i+1,i \right\} =\emptyset ; \end{array}\right. }\\ \varphi _{DB} \left( \sigma _{i}^{-1} \right) :{\left\{ \begin{array}{ll}u_{k, i}\mapsto u_{k,i+1},& k<i;\\ u_{k, i+1}\mapsto \frac{1}{q} u_{k, i}+\frac{q-1}{q} u_{k, i+1}, & k<i;\\ u_{i, i+1}\mapsto - \frac{1}{q-1} u_{i, i+1};& \\ u_{i, l} \mapsto u_{i+1, l},& i+1<l; \\ u_{i+1, l} \mapsto \frac{1}{q} u_{i, l}+\frac{q-1}{q} u_{i+1, l}, & i+1<l;\\ u_{k, l} \mapsto u_{k, l}, & \left\{ k,l \right\} \cap \left\{ i+1,i \right\} =\emptyset ; \end{array}\right. } \end{aligned}$$

for all $i=1, 2, \ldots ,n-1$.

Notice that the vector spaces on which act the representations $\varphi _{LKB}$ and $\varphi _{DB}$ are isomorphic. Further we will assume that both representations act on the vector space $V_m$ with the basis $\{ v_{ij} \}$, $1\le i < j \le n$. We are interested in investigating the connection between these two representations. We can reformulate the general definition of the defect as follows.

Definition 5.3

The additive defect of an element $w \in B_n$ is an element

$$\begin{aligned} d_w = \varphi _{LKB}(w)-\varphi _{DB}(w). \end{aligned}$$

The multiplicative defect of an element $w \in B_n$ is an element

$$\begin{aligned} k_w = \varphi _{DB}(w)^{-1} \varphi _{LKB}(w). \end{aligned}$$

Let us find the defect of the generators $\sigma _i$. Denote $h_i = \varphi _{LKB}(\sigma _i)$, $g_i = \varphi _{DB}(\sigma _i)$ and $g_i^{-1} = \varphi _{DB} \left( \sigma _{i}^{-1} \right)$, then the additive defect of $\sigma _i$ is equal to $d_i=h_i-g_i$, and the multiplicative defect is equal to $k_i=g_i^{-1}h_i$.

Proposition 5.4

The following formulas hold

$$\begin{aligned} & d_{i}:{\left\{ \begin{array}{ll} v_{k, l} \mapsto 0, & \{k,l\} \cap \{i,i+1\} = \emptyset ;\\ v_{i+1, l}\mapsto 0, & i +1< l;\\ v_{i, l} \mapsto tq(q-1) v_{i, i+1}, & i+1< l;\\ v_{i, i+1} \mapsto (tq^{2}+q-1)v_{i, i+1}; & \\ v_{k, i+1} \mapsto 0, & k<i;\\ v_{k, i} \mapsto q(q-1)v_{i,i+1}, & k< i;\ \end{array}\right. }\\ & k_i :{\left\{ \begin{array}{ll}v_{k, i}\mapsto v_{k,i},& k<i;\\ v_{k, i+1}\mapsto v_{k, i+1}+(q-1)v_{k+1, i+1}& k<i + 1;\\ v_{i, i+1}\mapsto -\frac{tq^{2}}{q-1} v_{i, i+1};& \\ v_{i, l} \mapsto v_{i, l},& i+1<l; \\ v_{i+1, l} \mapsto t(q-1) v_{i, i+1}+ v_{i+1, l}, & i+1<l;\\ v_{k, l} \mapsto v_{k, l}, & \left\{ k,l \right\} \cap \left\{ i+1,i \right\} =\emptyset . \end{array}\right. } \end{aligned}$$

Proof

To compute the additive defect, we rewrite (8) in the form:

$$\begin{aligned} & h_i = \varphi _{LKB} (\sigma _i):\\ & \qquad {\left\{ \begin{array}{ll} v_{k, l} \mapsto v_{k,l}, & \{k,l\} \cap \{i,i+1\} = \emptyset ;\\ v_{i+1, l}\mapsto v_{i,l}, & i +1< l;\\ v_{i, l} \mapsto tq(q-1)v_{i,i+1}+(1-q)v_{i,l}+qv_{i+1,l}, & i+1< l;\\ v_{i, i+1} \mapsto tq^{2}v_{i,i+1}; & \\ v_{k, i+1} \mapsto v_{k,i}, & k<i;\\ v_{k, i} \mapsto (1-q)v_{k,i}+qv_{k,i+1}+q(q-1)v_{i,i+1}, & k < i.\ \end{array}\right. } \end{aligned}$$

By Proposition 5.2, we have:

$$\begin{aligned} g_i = \varphi _{DB} ( \sigma _{i}) :{\left\{ \begin{array}{ll} v_{k, l}\mapsto v_{k,l}, & \{k,l\} \cap \{i,i+1\} = \emptyset ;\\ v_{i+1, l} \mapsto v_{i,l}, & i +1< l;\\ v_{i, l} \mapsto (1-q)v_{i,l}+q v_{i+1,l}, & \, i+1< l;\\ v_{i, i+1} \mapsto (1-q) v_{i,i+1}; & \\ v_{k, i+1} \mapsto v_{k,i}, & k<i;\\ v_{k, i} \mapsto (1-q)v_{k,i}+qv_{k,i+1}, & k < i.\\ \end{array}\right. } \end{aligned}$$

Thus, the additive defect is given by the following formula,

$$\begin{aligned} d_{i} = h_i - g_i :{\left\{ \begin{array}{ll} v_{k, l} \mapsto 0, & \{k,l\} \cap \{i,i+1\} = \emptyset ;\\ v_{i+1, l}\mapsto 0, & i +1< l;\\ v_{i, l} \mapsto tq(q-1) v_{i, i+1}, & i+1< l;\\ v_{i, i+1} \mapsto (tq^{2}+q-1)v_{i, i+1}; & \\ v_{k, i+1} \mapsto 0, & k<i;\\ v_{k, i} \mapsto q(q-1)v_{i,i+1}, & k < i.\ \end{array}\right. } \end{aligned}$$

The formula for the multiplicative defect $k_i$ can be proved in a similar way. $\square$

We shall now calculate the additive and multiplicative defects in the cases $n=3$ and $n=4$.

Example 5.5

In the case $n=3$, we have

$$\begin{aligned} & g_{1} =\begin{pmatrix} 1-q & \quad 0 & \quad 0 \\ 0 & \quad 1-q & \quad q \\ 0 & \quad 1 & \quad 0 \end{pmatrix}~~~ g_{1}^{-1} =\begin{pmatrix} \frac{-1}{q-1} & \quad 0 & \quad 0 \\ 0& \quad 0 & \quad 1 \\ 0& \quad \frac{1}{q}& \quad \frac{q-1}{q} \end{pmatrix} ~~~\\ & h_{1} =\begin{pmatrix} tq^2& \quad 0 & \quad 0\\ qt(q-1) & \quad 1-q & \quad q \\ 0 & \quad 1 & \quad 0 \end{pmatrix}. \end{aligned}$$

Hence, the multiplicative and additive defects are equal to

$$\begin{aligned} k_1 = g_{1}^{-1} h_{1} =\begin{pmatrix} \frac{-tq^2}{q-1}& 0& 0\\ 0& 1& 0\\ t(q-1)& 0& 1 \\ \end{pmatrix}, d_1 = \begin{pmatrix}q^2t+q-1& 0& 0\\ qt(q-1)& \quad 0& \quad 0\\ 0& \quad 0& 0 \end{pmatrix}. \end{aligned}$$

For the image of $\sigma _2$, we have

$$\begin{aligned} & g_{2} =\begin{pmatrix} 1-q & \quad q& \quad 0\\ 1& \quad 0 & \quad 0& \quad \\ 0& \quad 0& \quad 1-q \end{pmatrix},~~ g_{2}^{-1} =\begin{pmatrix} 0& \quad 1& \quad 0 \\ \frac{1}{q}& \quad \frac{q-1}{q}& \quad 0 \\ 0& \quad 0& \quad \frac{-1}{q-1} \end{pmatrix},~~~\\ & h_{2} =\begin{pmatrix} 1-q& \quad q & \quad q(q-1)\\ 1& \quad 0& \quad 0\\ 0& \quad 0& \quad tq^2 \end{pmatrix} \end{aligned}$$

Hence,

$$\begin{aligned} k_2= & g_{2}^{-1} h_{2} =\begin{pmatrix} 1& \quad 0& \quad 0 \\ 0& \quad 1& \quad q-1\\ 0& \quad 0& \quad \frac{-tq^2}{q-1} \end{pmatrix},\\ d_2= & \begin{pmatrix} 0& \quad 0& \quad q(q-1)\\ 0& \quad 0& \quad 0\\ 0& \quad 0& \quad q^2t+q-1. \end{pmatrix} \end{aligned}$$

Example 5.6

In the case $n=4$, we have

$$\begin{aligned} g_{1}= & \begin{pmatrix} 1-q& \quad 0& \quad 0& \quad 0& \quad 0 & \quad 0 \\ 0& \quad 1-q & \quad 0& \quad q& \quad 0& \quad 0 \\ 0& \quad 0& \quad 1-q& \quad 0& \quad q& \quad 0 \\ 0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0 & \quad 1& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0 & \quad 1 \end{pmatrix},\\ g_{1}^{-1}= & \begin{pmatrix} \frac{-1}{q-1}& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0\\ 0& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0\\ 0& \quad \frac{1}{q}& \quad 0& \quad \frac{q-1}{q} & \quad 0& \quad 0\\ 0& \quad 0& \quad \frac{1}{q} & \quad 0& \quad \frac{q-1}{q}& \quad 0\\ 0& \quad 0& \quad 0 & \quad 0& \quad 0& \quad 1\\ \end{pmatrix}.\\ h_{1}= & \begin{pmatrix} tq^2 & \quad 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\ tq(q-1)& \quad 1-q & \quad 0& \quad q & \quad 0& \quad 0 \\ tq(q-1)& \quad 0& \quad 1-q & \quad 0 & \quad q & \quad 0\\ 0 & \quad 1 & \quad 0 & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 1 & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad 0 & \quad 0& \quad 1 \end{pmatrix}. \end{aligned}$$

Hence, the multiplicative and additive defects are equal to

$$\begin{aligned} k_1= & g_{1}^{-1} h_{1} =\begin{pmatrix} \frac{-tq^2}{q-1} & \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0\\ 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 0\\ t(q-1)& \quad 0& \quad 0 & \quad 1& \quad 0& \quad 0\\ t(q-1)& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0\\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 1 \end{pmatrix}, \,\,\,\\ d_1= & \begin{pmatrix} q^2t+q-1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ qt(q-1)& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ qt(q-1)& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \end{pmatrix}. \end{aligned}$$

Let us consider the image of $\sigma _2$. We have

$$\begin{aligned} g_{2}= & \begin{pmatrix} 1-q& \quad q& \quad 0& \quad 0& \quad 0& \quad 0\\ 1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0\\ 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 0\\ \ 0& \quad 0& \quad 0& \quad 1-q& \quad 0& \quad 0\\ 0& \quad 0& \quad 0& \quad 0& \quad 1-q& \quad q\\ 0& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0 \end{pmatrix},~~~\\ g_{2}^{-1}= & \begin{pmatrix} 0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0 \\ \frac{1}{q}& \quad \frac{q-1}{q}& \quad 0& \quad 0& \quad 0& \quad 0\\ 0& \quad 0 & \quad 1& \quad 0& \quad 0& \quad 0\\ 0& \quad 0 & \quad 0& \quad \frac{-1}{q-1}& \quad 0& \quad 0\\ 0& \quad 0 & \quad 0& \quad 0& \quad 0& \quad 1\\ 0& \quad 0 & \quad 0& \quad 0& \quad \frac{1}{q}& \quad \frac{q-1}{q} \end{pmatrix}\\ h_{2}= & \begin{pmatrix} 1-q& \quad q & \quad 0& \quad q(q-1)& \quad 0& \quad 0 \\ 1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0 & \quad 1& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad tq^2 & \quad 0 & \quad 0 \\ 0& \quad 0& \quad 0& \quad tq(q-1)& \quad 1-q& \quad q \\ 0& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0 \end{pmatrix}. \end{aligned}$$

Hence, the multiplicative and additive defects are equal to

$$\begin{aligned} k_2= & g_{2}^{-1} h_{2} =\begin{pmatrix} 1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 1& \quad 0& \quad q-1& \quad 0& \quad 0 \\ 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad \frac{-tq^2}{q-1} & \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0 \\ 0& \quad 0& \quad 0& \quad t(q-1)& \quad 0& \quad 1 \end{pmatrix}, \\ d_2= & \begin{pmatrix} 0& \quad 0& \quad 0& \quad q(q-1)& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad q^2t+q-1& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad qt(q-1)& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \end{pmatrix}. \end{aligned}$$

For the image of $\sigma _3$,

$$\begin{aligned} & g_{3} =\begin{pmatrix} 1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 1-q& \quad q& \quad 0& \quad 0& \quad 0 \\ 0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 1-q& \quad q& \quad 0 \\ 0& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 1-q \end{pmatrix},~~~\\ & g_{3}^{-1} =\begin{pmatrix} 1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 0\\ 0& \quad \frac{1}{q}& \quad \frac{q-1}{q}& \quad 0& \quad 0& \quad 0\\ 0& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0 \\ 0& \quad 0& \quad 0& \quad \frac{1}{q}& \quad \frac{q-1}{q}& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad \frac{-1}{q-1} \end{pmatrix},\\ & h_{3} =\begin{pmatrix} 1& \quad 0& \quad 0& \quad 0 & \quad 0& \quad 0 \\ 0& \quad 1-q & \quad q& \quad 0& \quad 0& \quad q(q-1) \\ 0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 1-q& \quad q& \quad q(q-1) \\ 0& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad tq^2 \end{pmatrix}. \end{aligned}$$

Hence, the multiplicative and additive defects are equal to

$$\begin{aligned} k_3= & g_{3}^{-1} h_{3} =\begin{pmatrix} 1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad q-1 \\ 0& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 1& \quad q-1 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad \frac{-tq^2}{q-1} \end{pmatrix},\\ d_3= & \begin{pmatrix} 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad q(q-1) \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad q(q-1) \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0 \\ 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad q^2t+q-1 \end{pmatrix}. \end{aligned}$$

Remark 5.7

According to Ref. [7], the monoid of singular braids $SM_n$ is isomorphic to the monoid of pseudo-braids $PM_n$ and the group of singular braids $SB_n$ is isomorphic to the group of pseudo-braids $PG_n$. Hence, all representations of $SM_n$ and $SB_n$ give representations of $PM_n$ and $PB_n$, respectively.

6 Open Problems and Directions for Further Research

6.1 From the Lawrence–Bigelow–Krammer Representation to Knot Invariants

Using the Burau representation of the braid groups, one can define the Alexander polynomial which is a knot invariant of classical knots. To the best of our knowledge, there are no knot invariants defined from the Lawrence–Bigelow–Krammer representation. We suggest the following construction of such invariants.

Let $B_{\infty } = \cup _{n=1}^{\infty } B_n$. For any $\beta \in B_{\infty }$, define the polynomial

$$\begin{aligned} f_{\beta } = f_{\beta }(q, t, \lambda ) = det(\varphi _{LKB}(\beta ) - \lambda \cdot id) \in \mathbb {Q}[q,t, \lambda ], \end{aligned}$$

that is the characteristic polynomial which corresponds to the image of $\beta$ by the Lawrence–Bigelow–Krammer representation $\varphi _{LKB}$. Let

$$\begin{aligned} F = \{ f_{\beta } ~|~\beta \in B_{\infty } \} \subseteq \mathbb {Q}[q,t, \lambda ] \end{aligned}$$

be the set of such characteristic polynomials. We define an equivalence relation on F as follows:

$$\begin{aligned} f_{\beta } \sim _M f_{\gamma } \Leftrightarrow ~\text{ there } \text{ is } \text{ a } \text{ sequence } \text{ of } \text{ Markov } \text{ moves } \text{ which } \text{ transforms } ~\beta ~ \text{ into }~ \gamma . \end{aligned}$$

Using the Markov theorem one can prove the following:

Proposition 6.1

The equivalence class $[f_{\beta }]$ under the equivalence relation $\sim _M$ is an invariant of the knot $\hat{\beta }$ that is the closure of the braid $\beta$.

Question 6.2

Which knots it is possible to distinguish using the invariant $[f_{\beta }]$?

By properties of characteristic polynomials, $f_{\beta }$ does not change under the first Markov move, i.e., $f_{\beta } =f_{\alpha ^{-1} \beta \alpha }$ for all $\alpha ,\beta \in B_n$. Let $L = \hat{\beta }$ be a link that is the closure of a braid $\beta$. Define the following set of polynomials

$$\begin{aligned} F_L = \{ f_{\gamma }~|~\gamma \in B_{\infty } ~\text{ can } \text{ be } \text{ constructed } \text{ from }~ \beta ~\text{ using } \text{ Markov } \text{ moves }\}. \end{aligned}$$

From Proposition 6.1, it follows.

Corollary 6.3

The set of polynomials $F_L$ is an invariant of the link $L = \hat{\beta }$.

It is interesting to investigate whether it is possible to find all polynomials in $F_L$. In the following example, we give some calculations.

Example 6.4

(1) (Trivial knot) Let $\beta = \sigma _1 \sigma _2 \in B_3$ be a three-strand braid. It is easy to check that its closure $\hat{\beta }$ is the trivial knot U. Also, one can see that the closure of any of the three-strand braids

$$\begin{aligned} \sigma _1^{-1} \sigma _2,~~\sigma _1 \sigma _2^{-1},~~\sigma _1^{-1} \sigma _2^{-1}, \end{aligned}$$

gives the trivial knot. The corresponding polynomials have the form,

$$\begin{aligned} f_{\sigma _1 \sigma _2}= & q^6t^2-w^3,\\ \quad f_{\sigma _1^{-1} \sigma _2}= & (q^2t-q^2tw^3-qw^2+q^4t^2w^2-q^3t^2w^2-q^3tw^2\\ & +2q^2tw^2-qtw^2+w^2+qw-q^4t^2w+q^3t^2w+\\ & +q^3tw-2q^2tw+qtw-w)/(q^2t),\\ f_{\sigma _1 \sigma _2^{-1}}= & (q^2t-q^2tw^3-qw^2+q^4t^2w^2-q^3t^2w^2-q^3tw^2\\ & +2q^2tw^2-qtw^2+w^2+qw-q^4t^2w+q^3t^2w+\\ & +q^3tw-2q^2tw+qtw-w)/(q^2t),\\ f_{\sigma _1^{-1} \sigma _2^{-1}}= & (-q^6t^2w^3+1)/(q^6t^2). \end{aligned}$$

Also, the closure of the four-strand braid $\sigma _1 \sigma _2 \sigma _3$ gives the trivial knot. For this braid,

$$\begin{aligned} f_{\sigma _1 \sigma _2 \sigma _3}=q^{12}t^3+w^6-q^4tw^4-q^8t^2w^2. \end{aligned}$$

(2) (Hopf link) Let $\beta = \sigma ^2_1 \sigma _2 \in B_3$ be a three-strand braid. It is easy to check that its closure $\hat{\beta }$ is the Hopf link H. We have

$$\begin{aligned} f_{\sigma ^2_1 \sigma _2}=-q^9t^3-w^3+q^3tw^2+q^6t^2w. \end{aligned}$$

(3) (Trefoil knot) Let $\beta = \sigma ^3_1 \sigma _2 \in B_3$ be a three-strand braid. It is easy to check that its closure $\hat{\beta }$ is the trefoil knot T. We have

$$\begin{aligned} f_{\sigma ^3_1 \sigma _2}=q^{12}t^4-w^3. \end{aligned}$$

6.2 Extensions of the Artin Representations

In Ref. [17], a family of extensions of the Artin representation of $B_n$ to the monoid of the singular braids $SM_n$ is constructed.

Question 6.5

Is it possible to construct non-trivial extensions of the Artin representation of $B_n$ to the group of the singular braids $SB_n$? Is it possible to construct a faithful such representation?

6.3 Representation into the Temperley–Lieb Algebra

For each integer $n \ge 2$, the n-strand Temperley–Lieb algebra, denoted $TL_n$, is the unital, associative algebra over the ring $\mathbb {Z}[t, t^{-1}]$ generated by $u_i$, for $1 \le i \le n-1$, and subject to the following relations:

• (1) $u_i^2 = (-t^2 - t^{-2}) u_i$, $1 \le i \le n-1$;

• (2) $u_i u_j u_i = u_i$, for all $1 \le i, j \le n-1$ with $|i - j| = 1$;

• (3) $u_i u_j = u_j u_i$, for all $1 \le i, j \le n-1$ with $|i - j| > 1$.

In Ref. [11], it was proved that for any $a, b \in \mathbb {Z}[t, t^{-1}]$ the map $\rho _{a, b} :SM_n \rightarrow TL_n$, which is defined on the generators by,

$$\begin{aligned} \rho _{a,b}(\sigma _i) = t^{-1} u_i + t e,~~~\rho _{a,b}(\sigma _i^{-1}) = t u_i + t^{-1} e,~~~\rho _{a,b}(\tau _i) = a u_i + b e,~~1 \le i \le n-1, \end{aligned}$$

where e is the unit element of $TL_n$, is a representation of the singular braid monoid.

Question 6.6

Is it possible to extend $\rho _{a,b}$ to a representation of the group $SB_n$?