Polynomial invariants for links and tied links

TL;DR

This paper discusses polynomial invariants used to distinguish links and tied links, based on lessons from specialized courses and conferences, highlighting their theoretical foundations and applications.

Contribution

It introduces or analyzes polynomial invariants specifically for links and tied links, expanding the understanding of their algebraic and topological properties.

Findings

Development of new polynomial invariants for tied links

Application of invariants to classify complex link structures

Insights into algebraic relations among invariants

Abstract

Notes based on lessons given at {\sc Escuela " Fico Gonz\'alez Acu\~na" de Nudos y 3-variedades}, M\'erida Yucat\'an, M\'exico, 7--10 (2015) and {\sc Encuentro de nudos, trenzas y \'algebras}, Oaxaca--M\'exico, 3--10 October (2018).

Figures

Equations

$$\text{If}{L}_1\sim L_2,\text{then}\mathrm{I}(L_1)=\mathrm{I}(L_2).$$

$$w(D):=\sum \mathrm{sig}(p),$$

$$L\sim L^{\prime }\text{ if and only if}D{\sim }_{\mathrm{R}}{D}^{\prime }.$$

$$f(\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler1.pdf})=af(\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler0.pdf})\quad\text{and}\quad f(\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler2.pdf})=a^{-1}f(\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler0.pdf}).$$

$$F(L):=a^{-w(D)}f(|D|)$$

$$⟨D⟩=\sum _{S\in \mathrm{St}(D)}{z}^{|S|}⟨D,S⟩.$$

$$\mathrm{St}(H)=\left\{\includegraphics[scale={0.2},trim=0.0pt 42.67912pt 0.0pt 0.0pt]{State1HopfLink.pdf},\includegraphics[scale={0.2},trim=0.0pt 42.67912pt 0.0pt 0.0pt]{State2HopfLink.pdf},\includegraphics[scale={0.2},trim=0.0pt 42.67912pt 0.0pt 0.0pt]{State3HopfLink.pdf},\includegraphics[scale={0.2},trim=0.0pt 42.67912pt 0.0pt 0.0pt]{State4HopfLink.pdf}\right\}.$$

$$⟨D⟩=z^1{A}^2+z^{0}AB+z^{0}AB+z^1{B}^2=z{A}^2+2AB+z{B}^2.$$

$$B=A^{-1}\text{and}z=-(A^2+A^{-2}).$$

$$\begin{array}[]{ccll}\langle\hskip 14.22636pt\rangle:&|\mathfrak{D}|&\longrightarrow&\mathbb{Z}[A,A^{-1}]\\ &D&\mapsto&\langle D\rangle\end{array}$$

$$f(L):=(-A^3{)}^{-w(D)}⟨|D|⟩\in \mathbb{Z}[A,A^{-1}].$$

$${\mathtt{s}}_i{\mathtt{s}}_j$$

$${\mathtt{s}}_i{\mathtt{s}}_j{\mathtt{s}}_i$$

$$[\alpha ][\beta ]=[\alpha \beta ](\alpha ,\beta \in B_n).$$

$${\sigma }_i{\sigma }_j$$

$${\sigma }_i{\sigma }_j{\sigma }_i$$

$$\mathsf{p}:B_n⟶{\mathtt{S}}_n,\text{defined by mapping σi↦si}.$$

$$\hat{}:B_{\infty }⟶\mathfrak{L},\alpha \mapsto \hat{\alpha }.$$

$$h_{i}hj$$

$$h_i{h}_j{h}_i$$

$$h_i^2$$

$$h_i^{-1}=(\mathsf{u}{}^{-1}-1)+\mathsf{u}{}^{-1}{h}_i.$$

$${\sigma }_i^2-\mathsf{u}-(\mathsf{u}-1){\sigma }_i\text{for}1\le i\le n-1.$$

$$M=M_1{h}_n{M}_2{h}_n{M}_3,$$

$${\mathcal{C}}_n={\mathcal{C}}_{n-1}\cup \{x{h}_{n-1}{h}_{n-2}\cdots h_i;x\in {\mathcal{C}}_{n-1}1\le i\le n-1\}.$$

$$\text{(i)}{M}_0\text{and}\text{(ii)}{M}_1{h}_n{M}_2,$$

$$M_1{h}_{n-1}{M}_2=M_1{h}_{n-1}{\mathfrak{n}}_1{\mathfrak{n}}_2\cdots {\mathfrak{n}}_{n-2}=M_1{\mathfrak{n}}_1{\mathfrak{n}}_2\cdots h_{n-1}{\mathfrak{n}}_{n-2}.$$

$${\mathrm{H}}_1=\mathbb{K}\subset {\mathrm{H}}_2\subset \cdots \subset {\mathrm{H}}_n\subset {\mathrm{H}}_{n+1}\subset \cdots$$

$$x+\sum _i{y}_i\otimes z_i\mapsto x+\sum _i{y}_i{h}_n{z}_i$$

$${\tau }_{n+1}(a):={\tau }_n(x)+\mathsf{z}\sum _i{\tau }_n(y_i{z}_i).$$

$${\tau }_n\circ {\pi }_{\theta }:B_n⟶\mathbb{K}(\mathsf{z})(n\in \mathbb{N}).$$

$$({\tau }_n\circ {\pi }_{\theta })({\sigma }_n)=({\tau }_n\circ {\pi }_{\theta })({\sigma }_n^{-1})\text{for all}n.$$

Full text

Notes based on lessons given at Escuela ‘Fico González Acuña’ de Nudos y 3-variedades, Mérida Yucatán, México, 7–10 (2015) and Encuentro de nudos, trenzas y álgebras, Oaxaca–México, 3–10 October (2018).

I would like to thank for the invitation and hospitality received from Mario Eudave at Merida and Bruno Cisneros at Oaxaca. Also, I would like to thank Francesca Aicardi and Nicoletta Zar for their contribution in the writing of these notes.

1. 1 Background
2. 2 Planar diagrams of links
3. 3 Bracket polynomial
4. 4 Links via braids
5. 5 Hecke algebra
6. 6 The Homflypt polynomial
7. 7 Hecke algebras in representation theory
8. 8 Yokonuma–Hecke algebra
9. 9 The $E$–system
10. 10 The invariants $\Delta_{m}$ and $\Theta_{m}$
11. 11 The bt–algebra
12. 12 Tied links
13. 13 The invariant $\mathcal{F}$

1. Background

Definition 1**.**

A knot is a subset of $\mathbb{R}^{3}$ homeomorphic to $S^{1}$. A link is a disjoint union of $n$ knots, $n$ is called the number of components of the link.

In Fig. 1: the first knot is called the unknot, the second is the left trefoil knot, the third is the Hopf link, the fourth is a link with two components and the fifth is a link with three components.

Since a knot is a simple closed curve we can provide it with an orientation, in such case we say that the knot is oriented. If each component of a link is oriented, we say that the link is oriented.

Definition 2**.**

Two links $L_{1}$ and $L_{2}$ (both oriented or not) are called ambient isotopic (or equivalent) if there exists an ambient isotopy of $\mathbb{R}^{3}$ that carries $L_{1}$ in $L_{2}$, that is, there is a continue map $\phi:\mathbb{R}^{3}\times\left[0,1\right]\longrightarrow\mathbb{R}^{3}$ such that:

1. (1)

$\phi(L_{1},0)=L_{1}$* and $\phi(L_{1},1)=L_{2}$,* 2. (2)

For every $t\in[0,1]$, the maps $\phi_{t}$’s are continues and injective, where $\phi_{t}$ denotes to $\phi$ restricted to $\mathbb{R}^{3}\times\{t\}$.

The fact that $L_{1}$ and $L_{2}$ are ambient isotopic is denoted by $L_{1}\sim L_{2}$.

From now on we denote, respectively, by $\mathfrak{L}$ and $|\mathfrak{L}|$ the set of all oriented links and unoriented links in $\mathbb{R}^{3}$.

Proposition 1**.**

The relation $\sim$ is an equivalence relation on $\mathfrak{L}$.

The main problem in knot theory is to find a subset of representatives of equivalence class of $\mathfrak{L}$ under the relations $\sim$, or equivalently, given two links to decide if they are ambient isotopic or not. This problem is far from solution; however, we have a tool that helps to decide when two links are not ambient isotopic: the invariant of links.

Definition 3**.**

Let $\mathrm{Set}$ be a set, an invariant of link is a function $\mathrm{I}:\mathfrak{L}\longrightarrow\mathrm{Set}$ such that:

[TABLE]

In the case that $\mathrm{Set}$ is a polynomial ring or a field of rational functions, we say that $\mathrm{I}$ is a polynomial invariant of links.

To have an invariant when $\mathrm{Set}$ is well understood allows to compare easily the images of the links by $\mathrm{I}$; therefore, $\mathrm{I}$ is useful in the sense that if the images of the links are different then the links are not equivalent. Some famous classical invariants are: the linking number, the 3–coloration, the fundamental group of a link and the following polynomial invariants:

1. (1)

The Alexander polynomial, 2. (2)

The Jones polynomial, 3. (3)

The Homflypt polynomial, 4. (4)

The Kauffman polynomial.

The Alexander and Jones invariants are polynomials in one variable. The Homflypt and Kauffman polynomial are in two variables. The Jones polynomial can be obtained as a specialization both of the Kauffman and Homflypt polynomials; the Alexander polynomial can be obtained also as a specialization of the Homflypt polynomial, but not as a specialization of the normalized Kauffman polynomial . We note that all these polynomial invariants are defined for oriented links.

2. Planar diagrams of links

The links can be studied through diagrams in $\mathbb{R}^{2}$: we associate to each link a generic projection on a plane, this projection is provided with codes to indicate, at every double point, what portion of the projected curve (shadows) is coming from an over or under crossing. Thanks to a theorem of Kurt Reidemeister, the study of links can be translated to the study of diagrams of links, allowing thus a combinatoric treating of the study of links.

Definition 4**.**

A generic projection of a curve of $\mathbb{R}^{3}$ on a plane is one that only admits simple crossings.

In Fig. 3, the first figure is a simple crossing, the second figure is a generic projection of a curve and the third and fourth figures are not generic projections of a curve.

Definition 5**.**

A diagram of a link is a generic projection (or shadow) of it on a plane, where each further simple crossing is codified, depending if it is originated from an over or under crossing in the link, by one of the two crossing codifications below.

More precisely, in Fig. 4, the first picture is a simple crossing in the diagram produced by the projection of the curves $c$ and $d$ of the link, the second picture shows the code used to indicate that the curve $c$ is under the curve $d$ in the link and the last picture is to indicate that the curve $c$ is over the curve $d$ in the link.

Example 1**.**

Fig. 5 shows the procedure to obtain a projection of a left trefoil knot.

Oriented links yields oriented diagrams; thus a diagram is oriented if every crossing codification is in one of the following situations:

The first codification is called (by convention) positive crossing and the other one negative crossing. The sign of the crossing is $1$ if it is a positive crossing, otherwise is $-1$.

Definition 6**.**

Let $D$ be an oriented diagram, the writhe of $D$, denoted by $w(D)$, is the sum over the sign of all crossings of $D$:

[TABLE]

where $p$ running on the crossing of $D$ and $\mathrm{sig}(p)$ is the sign of the crossing $p$.

Definition 7**.**

*Two diagrams of unoriented links are R–isotopic if one of them can be transformed in the other, by so–called Reidemeister moves *R0, R1, R2 and/or R3, where in the case unoriented links are:

We have the analogous definition for diagrams of oriented links but by adding now all possible orientations to the moves R1–R3 above. For instance, some of these oriented Reidemeister are shown in Fig. 8.

Proposition 2**.**

The relation of R–isotopic, denoted by $\sim_{\rm{R}}$, defines an equivalence relation on the set of planar diagrams.

Notation 1**.**

1. (1)

If $D$ is an oriented diagram, we denote by $|D|$ the unoriented diagram obtained by forgetting the orientation in $D$. 2. (2)

We denote, respectively, by $\mathfrak{D}$ and $|\mathfrak{D}|$ the set of diagrams of, respectively, oriented and unoriented links in $\mathbb{R}^{3}$.

Theorem 1** (Reidemeister, 1932).**

Let $L$ and $L^{\prime}$ be two links (oriented or not) and set $D$ and $D^{\prime}$ diagrams, respectively, of $L$ and $L^{\prime}$, we have:

[TABLE]

So, $\mathfrak{L}/\sim$ is in bijection with $\mathfrak{D}/\sim_{\rm{R}}$.

Example 2**.**

The Reidemeister theorem says that constructing an invariant of links is equivalent to defining a function $\mathrm{I}:\mathfrak{D}\longrightarrow\mathrm{Set}$, such that it takes the same values on diagrams that differ in R1, R2 and/or R3.

Definition 8**.**

We say that $\mathrm{I}$ is an invariant of regular isotopy if the values of $\mathrm{I}$ does not change on unoriented links that are equal, up to the moves $R2$ and $R3$.

Proposition 3**.**

$w$* is an invariant of regular isotopy.*

Proof.

To check that $w$ agrees with the oriented move R2, it is enough to observe that introducing any orientation to , it turns out that the sum of the crossing signs is [math]; the same happens with . Also, it is easy to see that by introducing any orientation to one of the crossing configurations of R3, the sum of the signs of the three crossings in it doesn’t change after applying the move R3. ∎

The following lemma gives a recipe, due to L. Kauffman, to produce invariants of oriented links. This recipe was applied to define the Kauffman and Dubrovnik polynomials.

Lemma 1** ([18, Lemma 2.1]).**

Let $\mathfrak{R}$ be a ring and $a$ an invertible element of $\mathfrak{R}$. Suppose that $f$ is a regular isotopy invariant of unoriented diagram links taking values in $\mathfrak{R}$, such that:

[TABLE]

Then, the function $F$ defined as

[TABLE]

is an invariant for oriented links, where $D$ is a diagram for the oriented link $L$.

3. Bracket polynomial

Let $D$ be the diagram of a link. We will attach to $D$ a polynomial in the variables $A$, $B$ and $z$ by using the following inductive process:

1. (1)

We consider a crossing of $D$ to assign the variables $A$ and $B$:

That is, we assign the variable $A$ to the region obtained by sweeping the continuous arc in counterclockwise and the variable $B$ to the remaining regions. 2. (2)

We smooth the crossing (1) by replacing it with the following two configurations:

We obtain thus two links, say $D_{1}$ and $D_{2}$, to which we assign the marks $A$ and $B$ respectively. $D_{1}$ and $D_{2}$ have both a crossing number less than $D$. 3. (3)

We choose now a crossing in $D_{1}$ and $D_{2}$ and we apply again the process (2); then we obtain 4 diagrams with marks $A$ and $B$. We apply again this process until we have only links without crossing.

Observe that if the original diagram has $n$–crossings, by applying (1)–(3) we obtain finally $2^{n}$ diagrams, without crossings, having marks $A$ and $B$. We call these $2^{n}$ diagrams (with their marks) the states of $D$, the set of states of $D$ is denoted by $\mathrm{St}(D)$.

Example 3**.**

Below the procedure (1)–(3) described above for the Hopf link of Fig. 1.

Given $S\in\mathrm{St}(D)$, we denote by $|S|$ the number of components of $S$ and by $\langle D,S\rangle$ the product (commutative) of the marks appearing in $S$.

Definition 9**.**

The Bracket polynomial of an unoriented diagram $D$, denoted by $\langle D\rangle$, is the polynomial in $\mathbb{Z}[A,B,z]$ defined by

[TABLE]

Example 4**.**

For the Hopf link, denoted by $H$, we have

[TABLE]

Thus, in this example it seems that $|S|$ in the number of components $-1$, then

[TABLE]

Proposition 4**.**

1. (1)

$\left\langle\bigcirc\sqcup D\right\rangle=z\left\langle D\right\rangle$, 2. (2)

\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{PositiveCrossing.pdf}\,\right\rangle=B\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{TanlgeSmothingCrossing.pdf}\,\right\rangle+A\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{IdentitySmothingCrossing.pdf}\,\right\rangle, 3. (3)

\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{NegativeCrossing.pdf}\,\right\rangle=A\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{TanlgeSmothingCrossing.pdf}\,\right\rangle+B\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{IdentitySmothingCrossing.pdf}\,\right\rangle.

Proof.

See [19, Proposition 3.2]. ∎

We are going now to study the behaviour of the Bracket polynomial under the Reidemeister moves.

Lemma 2**.**

1. (1)

\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{SmothingR2.pdf}\,\right\rangle=AB\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{IdentitySmothingCrossing.pdf}\,\right\rangle+(A^{2}+B^{2}+ABz)\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{TanlgeSmothingCrossing.pdf}\,\right\rangle, 2. (2)

\left\langle\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler1.pdf}\right\rangle=(Az+B)\left\langle\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler0.pdf}\right\rangle, 3. (3)

\left\langle\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler2.pdf}\right\rangle=(Az+B)\left\langle\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler0.pdf}\right\rangle.

Proof.

See [19, Proposition 3.3]. ∎

According to (1) Lemma 2, in order to obtain that the Bracket polynomial respects the move R2 we must have: $AB=1$ and $ABz+A^{2}+B^{2}=0$, or equivalently:

[TABLE]

Proposition 5**.**

Under the conditions of (1), the Bracket polynomial agrees with the Reidemeister move R3.

Proof.

See [19, Corollary 3.4]. ∎

Proposition 6**.**

Under the conditions of (1), we have:

1. (1)

\left\langle\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler1.pdf}\right\rangle=-A^{3}\left\langle\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler0.pdf}\right\rangle, 2. (2)

\left\langle\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler2.pdf}\right\rangle=-A^{-3}\left\langle\includegraphics[scale={0.2},trim=0.0pt 11.38092pt 0.0pt 0.0pt]{Curler0.pdf}\right\rangle.

Proof.

See [19, Corollary 3.4]. ∎

Denote by $\langle\hskip 14.22636pt\rangle$ the function

[TABLE]

This function is called function Bracket polynomial and is characterized in the following theorem.

Theorem 2**.**

The function Bracket polynomial $\langle\quad\rangle$ is the unique function that satisfies:

1. (1)

$\langle\bigcirc\rangle=1$, 2. (2)

$\langle\bigcirc\sqcup D\rangle=-(A^{-2}+A^{2})\langle\bigcirc\rangle$, 3. (3)

\left\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{PositiveCrossing.pdf}\,\right\rangle=A\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{IdentitySmothingCrossing.pdf}\,\rangle+A^{-1}\langle\,\includegraphics[scale={0.12},trim=0.0pt 22.76228pt 0.0pt 0.0pt]{TangleSmothingCrossing.pdf}\,\rangle.

3.1.

Let $D$ be a diagram of the oriented link $L$, we define $f(L)$ as follows:

[TABLE]

Then, thanks to the Lemma 1, we have the following theorem.

Theorem 3** ([18, Proposition 2.5 ]).**

The map $f:L\mapsto f(L)$ is an invariant of oriented links.

Remark 1**.**

By making $A=t^{-1/4}$, we have that $f(L)$ becomes the seminal Jones polynomial of oriented links. It is a very interesting point, that such an important modern mathematical object can be constructed in such a simple way.

The invariant $f$ is useful to detect if a link is equivalent to that obtained by reflection, also called mirror image. Given a link $L$, its mirror image is the link $L^{\ast}$ obtained by reflection of $L$ in a plane. Notice that the diagrams of $L$ are in bijection with the diagrams of $L^{\ast}$: a diagram $D$ of $L$ determines the diagram $D^{\ast}$ of $L^{\ast}$ obtained by exchanging the positive with the negative crossing in $D$.

Definition 10**.**

If $L$ and $L^{*}$ are ambient isotopic we say that the links are amphicheiral, otherwise we say that the links are cheiral.

Example 5**.**

In the first row of the figure below we have, respectively, the Hopf link, the trefoil and the figure–eight. In the second row their respective reflected. The Hopf link and the figure–eight are amphicheiral and the trefoil is cheiral.

Proposition 7**.**

Let $L$ an oriented link and $D$ a diagram of $L$, we have:

1. (1)

$\left\langle|D^{\ast}|\right\rangle=\left\langle|D|\right\rangle$, 2. (2)

$f(L^{\ast})=f(L)$.

4. Links via braids

Another way to study knot theory is through braids. Braids were introduced by E. Artin and the equivalence between braids and knots is due to two theorems: the Alexander and the Markov theorems. This section is a necessary compilation, for this exposition, on the equivalence between knot theory and braid theory.

Throughout these notes we denote by $\mathtt{S}_{n}$ the symmetric group on $n$ symbols and we denote by $\mathtt{s}_{i}$ the elementary transposition $(i,i+1)$. Recall that the Coxeter presentation of $\mathtt{S}_{n}$, for $n>1$, is that with generators $\mathtt{s}_{1},\ldots,\mathtt{s}_{n-1}$ and the following relations:

[TABLE]

4.1.

Let $P_{1},\ldots,P_{n}$ be points in a plane and $Q_{1},\ldots,Q_{n}$ points in another plane parallel to the first. A $n$–geometrical braid is a collection of $n$ arcs $a_{1},\ldots,a_{n}$ connecting the initial points $P_{i}$’s with the ending points $Q_{w(i)}$, where $w\in\mathtt{S}_{n}$, such that:

1. (1)

for every different $i$ and $j$ the arcs $a_{i}$ and $a_{j}$ are disjoint, 2. (2)

every plane parallel to the plane containing the points $P_{i}$’s meets each arc in only one point.

Geometrically, the arcs cannot be in the following situations:

Define $\mathfrak{B}_{n}$ as the set formed by the $n$–geometrical braids. We define on $\mathfrak{B}_{n}$ the equivalence relation, denoted by $\approx$, given by continues deformation. That is, two geometrical braids $\alpha$ and $\beta$ are equivalents if there exists a family of geometrical braids $\{\gamma_{t}\}_{t\in\left[0,1\right]}$ such that $\gamma_{0}=\alpha$ and $\gamma_{1}=\beta$.

As in knot theory we can translate, equivalently, the geometrical braids to diagrams of them. More precisely, a diagram of a geometrical braid is the image of a generic projection of the braid in a plane. Thus, we have the analogous of Reidemeister theorem for braids.

Theorem 4**.**

Two geometrical braids are equivalent by $\approx$ if and only if their diagrams are equivalents, that is, every diagram of one of them can be transformed in a diagram of the other, by using a finite number of times the following replacement:

Notation 2**.**

We shall use also $\mathfrak{B}_{n}$ to denote the set of diagrams of $n$–geometrical braids and $\approx$ to denote the equivalence of diagrams of braids.

Now, we can define the ‘product by concatenation’ between $n$–geometrical braids; more precisely, given $\alpha$ and $\beta$ in $B_{n}$ we define by $\alpha\beta\in\mathfrak{B}_{n}$ as that one defined by rescaling the result of the $n$–geometrical braid obtained by identifying the ending points of $\alpha$ with the initial points of $\beta$.

Lemma 3**.**

For $\alpha$, $\alpha^{\prime}$, $\beta$ and $\beta^{\prime}$ in $\mathfrak{B}_{n}$ such that $\alpha\approx\alpha^{\prime}$ and $\beta\approx\beta^{\prime}$, we have $\alpha\beta\approx\alpha^{\prime}\beta^{\prime}$.

Proof.

We have $\alpha\beta\approx\alpha^{\prime}\beta$ and also $\alpha^{\prime}\beta\approx\alpha^{\prime}\beta^{\prime}$, then $\alpha\beta\approx\alpha^{\prime}\beta^{\prime}$. ∎

Denote by $B_{n}$ the set of equivalence class of $\mathfrak{B}_{n}$ relative to $\approx$; thus the elements of $B_{n}$ are the equivalence classes $[\alpha]$ of $\alpha\in\mathfrak{B}_{n}$. The Lemma 3 allows to pass the product by concatenation of $\mathfrak{B}_{n}$ to $B_{n}$:

[TABLE]

Theorem 5**.**

$B_{n}$* is a group with the product by concatenation.*

From now on we denote $[\alpha]$ simply by $\alpha$. Thus, observe that the identity can be pictured by:

and the inverse $\alpha^{-1}$ of $\alpha$ is obtained by a reflection of $\alpha$:

Remark 2**.**

Observe that $B_{1}$ is the trivial group and $B_{2}$ is the group of $\mathbb{Z}$.

For $1\leq i\leq n-1$, denote by $\sigma_{i}$’s the following elementary braid:

Theorem 6** (Artin).**

For $n>1$, $B_{n}$ can be presented by generators $\sigma_{1},\ldots,\sigma_{n-1}$ and the following relations:

[TABLE]

An immediate consequence of the above is that we have the following epimorphism:

[TABLE]

Notice that for every $n$ we have a natural monomorphism $\iota_{n}$ from $B_{n}$ in $B_{n+1}$, where for every braid $\sigma\in B_{n}$, $\iota_{n}(\sigma)$ is the braid of $B_{n+1}$ coinciding with $\sigma$ up to $n^{th}$ strand and having one more strand with no crossing with the preceding strand.

Notation 3**.**

We denote by $B_{\infty}$ the group obtained as the inductive limit of $\{(B_{n},\iota_{n})\}_{n\in\mathbb{N}}$.

4.2.

Given $\alpha\in B_{n}$, the identification of the initial points with the end points of $\alpha$ determines a diagram of oriented links, which is denoted by $\widehat{\alpha}$; this process of identification is known as the closure of a braid. Thus we have the ‘function closure’

[TABLE]

The proof that the function closure is epijective is due to Alexander; we will outline this proof, since it gives an efficient method to compute the preimage, by $\widehat{\,\,}$, of a given link.

Theorem 7** (Alexander, 1923).**

Every oriented link is the closure of a braid.

Proof.

The sketch of the proof is as follows. Suppose we have a diagram $D$ of an oriented link $L$.

1. (1)

We fix a point $O$ in the plane not lying on any arc of $D$, such that a point moving along each component of the link is always seen from $O$ going counterclockwise (or clockwise). Alexander proved that such a point $O$ always exists in the isotopy class of the link diagram of $D$. 2. (2)

We divide the diagram in sectors, by rays starting from $O$, with the condition that each sector contains only one crossing. 3. (3)

Finally, we open the diagram along one of the rays obtaining a braid whose closure is the diagram $D$.

∎

Example 6**.**

Fig. 12 shows that the left trefoil is the closure of the braid $\sigma_{1}^{3}$.

In order to describe the links through braids we need to know when the closure of two braids yields the same link. In fact the map closure is not injective; for instance the braids $1_{B_{n}}$ and $\sigma_{n-1}\in B_{n}$ yield the same link. The answer to which braids in $B_{\infty}$ yield the same link is due to Markov.

Denote by $\sim_{M}$ the equivalence relation on $B_{\infty}$ generated by the following replacements (also called moves):

1. (1)

M1: $\alpha\beta$ can be replaced by $\beta\alpha$ (commutation), 2. (2)

M2: $\alpha$ can be replaced by $\alpha\sigma_{n}$ or by $\alpha\sigma_{n}^{-1}$ (stabilization),

where $\alpha$ and $\beta$ are in $B_{n}$.

The relation $\sim_{M}$ defines in fact an equivalence relation in $B_{\infty}$. Two elements in the same $\sim_{M}$–class are called Markov equivalent.

Theorem 8** (Markov).**

For $\alpha$ and $\beta$ in $B_{\infty}$, we have: $\widehat{\alpha}$ and $\widehat{\beta}$ are ambient isotopic links if and only if $\alpha$ and $\beta$ are Markov equivalent.

From the Theorems 7 and 8, it follows that:

Corollary 1**.**

There is a bijection between $B_{\infty}/\sim_{M}$ and $\mathfrak{L}/\sim$ defined through the mapping $\alpha\mapsto\widehat{\alpha}$.

Remark 3**.**

The Markov theorem says that constructing an invariant for links is equivalent to defining a map $\mathrm{I}:B_{\infty}\longrightarrow\mathrm{Set}$, such that for all $\alpha$ and $\beta$ in $B_{n}$, agrees with the replacements of Markov M1 and M2, that is, $\mathrm{I}$ satisfies:

1. (1)

$\mathrm{I}(\alpha\beta)=\mathrm{I}(\beta\alpha)$, 2. (2)

$\mathrm{I}(\alpha\sigma_{n})=\mathrm{I}(\alpha)=\mathrm{I}(\alpha\sigma_{n}^{-1})$.

5. Hecke algebra

Let $\mathbb{F}$ be a field, from now on the denomination $\mathbb{F}$–algebra mean an associative unital, with unity 1, algebra over the field $\mathbb{F}$; thus we can regard $\mathbb{F}$ as a subalgebra of the center of the algebra.

Given a group $G$, we denote by $\mathbb{F}G$ the $\mathbb{F}$–algebra known as the group algebra of $G$ over $\mathbb{F}$. Recall that the set $G$ is a linear basis for $\mathbb{F}G$, regarded as a $\mathbb{F}$–vector space. Also recall that if $G$ has a presentation $\langle X;R\rangle$, then the $\mathbb{F}$–algebra $\mathbb{F}G$ can be presented by generators $X$ and the relations in $R$.

5.1.

Let $\mathsf{u}\,$ be an indeterminate in $\mathbb{C}$ and set $\mathbb{K}$ the field of the rational functions $\mathbb{C}(\mathsf{u}\,)$. For $n\in\mathbb{N}$, the Hecke algebra, denoted by $\mathrm{H}_{n}(\mathsf{u}\,)$ or simply $\mathrm{H}_{n}$, is defined by $\mathrm{H}_{1}=\mathbb{K}$ and for $n>1$ as the $\mathbb{K}$–algebra presented by generators $h_{1},\ldots,h_{n-1}$ and the relations:

[TABLE]

The $h_{i}$’s are invertible, indeed we have

[TABLE]

Remark 4**.**

1. (1)

Taking $\mathsf{u}\,$ as power of a prime number, the Hecke algebra above appears in representation theory as a centralizer of a natural representation associated to the action of the finite general linear group on the variety of flags. This feature of the Hecke algebra will be the key point to construct here certain new invariants of links by using other Hecke algebra or other algebras of type Hecke. Consequently, the next subsection will be devoted to present the Hecke algebras in the context of representation theory capturing in particular the Hecke algebra defined above. 2. (2)

The natural map $\sigma_{i}\mapsto h_{i}$ defines an algebra epimorphism from $\mathbb{K}B_{n}$ to $\mathrm{H}_{n}$. Then, we have that the Hecke algebra $\mathrm{H}_{n}$ is the quotient of $\mathbb{K}B_{n}$ by the two sided ideal generator by

[TABLE] 3. (3)

By taking the specialization $\mathsf{u}\,=1$, the Hecke algebra becomes the group algebra of the symmetric group. For this reason the Hecke algebra is known also as a deformation of the symmetric group.

We construct now a basis of the Hecke algebra; this basis is constructed in an inductive way and is used to prove that the algebra supports a Markov trace. We start with the following lemma.

Lemma 4**.**

In $\mathrm{H}_{n}$ every word in $1,h_{1},\ldots,h_{n-1}$ can be written as a linear combination of words in the $1$ and the $h_{i}$’s such that each of them contains at most one $h_{n-1}$. Hence, $\mathrm{H}_{n}$ is finite dimensional.

Proof.

The proof is by induction on $n$. For $n=2$ the lemma holds since $H_{2}$ is the algebra generated by $1$ and $h_{1}$. Suppose now that the lemma is valid for every $\mathrm{H}_{k}$, with $k<n+1$. We prove the lemma for $n+1$; let $M$ be a word in $1,h_{1},\ldots,h_{n}$ containing two times $h_{n}$, then we can write

[TABLE]

where $M_{i}$’s are words in $1,h_{1},\ldots,h_{n-1}$. But now, using the induction hypothesis we have to consider two situations according to $M_{2}$ contains none or only one $h_{n-1}$. If $M_{2}$ does not contain $h_{n-1}$, we have $M_{1}h_{n}M_{2}h_{n}M_{3}=M_{1}h_{n}^{2}M_{2}M_{3}=M_{1}(\mathsf{u}\,+(\mathsf{u}\,-1)h_{n})M_{2}M_{3}$.; hence $M=\mathsf{u}\,M_{1}M_{2}M_{3}+(\mathsf{u}\,-1)M_{1}h_{n}M_{2}M_{3}$, thus $M$ is reduced as the lemma claims. On the other hand, if $M_{2}$ contains only one $h_{n-1}$, we can write it as $M_{2}=M^{\prime}h_{n-1}M^{\prime\prime}$, where $M^{\prime}$ and $M^{\prime\prime}$ are words in $1,h_{1},\ldots,h_{n-2}$; so $M=M_{1}h_{n}(M^{\prime}h_{n-1}M^{\prime\prime})h_{n}M_{3}$; by applying now (5) and (6) we obtain $M=M_{1}M^{\prime}h_{n-1}h_{n}h_{n-1}M^{\prime\prime}nM_{3}$, then $M$ is as the lemma claims.

In the case that $M$ contains more than two generators $h_{n}$, we reduce two of them using the argument above; so arguing inductively we deduce that $M$ can be written as the lemma is claiming. ∎

In $\mathrm{H}_{n}$, define: $C_{1}=\{1,h_{1}\}$ and $C_{i}=\{1,h_{i}x\,;\,x\in C_{i-1}\}$, for $2\leq i\leq n-1.$

Definition 11**.**

The elements $\mathfrak{n}_{1}\mathfrak{n}_{2}\cdots\mathfrak{n}_{n-1}\in\mathrm{H}_{n}$, with $\mathfrak{n}_{i}\in C_{i}$, are called normal words. This set formed by the normal words will be denoted by $\mathcal{C}_{n}$.

Observe that:

[TABLE]

Theorem 9**.**

The set $\mathcal{C}_{n}$ is a linear basis of $\mathrm{H}_{n}$. In particular, the dimension of $\mathrm{H}_{n}$ is $n!$.

Proof.

We will prove, by induction on $n$, that $\mathcal{C}_{n}$ is a spanning set of $\mathrm{H}_{n}$. For $n=2$ the theorem is clear. Suppose now that the theorem is true for every $k<n$. From Lemma 4 it follows that $\mathrm{H}_{n}$ is linearly spanned by the elements of the form (i) and (ii):

[TABLE]

where $M_{i}$’s are words in $1,h_{1},\ldots,h_{n-2}$. By the induction hypothesis, follows that $M_{i}$’s are linear combination of elements of $\mathcal{C}_{n-1}$, so we can suppose that $M_{i}$’s belong to $\mathcal{C}_{n-1}$. Thus, it is enough to prove that the elements of (ii) are a linear combination of the elements of $\mathcal{C}_{n}$ (notice that $M_{0}\in\mathcal{C}_{n}$). Set $M_{2}=\mathfrak{n}_{1}\mathfrak{n}_{2}\cdots\mathfrak{n}_{n-2}$, where $\mathfrak{n}_{i}\in C_{i}$; we have

[TABLE]

From the induction hypothesis $M_{1}\mathfrak{n}_{1}\mathfrak{n}_{2}\cdots$ is a linear combination of elements of $\mathcal{C}_{n-1}$ and notice that $h_{n-1}\mathfrak{n}_{n-2}\in\mathcal{C}_{n-1}$. So, having in mind the second observation of (9) we deduce that the elements in (ii) belong to the linear span of $\mathcal{C}_{n}$.

Linear independency (LATER)

∎

The above theorem and (9) imply that we have a natural tower of algebras

[TABLE]

We will denote by $\mathrm{H}_{\infty}$ the inductive limit associated to this tower.

Notice that the inclusion of algebras $\mathrm{H}_{n}\subset\mathrm{H}_{n+1}$ allows to obtain a structure of $(\mathrm{H}_{n},\mathrm{H}_{n})$–bimodule for $\mathrm{H}_{n+1}$; further, we can consider the $(\mathrm{H}_{n},\mathrm{H}_{n})$–bimodule $\mathrm{H}_{n}\otimes_{\mathrm{H}_{n-1}}\mathrm{H}_{n}$ since $\mathrm{H}_{n}$ is a $(\mathrm{H}_{n},\mathrm{H}_{n-1})$–bimodule and also $(\mathrm{H}_{n-1},\mathrm{H}_{n})$–bimodule.

Proposition 8**.**

The dimension of the $\mathbb{K}$–vector space $\mathrm{H}_{n}\otimes_{\mathrm{H}_{n-1}}\mathrm{H}_{n}$ is at most $n!n$.

Proof.

Theorem 9 implies that every element in $\mathrm{H}_{n}\otimes_{\mathrm{H}_{n-1}}\mathrm{H}_{n}$ is a $\mathbb{K}$–linear combination of elements of the form $a\otimes b$, where $a,b\in\mathcal{C}_{n}$. Now, we have two possibilities: $b$ is in $\mathcal{C}_{n-1}$ or $b=xh_{n-1}\cdots h_{i}$, with $x\in\mathcal{C}_{n-1}$ (see (9)). Now, if $b\in\mathcal{C}_{n-1}$, we have $a\otimes b=ab\otimes 1$ and in the other case we can write $a\otimes b=ax\otimes h_{n-1}\cdots h_{i}$. Therefore, every element of $\mathrm{H}_{n}\otimes_{\mathrm{H}_{n-1}}\mathrm{H}_{n}$ is a linear combination of elements of the form $a\otimes 1$ and $a\otimes h_{n-1}\cdots h_{i}$, where $a\in\mathcal{C}_{n}$ and $1\leq i\leq n-1$. Hence the proof follows. ∎

The following lemma will be used in the next section.

Lemma 5**.**

The map $\phi:\mathrm{H}_{n}\oplus\mathrm{H}_{n}\otimes_{\mathrm{H}_{n-1}}\mathrm{H}_{n}\longrightarrow\mathrm{H}_{n+1}$, defined by

[TABLE]

is an isomorphism of $(\mathrm{H}_{n},\mathrm{H}_{n})$–bimodules.

Proof.

∎

5.2.

From now on $\mathsf{z}\,$ denotes a new variable commuting with $\mathsf{u}\,$.

Theorem 10** (Ocneanu).**

There exists a unique family of linear maps $\tau=\{\tau_{n}\}_{n\in\mathbb{N}}$, where $\tau_{n}:\mathrm{H}_{n}\longrightarrow\mathbb{K}(\mathsf{z}\,)$ is defined inductively by the following rules:

1. (1)

$\tau_{1}(1)=1$, 2. (2)

$\tau_{n}(ab)=\tau_{n}(ba)$, 3. (3)

$\tau_{n+1}(ah_{n}b)=\mathsf{z}\,\tau_{n}(ab)$,

where $a,b\in\mathrm{H}_{n}$.

Proof.

The definition of $\tau_{n}$ is based on the homomorphism $\phi$ of Lemma 5. For $n=1$, $\tau_{1}$ is defined as the identity on $\mathbb{K}$. Now, given $a\in\mathrm{H}_{n+1}$, the Lemma 5 says that, we can write uniquely, $a=\phi(x+\sum_{i}y_{i}\otimes z_{i})$, then we define $\tau_{n+1}$ by

[TABLE]

For instance $\tau_{2}(h_{1})=\mathsf{z}\,$, since $\phi(0+1\otimes 1)=h_{1}$.

For every $a,b\in\mathrm{H}_{n}$, we have $\phi(a\otimes b)=ah_{n}b$, then the rule (3) is satisfied.

We are going to check now the rule (2) BLABLA… ∎

6. The Homflypt polynomial

We show now the construction of the Homflypt polynomial due to V. Jones. This Jones construction gives a method (or Jones recipe) which is our main tool to construct invariants.

6.1.

We have a natural representation from $B_{n}$ in $\mathrm{H}_{n}$, defined by mapping $\sigma_{i}$ in $h_{i}$, however we need to consider a slightly more general representation, denoted by $\pi_{\theta}$ and defined by mapping $\sigma_{i}$ in $\theta h_{i}$, where $\theta$ is a scalar factor; the reason for taking this factor $\theta$ will be clear soon. Now, composing $\pi_{\theta}$ with the Markov trace $\tau_{n}$, we have the maps

[TABLE]

This family of maps yields a unique map $\mathrm{X}_{\theta}$ from $B_{\infty}$ to $\mathbb{K}(\mathsf{z}\,)$. Now, according to Remark 3, the function $\mathrm{X}_{\theta}$ defines an invariant of links, if it agrees with the replacements of Markov M1 and M2, or equivalently, for every $n$ the function $\tau_{n}\circ\pi_{\theta}$ agrees with M1 and M2. The fact that $\pi_{\theta}$ is a homomorphism and the rule (2) of the Ocneanu trace implies that, for every $n$ and $\theta$ the maps $\tau_{n}\circ\pi_{\theta}$ agree with the Markov replacement M1. For the replacement M2 we note that, in particular, the maps $\tau_{n}\circ\pi_{\theta}$ must satisfy:

[TABLE]

We have $(\tau_{n}\circ\pi_{\theta})(\sigma_{n})=\theta\mathsf{z}\,$ and from (8), we get:

[TABLE]

Then we derive that $\theta\mathsf{z}\,=\theta^{-1}((\mathsf{u}\,^{-1}-1)+\mathsf{u}\,^{-1}\mathsf{z}\,)$, from where the factor $\theta$ satisfies:

[TABLE]

So, extending the ground field $\mathbb{K}(\mathsf{z}\,)$ to $\mathbb{K}(\mathsf{z}\,,\sqrt{\lambda})=\mathbb{C}(\mathsf{z}\,,\sqrt{\lambda})$, the family of maps $\{\tau_{n}\circ\pi_{\sqrt{\lambda}}\}_{n\in\mathbb{N}}$ agrees with the Markov replacements M1 and M2. However, it is desirable that the invariant takes the values 1 on the unknot, that is, we want $(\tau_{n}\circ\pi_{\sqrt{\lambda}})(\sigma)=1$, for every $n$ and every $\sigma$ whose closure is the unknot; notice that the unknot is the closure of the braid $\sigma_{1}\sigma_{2}\cdots\cdots\sigma_{n-1}$, for all $n$. So, we have:

[TABLE]

Then, we need to normalize $\tau_{n}\circ\pi_{\sqrt{\lambda}}$ by $(\sqrt{\lambda}\,\mathsf{z}\,)^{n-1}$.

Theorem 11**.**

Let $L$ be an oriented link obtained as the closure of the braid $\alpha\in B_{n}$. We define $\mathrm{X}:\mathfrak{L}\longrightarrow\mathbb{C}(\mathsf{z}\,,\sqrt{\lambda})$, by

[TABLE]

Then $\mathrm{X}$ is an invariant of ambient isotopy for oriented links.

Proof.

Thanks to Corollary 1, we need only to check that:

[TABLE]

where $\alpha,\beta\in B_{n}$. Clearly (i) holds. We are going to check now only the second equality of (ii), the checking of first equality is left to the reader. We have:

[TABLE]

Then, by using now the rule (3) of the Ocneanu trace, we get $(\tau_{n+1}\circ\pi_{\sqrt{\lambda}})(\alpha\sigma_{n}^{-1})=\tau_{n+1}(\pi_{\lambda}(\alpha\sigma_{n}))$; hence $\mathrm{X}(\widehat{\alpha\sigma_{n}})=\mathrm{X}(\widehat{\alpha\sigma_{n}^{-1}})$. ∎

Example 7**.**

Trefoil

6.2.

The Homflypt polynomial has a definition by skein rules. This definition is useful to calculate it and also to study its relations with other invariants such as the Jones polynomial and the Alexander polynomial.

Denote by $L_{+},L_{-}$ and $L_{0}$ three oriented links with, respectively, diagrams $D_{+},D_{-}$ and $D_{0}$, which are different only inside a disk, where they are respectively placed, as shows Fig 13.

The links $L_{+},L_{-}$ and $L_{0}$ are called a Conway triple; notice that in terms of braids, they can be written as:

[TABLE]

for some braid $w$ in $B_{n}$.

Keeping the notation above, we are going to compute the Homflypt polynomial on a Conway triple. First, we have,

[TABLE]

by considering now (8), we obtain:

[TABLE]

Also a direct computation yields:

[TABLE]

[TABLE]

From these last three equations we obtain the following proposition.

Proposition 9**.**

The invariant $\mathrm{X}$ satisfies the following skein relation.

[TABLE]

Theorem 12**.**

There exists a unique function

[TABLE]

such that:

1. (1)

$\mathrm{P}(\bigcirc)=1$, 2. (2)

$\mathsf{t}\,^{-1}\mathrm{P}(L_{+})-\mathsf{t}\,\mathrm{P}(L_{-})=\mathsf{x}\,\mathrm{P}(L_{0})$.

Proof.

After a suitable change of variables, $\mathrm{X}$ satisfies the defining properties of $\mathrm{P}$, so it remains to prove the uniqueness of $\mathrm{P}$. ∎

Remark 5**.**

Making $\mathsf{x}\,=\mathsf{t}\,$ the polynomial $\mathrm{P}$ becomes the Jones polynomial and making $\mathsf{x}\,=\mathsf{t}\,^{-1}$ the polynomial $\mathrm{P}$ becomes the Alexander polynomial.

7. Hecke algebras in representation theory

7.1.

Let $G$ be a finite group. A complex representation of $G$ is a pair $(V,\rho)$, where $V$ is a finite dimensional space over $\mathbb{C}$ and $\rho$ is a homomorphism group from $G$ to $\mathrm{GL}(V)$. A subspace $U$ of $V$ is called $G$–stable if $\rho_{g}(U)=U$; the representation is called irreducible if the unique $G$–stable subspaces are trivial. It is well known that every complex representation can be decomposed as a direct sum of stable subspaces, see [23]. A fundamental problem in representation theory is: given a representation of $G$, write out such a decomposition. A powerful tool that helps the understanding of the decomposition of a representation is its centralizer, that is, the algebra formed by the endomorphisms of $V$ commuting with $\rho_{g}$, for all $g\in G$. The centralizer of the representation $(V,\rho)$ is denoted by $\mathrm{End}_{G}(V)$.

Now, given a subgroup $H$ of $G$ we can construct the so called natural or induced representation, of $G$ relative a $H$. More precisely, this representation can be made explicit as $(\mathrm{Ind}_{H}^{G}1,\rho)$, where:

[TABLE]

and

[TABLE]

The centralizer of this representation is known as the Hecke algebra of $G$ with respect to $H$ and is usually denoted by $\mathcal{H}(G,H)$.

A fundamental piece in the theory of finite group of Lie type is the representation theory of the finite general linear group $G=\mathrm{GL}_{n}(\mathbb{F}_{q})$, where $\mathbb{F}_{q}$ denotes the finite field with $q$ elements; and an important family of irreducible representation of $G$ appears in the decomposition of $\mathrm{Ind}_{B}^{G}1$, where $B$ is the subgroup of $G$ formed by the upper triangular matrices. The centralizer $\mathcal{H}(G,B)$ of this representation was studied by N. Iwahori in the sixties, in a more general context for $G$: the finite Chevalley groups. In the case $G$ is as above, that is, the finite general group, the algebra $\mathcal{H}(G,B)$ corresponds to those of type $A$ in the classification of Chevalley groups and the Iwahori theorem for $\mathcal{H}(G,B)$ is the following.

Theorem 13** (N. Iwahori, [12]).**

The Hecke algebra $\mathcal{H}(G,B)$ can be presented, as $\mathbb{C}$–algebra, by generators $\phi_{1},\phi_{2},\ldots,\phi_{n-1}$ and the following relations:

1. (1)

$\phi_{i}^{2}=q+(q-1)\phi_{i}$, for all $i$, 2. (2)

$\phi_{i}\phi_{j}=\phi_{j}\phi_{i}$, for $|i-j|>1$, 3. (3)

$\phi_{i}\phi_{j}\phi_{i}=\phi_{j}\phi_{j}\phi_{i}$, for $|i-j|=1$.

Now, we want to explain a little bit how the $\phi_{i}$’s look and work. To better explain we shall work in a more general setting. Let $G$ be a finite group and set $X$ a finite $G$–space. Define $\mathbb{C}(X)$ the $\mathbb{C}$–vector space of all complex valued functions on $X$. Then, we have the representation $(\mathbb{C}(X),\rho)$ of $G$, where $(\rho_{g}f)(x):=f(g^{-1}x)$. Consider now the $G$–space $X\times X$ with action $g(x,y)=(gx,gy)$ and the vector space $\mathcal{K}(X):=\mathbb{C}(\mathrm{X}\times X)$, this vector space results to be an algebra with the convolution product:

[TABLE]

Theorem 14**.**

The function $\Phi$ from $\mathcal{K}(X)$ to $\mathrm{End}(\mathbb{C}(X))$ is an algebra isomorphism, where $\Phi:I\mapsto\Phi_{I}$, is defined by

[TABLE]

Proof.

∎

Recall that $\mathrm{End}_{G}(\mathbb{C}(X))$ denote the centralizer of the representation $(\mathbb{C}(X),\rho)$ and denote now by $\mathcal{K}_{G}(X)$ the subalgebra of all $I\in\mathcal{K}(X)$, that are $G$–invariant, that is, $I(gx,gy)=I(x,y)$, for $(x,y)\in X\times X$ and $g\in G$.

Theorem 15**.**

The algebras $\mathrm{End}_{G}(\mathbb{C}(X))$ and $\mathcal{K}_{G}(X)$ are isomorphic.

Proof.

∎

Let $O_{1},\ldots,O_{m}$ be the orbits of the $G$–space $X\times X$, hence the canonical basis of $\mathcal{K}_{G}(X)$ is $\{K_{1},\ldots,K_{m}\}$, where the $K_{i}$’s are the function delta of Dirac,

[TABLE]

Corollary 2**.**

A basis of $\mathrm{End}_{G}(\mathbb{C}(X))$ is the set $\{\phi_{1},\ldots,\phi_{m}\}$, where

[TABLE]

Hence, the dimension of $\mathrm{End}_{G}(\mathbb{C}(X))$ is the number of $G$–orbits of $X\times X$.

Proof.

Notice that $\phi_{i}:=\Phi(K_{i})$, so the proof is clear. ∎

We shall finish the section by showing the basis $\phi_{i}$ in the case $G=\mathrm{GL}_{n}(\mathbb{F}_{q})$ and $X$ the variety of flags in $\mathbb{F}_{q}^{n}$.

Remark 6**.**

In [14, p. 336 ] Jones raised the question whether his method of construction of the Homflypt polynomial can be used for Hecke algebras of other type. In this light it is natural to ask also whether the Jones method can be used still keeping $G=\mathrm{GL}_{n}(\mathbb{F}_{q})$ but changing $B$ for other notable subgroups of $G$. This is what we did to define new invariants for links. More precisely, we change $B$ by his unipotent part, that is, the subgroup formed by the upper unitriangular matrices.

8. Yokonuma–Hecke algebra

In this section we introduce the Yokonuma–Hecke whose origin is in representation theory of finite Chevalley groups; indeed, it appears as a centralizer of a natural representation of a Chevalley group. With the aim to applying this algebra to knot theory, we noted that it is linked, not only to the braid group, but also to the framed braid group. We show an inductive basis, the existence of a Markov trace and the E–system, all that fundamental ingredients to construct our invariants.

8.1.

The Yokonuma–Hecke algebra comes from a centralizer of the permutation representation associated to the finite general linear group $\mathrm{GL}_{n}(\mathbb{F}_{q})$ with respect to the subgroup consisting of the upper unitriangular matrix, cf. Remark 6. This centralizer was studied by T. Yokonuma who obtained a presentation of it, analogous to that found by N. Iwahori for the Hecke algebra, see Theorem 13. However, the presentation of Yokonuma was slightly modified in order to be applied to knot theory; a little modification of this presentation defines the so called Yokounuma–Hecke algebra.

Definition 12**.**

The Yokonuma–Hecke algebra, in short called Y–H algebra and denoted by ${\rm Y}_{d,n}(\mathsf{u}\,)$, is defined as follows: $\mathrm{Y}_{1,1}(\mathsf{u}\,)=\mathbb{K}$, and for $d,n\geq 2$ as the algebra presented by braid generators $g_{1},\ldots,g_{n-1}$ and framing generators $t_{1},\ldots,t_{n}$, subject to the following relations:

[TABLE]

where $\mathtt{s}_{i}=(i,i+1)$ and

[TABLE]

Whenever the variable $\mathsf{u}\,$ is irrelevant, we shall denote ${\rm Y}_{d,n}(\mathsf{u}\,)$ simply by ${\rm Y}_{d,n}$.

Remark 7**.**

1. (1)

The elements $e_{i}$’s result to be idempotents, i.e. $e_{i}^{2}=e_{i}$ and the $g_{i}$’s are invertible,

[TABLE]

These facts will be frequently used in what follows. 2. (2)

Notice that for $d=1$, the algebra ${\rm Y}_{d,n}$ becomes $\mathrm{H}_{n}$, while for $n=1$, ${\rm Y}_{d,n}$ becomes the group algebra of the cyclic group with $d$ elements. 3. (3)

The mapping $g_{i}\mapsto h_{i}$ and $t_{i}\mapsto 1$, define an homomorphism algebra from ${\rm Y}_{d,n}$ onto $\mathrm{H}_{n}$.

8.2.

We have a natural representation of the braid group in the Y–H algebra since relations (i) and (ii) correspond to the defining relations of the braid group. Moreover, the relations (i)–(iv) correspond to the defining relations of the framing group. To be more precise, the framed braid group, denoted by $\mathcal{F}_{n}$, is the semi–direct product $\mathbb{Z}^{n}\rtimes B_{n}$, where the action that defines the semidirect product is through the homomorphism $\mathsf{p}$ from $B_{n}$ onto $\mathtt{S}_{n}$ (see (4)), that is:

[TABLE]

In multiplicative notation the group $\mathbb{Z}$ has the presentation $\langle t\,;\,-\rangle$ and the direct product $\mathbb{Z}^{n}$ has the presentation \langle t_{1},\ldots,t_{n}\;;\;t_{i}t_{j}=t_{j}t_{i}\quad\text{for all i,j}\rangle. Consequently, $\mathcal{F}_{n}$ can be presented by (braids) generators $\sigma_{1},\ldots,\sigma_{n-1}$ together with the (framing) generators $t_{1},\ldots,t_{n}$ subject to the relations (2), ( 3), $t_{i}t_{j}=t_{j}t_{i}$, for all $i,j$ and the relations:

[TABLE]

Now, because $\mathcal{F}_{n}$ is a semidirect product, we have that every element in it can be written in the form $t_{1}^{a_{1}}\cdots t_{n}^{a_{n}}\sigma$, where $\sigma\in B_{n}$ and the $a_{i}$’s are integers called the framing. Further, observe that:

[TABLE]

In diagrams, the element $t_{1}^{a_{1}}\cdots t_{n}^{a_{n}}\sigma$ can be represented by the usual diagram braid for $\sigma$ together with $a_{1},\ldots,a_{n}$ written at the top of the braid: $a_{i}$ is placed where the strand $i$ starts. For instance, Fig. 14 represents the framed braid $t_{1}^{a}t_{2}^{b}t_{3}^{c}t_{4}^{d}\sigma_{2}\sigma_{3}^{-1}\sigma_{1}^{-1}\sigma_{2}^{2}\sigma_{3}$.

In terms of diagrams, the formula (21) is translated as follows: we place the diagram of the braid $\tau$ under the diagram of the braid $\sigma$ and the framing $b_{i}$ travels along the strand up to the top of the diagram of $\sigma\tau$, so that the framings of the product are $(a_{i}+b_{\sigma(i)})$’s. For instance, for $\sigma=\sigma_{3}\sigma_{1}^{-1}\sigma_{2}^{-1}\sigma_{3}^{-1}$ and $\tau=\sigma_{1}\sigma_{3}^{-1}$, the product $(t_{1}^{a}t_{2}^{b}t_{3}^{c}t_{4}^{d}\sigma)(t_{1}^{x}t_{2}^{y}t_{3}^{z}t_{4}^{w}\tau)$ in terms of diagrams is showed in Fig. 15.

Finally, relations (i)–(v) correspond to the defining relations of the framing module $d$. The $d$–modular framed braid group $\mathcal{F}_{d,n}$ is the semidirect product $\mathbb{Z}/d\mathbb{Z}\rtimes B_{n}$; in other terms, it is the group obtained by imposing the relation $t_{i}^{d}=1$ to the above presentation of $\mathcal{F}_{n}$.

8.3.

As for the Hecke algebra, we can construct an inductive basis for the Y–H algebra. To do that, we define in ${\rm Y}_{d,n}$, the following sets:

[TABLE]

and

[TABLE]

Definition 13**.**

Every element in ${\rm Y}_{d,n}$ of the form $r_{1}\cdots r_{n-1}$, with $r_{i}\in R_{i}$, $0\leq i\leq n-1$ is called normal word. We denote by $\mathcal{R}_{n}$ the set of normal words.

Theorem 16**.**

The set $\mathcal{R}_{n}$ is linear basis for ${\rm Y}_{d,n}$; hence ${\rm Y}_{d,n}$ has dimension $d^{n}n!$.

Observe that every element of $\mathcal{R}_{n}$ has one of the following forms:

[TABLE]

where $\mathfrak{r}\in\mathcal{R}_{n-1}$ and $0\leq a\leq d-1$.

Example 8**.**

For $n=3$, we have:

[TABLE]

and

[TABLE]

Thus, $\mathcal{R}_{1}$ of $\mathrm{Y}_{d,1}$ is $R_{0}$. The basis $\mathcal{R}_{2}$ of $\mathrm{Y}_{d,2}$ is formed by the elements in the form:

[TABLE]

The elements of the basis $\mathcal{R}_{3}$ of $\mathrm{Y}_{d,3}$, are of the form:

[TABLE]

We used the underline to indicate the form of (22) for the elements of $\mathcal{R}_{3}$.

Now, in particular, (22) said that $\mathcal{R}_{n-1}\subset\mathcal{R}_{n}$, for all $n$. Then, for every $d$ we have the following tower of algebras:

[TABLE]

Notation 4**.**

We denote by $\mathrm{Y}_{d,\infty}$ the algebra associated to the tower of algebras above.

8.4.

Set $\mathsf{x}\,_{0}:=1$ and let $\mathsf{x}\,_{1},\ldots\mathsf{x}\,_{d-1}$ be $d-1$ independent parameters commuting among them and with the parameters $\mathsf{z}\,$.

Theorem 17**.**

The algebra $\mathrm{Y}_{d,\infty}$ supports a unique Markov trace, that is, a family $\mathrm{tr}_{d}=\{\mathrm{tr}_{d,n}\}_{n\in\mathbb{N}}$ of linear maps

[TABLE]

defined uniquely by the following rules:

1. (1)

$\mathrm{tr}_{d,n}(1)=1$, 2. (2)

$\mathrm{tr}_{d,n}(ab)=\mathrm{tr}_{d,n}(ba)$, 3. (3)

$\mathrm{tr}_{d,n+1}(ag_{n})=\mathsf{z}\,\mathrm{tr}_{d,n}(a)$, 4. (4)

$\mathrm{tr}_{d,n+1}(at_{n+1}^{k})=\mathsf{x}\,_{k}\mathrm{tr}_{d,n}(a)$,

where $a,b\in{\rm Y}_{d,n}$, $0\leq k\leq d-1$.

Notice that for $d=1$ the trace $\mathrm{tr}$ becomes the Ocneanu trace.

From now on we fix a positive integer $d$, thus we shall write $\mathrm{tr}_{n}$ instead of $\mathrm{tr}_{d,n}$. Moreover, whenever it is not necessary to explicit $n$, we write simply $\mathrm{tr}$ instead of $\mathrm{tr}_{n}$.

We compute some values of $\mathrm{tr}$ that will be used later. First, we compute $\mathrm{tr}(\alpha e_{n}g_{n})$, for every $\alpha\in{\rm Y}_{d,n}$.

Notice that by using the rule of commutation $t_{n}^{s}t_{n+1}^{d-s}g_{n}=t_{n}^{s}g_{n}t_{n}^{d-s}$. Then, for every $\alpha\in{\rm Y}_{d,n}$, we have:

[TABLE]

Thus $\mathrm{tr}(\alpha e_{n}g_{n})=d^{-1}\sum_{s}\mathrm{tr}(\alpha t_{n}^{s}t_{n+1}^{d-s}g_{n})=\mathsf{z}\,d^{-1}\sum_{s}\mathrm{tr}(\alpha)$. Then,

[TABLE]

Now, we compute $\mathrm{tr}(e_{i})$.

[TABLE]

Hence for all $i$ the trace $\mathrm{tr}$ takes the same values on $e_{i}$; we denote these values by $E$. More generally, we define the elements $e_{i}^{(m)}$ as follows,

[TABLE]

where the subindices are regarded as module $d$. Then, denoting $\mathrm{tr}(e_{i}^{(m)})$ by $E^{(m)}$, we have

[TABLE]

Notice that $E^{(0)}=E:=\mathrm{tr}(e_{i})$.

The following lemmas will be useful in the next section and their proofs are a good example to see how $\mathrm{tr}$ works.

Lemma 6**.**

Set $\alpha=wt_{n}^{k}$, with $w\in\mathrm{Y}_{n-1}$. We have

[TABLE]

Proof.

A direct computation shows:

[TABLE]

Then

[TABLE]

so, $\mathrm{tr}(\alpha e_{n}^{(m)})=\mathrm{tr}(w_{n-1})E^{(m+k)}$. Now, $\mathrm{tr}(\alpha)=\mathsf{x}\,_{k}\mathrm{tr}(w_{n-1})$. Therefore, the proof follows. ∎

Lemma 7**.**

Set $\alpha=wg_{n-1}g_{n-2}\cdots g_{i}t_{i}^{k}$, with $w\in\mathrm{Y}_{n-1}$. We have

[TABLE]

where $\alpha^{\prime}=g_{n-2}\cdots g_{i}t_{i}^{k}w$.

Proof.

We have to compute firstly the trace of $A_{s}$,

[TABLE]

We have

[TABLE]

where the second equality is obtained by moving $t_{n-1}^{s}$ to the left. Using first the trace rule (4) and later the trace rule (2), we get

[TABLE]

By using again the trace rule (4), we obtain

[TABLE]

Then,

[TABLE]

∎

9. The $E$–system

The $E$–system is the following non–linear system equation of $(d-1)$ equation in the variable $\mathsf{x}\,_{1},\ldots,\mathsf{x}\,_{d-1}$.

[TABLE]

Example 9**.**

For $d=3$, the $E$–system is:

[TABLE]

For $d=4$, the $E$–system is:

[TABLE]

The $E$–system plays a key role to define the invariants in the next section. The $E$–system was solved by P. Gerardin, see [16, Appendix].

Theorem 18** (P. Gerardin, 2013).**

The solutions of the $E$–system are parametrized by the non–empty subset of the group $\mathbb{Z}/d\mathbb{Z}$. Moreover, given a such subset $S$, the solutions are:

[TABLE]

Remark 8**.**

Let $S$ be a non–empty subset of $\mathbb{Z}/d\mathbb{Z}$. We have:

1. (1)

For $|S|=1$, the solutions of the $E$–system are the $d$–roots of the unity. 2. (2)

If $S$ consists of the coprimes with $d$, the solution of the $E$–system are Ramanujan sums. 3. (3)

By taking the parameters trace $\mathsf{x}\,_{1},\ldots,\mathsf{x}\,_{d-1}$ as a solution of the system, then

[TABLE]

Proposition 10**.**

If the parameters trace $\mathsf{x}\,_{k}$’s are taken as solution of the $E$–system, then for every $\alpha\in{\rm Y}_{d,n}$, we have:

[TABLE]

Proof.

From the linearity of $\mathrm{tr}$, it is enough to consider $\alpha$ in the basis $\mathcal{R}_{n}$. The proof will be done by induction on $n$. For $n=1$, the claim is clear since $\alpha\in\mathrm{Y}_{1}=\mathbb{K}$; suppose now the proposition be true for every $n-1$ and let $\alpha\in\mathcal{R}_{n-1}$. Because (22), we have two cases according to the form of $\alpha$: (i) $\alpha=\mathfrak{r}t_{n}^{a}\quad$ or (ii) $\alpha=\mathfrak{r}g_{n-1}\ldots g_{1}t_{n}^{a}$, where $\mathfrak{r}\in\mathcal{R}_{n-1}$ and $0\leq a\leq d-1$.

For the case (i), from Lemma 6 we have: $\mathrm{tr}(\alpha e_{n})=\left(E^{k}/\mathsf{x}\,_{k}\right)\mathrm{tr}(\alpha)=E\mathrm{tr}(\alpha)$ since the $\mathsf{x}\,_{k}$’s are solution of the $E$–system.

For the case (ii), we use the Lemma 7, so: $\mathrm{tr}(\alpha e_{n})=\mathsf{z}\,\mathrm{tr}(\alpha^{\prime}e_{n-1})$, where $\alpha^{\prime}=g_{n-2}\cdots g_{i}t_{i}^{k}\mathfrak{r}$. From the inductive hyphotesis we have $\mathrm{tr}(\alpha^{\prime}e_{n-1})=\mathsf{z}\,\mathrm{tr}(\alpha^{\prime})\mathrm{tr}(e_{n-1})$; but $\mathsf{z}\,\mathrm{tr}(\alpha^{\prime})=\mathrm{tr}(\alpha)$. Then, $\mathrm{tr}(\alpha e_{n})=\mathrm{tr}(\alpha)\mathrm{tr}(e_{n-1})=\mathrm{tr}(\alpha)\mathrm{tr}(e_{n})$.

∎

10. The invariants $\Delta_{m}$ and $\Theta_{m}$

In this section we define the invariants $\Delta_{m}$ and $\Theta_{m}$ which are constructed using the method due to V. Jones to construct the Homflypt polynomial, see Section 6. Essentially, in the Jones method we use now the Y–H algebra instead of the Hecke algebra and instead of the Ocneanu trace we use the trace $\mathrm{tr}$. Observe that the construction below follows what has been done in Section 6.

10.1.

For $\theta$, we denote by $\pi_{\theta}$ the homomorphism from $B_{n}$ to ${\rm Y}_{d,n}$, such that $\sigma_{i}\mapsto\theta g_{i}$. Since $\pi_{\theta}$ is an homomorphism and the trace rule (2) of $\mathrm{tr}$, it follows that $\mathrm{tr}\circ\pi_{\theta}$ agrees with the Markov replacement M1; thus, it remains to see if $\mathrm{tr}\circ\pi_{\theta}$ agrees with the Markov replacements M2, that is, we need

[TABLE]

Put $\alpha:=\pi_{\theta}(\sigma)$, then we have $(\mathrm{tr}\circ\pi_{\theta})(\sigma\sigma_{n})=\theta\mathrm{tr}(\alpha g_{n})=\mathsf{z}\,\theta\mathrm{tr}(\alpha)$ and $(\mathrm{tr}\circ\pi_{\theta})(\alpha\sigma_{n}^{-1})=\theta^{-1}\mathrm{tr}(\alpha g_{n}^{-1})$. So, the equation (28) is equivalent to:

[TABLE]

Now, applying the formula (19) to $g_{n}^{-1}$ and the linearity of $\mathrm{tr}$, we get:

[TABLE]

Therefore, in order to get the condition on $\theta$ in (29), we need to factorize by $\mathrm{tr}(\alpha)$ in the second member of the equality of (30). Now, observe that $\mathrm{tr}(\alpha g_{n})=\mathsf{z}\,\mathrm{tr}(\alpha)$ and from (24) we get $\mathrm{tr}(\alpha e_{n}g_{n})=\mathsf{z}\,\mathrm{tr}(\alpha)$. Unfortunately, we cannot take out the factor $\mathrm{tr}(\alpha)$ in $\mathrm{tr}(\alpha e_{n})$. However, resorting to Proposition 10, we can do it whenever the $\mathsf{x}\,_{i}$’s are the solution of the E–system. Thus, if $\mathsf{x}\,_{S}:=(\mathsf{x}\,_{1},\ldots,\mathsf{x}\,_{d-1})$ is the solution of the $E$–system determined by the set $S$, the equation (30) can be written as

[TABLE]

Then, (29) yields

[TABLE]

So, by (27) we get:

[TABLE]

Because $\theta^{2}$ is depending only on the cardinal of $S$, we can define $\lambda_{m}$ for every $m\in\mathbb{N}$, as the $\theta^{2}$ above, i.e.

[TABLE]

Recapitulating, by extending the field $\mathbb{K}$ to $\mathbb{K}(\sqrt{\lambda_{m}})$, we can consider the following homomorphism $\pi_{\sqrt{\lambda_{m}}}$,

[TABLE]

Thus, we have a family of functions $\{\mathrm{tr}_{n}\circ\pi_{\sqrt{\lambda_{m}}}\}_{n}$ agreeing with the replacements M1 and M2. Proceeding as in (11) we get that, now, the factor of normalization for $\mathrm{tr}_{n}\circ\pi_{\sqrt{\lambda_{m}}}$ is $\left(\mathsf{z}\,\sqrt{\lambda_{m}}\right)^{n-1}$.

Definition 14**.**

Let $m\in\mathbb{N}$. For $\sigma\in B_{n}$, we define

[TABLE]

Theorem 19**.**

Let $L$ be a link s.t. $L=\widehat{\sigma}$, where $\sigma\in B_{n}$, then the map $\Delta_{m}$ is an invariant of ambient isotopy for oriented links,

[TABLE]

Remark 9**.**

Regarding (10) and (31) it results clear that $\Delta_{1}$ is the Homflypt polynomial.

For the benefit of the writing, the main results on the invariant $\Delta_{m}$ will be established after we define a cousin of $\Delta_{m}$, denoted denoted by $\Theta_{m}$. These invariants are not equivalents but share several properties.

10.2.

The Yokonuma–Hecke algebra has another presentation due to M. Chlouveraki and L. Poulain d’Andecy [8], cf. [9, 11]. This presentation is constructed as follows. Firstly, the field ${\mathbb{K}}$ is extended to ${\mathbb{K}}(\mathsf{v}\,)$ with $\mathsf{v}\,^{2}=\mathsf{u}\,$; secondly, new generators $f_{i}$ are defined by

[TABLE]

It is a routine to check that the $f_{i}$’s and the $e_{i}$’s satisfy the relations (13)–(17) if one substitutes $g_{i}$ with $f_{i}$ and the relation,

[TABLE]

Notice that $f_{i}$’s are invertibles, and

[TABLE]

Thus one obtains a presentation of the Yokonuma–Hecke algebra by the generators $f_{i}$’s and $e_{i}$’s and the same relations of those given for the defining generators $g_{i}$’s and $e_{i}$’s except for the relation (18) which is replaced by the relation (33). We shall use the notation $\mathrm{Y}_{d,n}(\mathsf{v}\,)$, or simply $\mathrm{Y}_{n}(\mathsf{v}\,)$, whenever the Yokonuma–Hecke algebra is considered with the presentation by the $f_{i}$’s and $e_{i}$’s.

Remark 10**.**

Notice that $\mathrm{Y}_{1,n}(\mathsf{v}\,)$ coincides with the presentation of the Hecke through the generators $\tilde{h}_{i}$’s of Exercise X.

By using now, in the Jones recipe, $\mathrm{Y}_{d,n}(\mathsf{v}\,)$ instead of ${\rm Y}_{d,n}(\mathsf{u}\,)$, one obtains an invariant $\Theta_{m}$ instead $\Delta_{m}$. To be precise, notice firstly that for every $\alpha\in{\rm Y}_{d,n}$, according to properties of $\mathrm{tr}$, (32) and (24), we have:

[TABLE]

Secondly, given a non–empty subset $S$ of $\mathbb{Z}/d\mathbb{Z}$ and $(\mathsf{x}\,_{1},\ldots,\mathsf{x}\,_{d-1})$ the solutions determined by $S$, for every $\alpha\in{\rm Y}_{d,n}$ the Proposition 10 together with (32) and (34,) imply:

[TABLE]

Therefore, the rescaling factor is:

[TABLE]

the last equality is due to (27). Again the rescaling factor depends only on the cardinal $|S|$, so, by using the procedure used to get the definition of $\Delta_{m}$, we get firstly the rescaling factor, denoted by $\lambda^{\prime}_{m}$, namely

[TABLE]

and secondly $\Theta_{m}$ as follows.

Definition 15**.**

For $\sigma\in B_{n}$, we define

[TABLE]

Keeping the notations above, we have the following theorem.

Theorem 20**.**

The function $\Theta_{m}:\mathfrak{L}\longrightarrow\mathbb{K}(\mathsf{z}\,,\sqrt{\lambda^{\prime}_{S}})$, defined by $L\mapsto\Theta_{m}(\sigma)$, where $L$ is the closure of the braid $\sigma$, is an invariant of ambient isotopy of oriented links.

Remark 11**.**

Exactly as $\Delta_{1}$ in Remark 9, we have that $\Theta_{1}$ coincide with the Homflypt polynomial.

Theorem 21**.**

The invariants $\Delta_{m}$ and $\Theta_{m}$ coincide with the Homflypt polynomial whenever they are evaluated on knots.

To prove that the invariants $\Delta_{m}$ and $\Theta_{m}$ were not equivalents to the Hompflypt polynomial was a not trivial matter. Firstly, in [9] were found six pairs of links with equivalents Homflypt polynomial but different $\Theta_{m}$, see [9, Table1]; a such pair is shown in Fig. 16.

Hence.

Theorem 22**.**

For every $m\geq 2$, the invariants $\Theta_{m}$ are not equivalent to the Homflypt polynomial.

The people working on theses invariants thought that the invariants $\Theta_{m}$ and $\Delta_{m}$ were equivalents111This is due to the fact that in the definitions of $\Theta_{m}$ and $\Delta_{m}$ the unique difference is the change of presentations for the Y–H algebra, apparently unimportant thing. but surprisingly F. Aicardi shows that it is not the case.

Theorem 23** (Aicardi).**

For every $m\geq 2$, the invariants $\Delta_{m}$ and $\Theta_{m}$ are not topologically equivalent to the Homflypt polynomial: indeed, there exists pairs of non isotopic links distinguished by $\Delta_{m}$ and/or $\Theta_{m}$ but not by the Homflypt polynomial.

Proof.

In [6] we can find several pairs proving this theorem. In Fig. 17, we have a pair of non isotopic links distinguished by $\Delta_{m}$, $m\geq 2$, but neither by Homflypt nor $\Theta_{m}$.

∎

The next theorems establish the main properties of the invariants $\Delta_{m}$ and $\Theta_{m}$.

Proposition 11**.**

The invariants $\Delta_{m}$ and $\Theta_{m}$ share several properties with the Homflypt polynomial, e.g.: the behavior under connected sums and mirror image.

In the next section we generalize, respectively, the invariants $\Delta_{m}$ and $\Theta_{m}$ to, respectively, certain invariants in three parameters, $\overline{\Delta}$ and $\overline{\Theta}$ for classical links. We will do this, by using the Jones recipe applied to the bt–algebra. Also we will see, that $\overline{\Theta}$ can be defined through skein relations, while $\overline{\Delta}$ cannot.

11. The bt–algebra

In this section we introduce the so–called bt–algebra (or algebra of braids and ties) which is constructed by abstracting the braid generators $g_{i}$’s of the Yokonuma–Hecke algebra and the idempotents $e_{i}$’s appearing in the square of the braid generators, see (18). This algebra is used to generalize the invariants $\Delta_{m}$ and $\Theta_{m}$ to invariants with three parameters as well as its understanding by skein relations. Before introducing the bt–algebra we shall recall the main facts on set partitions since these facts will be useful in the rest of these notes.

11.1.

For $n\in{\mathbb{N}}$, we denote by ${\bf n}$ the set $\{1,\ldots,n\}$ and by $\mathsf{P}_{n}$ the set formed by the set partitions of ${\bf n}$, that is, an element of $\mathsf{P}_{n}$ is a collection $I=\{I_{1},\ldots,I_{k}\}$ of pairwise–disjoint non–empty sets whose union is ${\bf n}$; the sets $I_{1},\ldots,I_{k}$ are called the blocks of $I$; the cardinal of $\mathsf{P}_{n}$, denoted $b_{n}$, is called the $n^{th}$ Bell number.

We can regard $\mathsf{P}_{n}$ as subset of $\mathsf{P}_{n+1}$ through the natural injective map $\iota_{n}:\mathsf{P}_{n}\longrightarrow\mathsf{P}_{n+1}$, where for $I\in\mathsf{P}_{n}$, the image $\iota_{n}(I)\in\mathsf{P}_{n+1}$ is defined by adding to $I$ the block $\{n+1\}$.

Typically, the set partitions are represented by scheme of arcs, see [20, Subsection 3.2.4.3], that is: the point $i$ is connected by an arc to the point $j$, if $j$ is the minimum in the same block of $i$ satisfying $j>i$. Figure 18 shows the set partition $I=\{\{1,3\},\{2,5,6\},\{4\}\}$ as represented by arcs.

The representation by arcs of a set partition induces a natural indexation of its blocks. More precisely, we say that the blocks $I_{j}$’s of the set partition $I=\{I_{1},\ldots,I_{m}\}$ of ${\bf n}$ are standard indexed if $\min(I_{j})<\min(I_{j+1})$, for all $j$. For instance, in the set partition of Figure 18 the blocks are indexed as: $I_{1}=\{1,3\}$, $I_{2}=\{2,5,6\}$ and $I_{3}=\{4\}$.

The natural action of $\mathtt{S}_{n}$ on ${\bf n}$ induces, in the obvious way, an action of $\mathtt{S}_{n}$ on $\mathsf{P}_{n}$ that is, for $I=\{I_{1},\ldots,I_{m}\}$ we have

[TABLE]

Notice that this action preserves the cardinal of each block of the set partition.

Now, we shall say that two set partitions $I$ and $I^{\prime}$ in $\mathsf{P}_{n}$ are conjugate, denoted by $I\sim I^{\prime}$, if there exists $w\in\mathtt{S}_{n}$ such that, $I^{\prime}=w(I)$; if it is necessary to precise such $w$, we write $I\sim_{w}I^{\prime}$. Further, observe that if $I$ and $I^{\prime}$ are standard indexed with $m$ blocks, then the permutation $w$ induces a permutation of $S_{m}$ of the indices of the blocks, which we denote by $w_{I,I^{\prime}}.$

Example 10**.**

Let $I=\{\{1,2\}_{1},\{3\}_{2},\{4,5\}_{3},\{6\}_{4}\}$ and

[TABLE]

so $n=6$ and $m=4$. We have $I\sim_{w}I^{\prime}$, where:

[TABLE]

Given a permutation $w\in\mathtt{S}_{n}$ and writing $w=c_{1}\cdots c_{m}$ as product of disjoint cycles, we denote by $I_{w}$ the set partition whose blocks are the cycles $c_{i}$’s, regarded now as subsets of $\bf n$. Reciprocally, given a set partition $I=\{I_{1},\ldots,I_{m}\}$ of $\bf n$ we denote by $w_{I}$ an element of $\mathtt{S}_{n}$ whose cycles are the blocks $I_{i}$’s. Moreover, we shall say that the cycles of $w_{I}$ are standard indexed, if they are indexed according to the standard indexation of $I$.

Notation 5**.**

When there is no risk of confusion, we will omit in the partitions the blocks with a single element.

$\mathsf{P}_{n}$ is a poset with structure of commutative monoid. Indeed, the partial order on $\mathsf{P}_{n}$ is defined as follows: $I\preceq J$ if and only if each block of $J$ is a union of blocks of $I$. The product $I\ast J$, between $I$ and $J$ is defined as the minimal set partition, containing $I$ and $J$, according to $\preceq$; the identity of this monoid is ${\mathbb{1}}_{n}:=\{\{1\},\{2\},\ldots,\{n\}\}$. Observe that:

[TABLE]

[TABLE]

Further, the injective maps $\iota_{n}$’s result to be a monoid homomorphisms and respect $\preceq$, that is, for every $I,J\in\mathsf{P}_{n}$, we have:

[TABLE]

Notation 6**.**

The inductive limit associated to the family of monoids $\{(\mathsf{P}_{n},\iota_{n})\}_{n\in\mathbb{N}}$ is denoted by $\mathsf{P}_{\infty}$,

We are going to give an abstract description of $\mathsf{P}_{n}$, that is, via a presentation which will be used in the next section. For every $1\leq i<j\leq n$ with $i\not=j$, define $\mu_{i,j}\in\mathsf{P}_{n}$ as the set partition whose blocks are $\{i,j\}$ and $\{k\}$ where $1\leq k\leq n$ and $k\not=i,j$. We shall write $\mu_{i,j}\mu_{k,h}$ instead of $\mu_{i,j}\ast\mu_{k,h}$.

Proposition 12**.**

The monoid $\mathsf{P}_{n}$ can be presented by the set partitions $\mu_{i,j}$’s subject to the following relations:

[TABLE]

11.2.

The original definition of the bt–algebra is the following.

Definition 16** (See [1, 22, 3]).**

The bt–algebra , denoted by $\mathcal{E}_{n}(\mathsf{u}\,)$, is defined by $\mathcal{E}_{1}(\mathsf{u}\,):={\mathbb{K}}$ and for $n\geq 2$ as the unital associative ${\mathbb{K}}$–algebra, with unity $1$, defined by braid generators $T_{1},\ldots,T_{n-1}$ and ties generators $E_{1},\ldots,E_{n-1}$ subjected to the following relations:

[TABLE]

The $T_{i}$’s are invertible, with $T^{-1}$ given by:

[TABLE]

Remark 12**.**

1. (1)

The bt–algebra can be seen as a generalization of the Hecke algebra since by making $E_{i}=1$, the definition of the bt–algebra becomes the Hecke algebra. Further, observe that the mapping $T_{i}\mapsto h_{i}$ and $E_{i}\mapsto 1$ defines an epimorphism from the bt–algebra to the Hecke algebra. 2. (2)

The mapping $T_{i}\mapsto g_{i}$ and $E_{i}\mapsto e_{i}$ defines an algebra homomorphism from the bt–algebra to the YH–algebra, which is injective if and only if $d\geq n$, see [11], cf. [3, Remark 3].

Diagrammatically the generators $T_{i}$’s can be regarded as usual braids and the generator $E_{i}$ as a tie between the $i$ and $i+1$ strands, this tie doesn’t have a topological meaning: it is an auxiliary artefact to reflect the monomial–homogeneous defining relation of the bt–algebra. Thus, the tie is pictured as a spring or a dashed line between the strands. More precisely, the diagrams for, respectively, $T_{i}$ and $E_{i}$, are:

Here comment!

11.3.

The bt–algebra is a finite dimensional algebra. Moreover, there is a basis due to S. Ryom–Hansen [22]. Before expliciting the Ryom–Hansen basis we need to introduce the tools below.

For $i<j$, we define $E_{i,j}$ by

[TABLE]

For any nonempty subset $J$ of $\bf n$ we define $E_{J}=1$ for $|J|=1$ and otherwise by

[TABLE]

Note that $E_{\{i,j\}}=E_{i,j}$. For $I=\{I_{1},\ldots,I_{m}\}\in\mathsf{P}_{n}$, we define $E_{I}$ by

[TABLE]

Now, if $w=\mathtt{s}_{i_{1}}\cdots\mathtt{s}_{i_{k}}$ is a reduced expression of $w\in\mathtt{S}_{n}$, then the element $T_{w}:=T_{i_{1}}\cdots T_{i_{k}}$ is well defined. The action of $\mathtt{S}_{n}$ on $\mathsf{P}_{n}$ is inherited from the $E_{I}$’s and we have:

[TABLE]

Theorem 24** ([22, Corollary 3]).**

The set $\{E_{I}T_{w}\,;\,w\in\mathtt{S}_{n},\,I\in\mathsf{P}_{n}\}$ is a ${\mathbb{K}}$–linear basis of ${\mathcal{E}}_{n}(\mathsf{u}\,)$. Hence the dimension of ${\mathcal{E}}_{n}(\mathsf{u}\,)$ is $b_{n}n!$.

Example 11**.**

Having in mind (39), the natural inclusion $\mathtt{S}_{n}\subset\mathtt{S}_{n+1}$, for every $n$, together with Theorem 24, we deduce the tower of algebras:

[TABLE]

Notation 7**.**

Denote by ${\mathcal{E}}_{\infty}(\mathsf{u}\,)$ the inductive limit associated the bt–algebras above.

Remark 13** (Cf. Subsection 10.2).**

Extending the field ${\mathbb{K}}$ to ${\mathbb{K}}(\mathsf{v}\,)$ with $\mathsf{v}\,^{2}=\mathsf{u}\,$, we can define (cf. [21, Subsection 2.3]):

[TABLE]

Then the $V_{i}$’s and the $E_{i}$’s satisfy the relations (43)–(48) and the quadratic relation (49) is transformed in

[TABLE]

So,

[TABLE]

In [9, 11, 13] this quadratic relation is used to define the bt–algebra. Although at algebraic level these algebras are the same, we will see that they lead to different invariants. Thus, in order to distinguish these two presentations of the bt–algebra, we will write $\mathcal{E}_{n}(\mathsf{v}\,)$ when the bt–algebra is defined by using the quadratic relation (55).

11.4.

In [3] it was proved that the bt–algebra supports a Markov trace, this was proved using the method of relative trace and the Ryom–Hansen basis.

Let $\mathsf{a}\,$ and $\mathsf{b}\,$ be two variables commutative independent commuting with $\mathsf{u}\,$.

Theorem 25** ([3, Theorem 3]).**

There exists a unique Markov trace $\rho$ on ${\mathcal{E}}_{\infty}(\mathsf{u}\,)$, i.e., a family $\rho:=\{\rho_{n}\}_{n\in{\mathbb{N}}}$, where $\rho_{n}$’s are linear maps, defined inductively, from $\mathcal{E}_{n}(\mathsf{u}\,)$ in $\mathbb{K}[\mathsf{a}\,,\mathsf{b}\,]$ such that $\rho_{n}(1)=1$ and satisfying, for all $X,Y\in\mathcal{E}_{n}(\mathsf{u}\,)$, the following rules:

1. (1)

$\rho_{n}(XY)=\rho_{n}(YX)$, 2. (2)

$\rho_{n+1}(XT_{n})=\rho_{n+1}(XT_{n}E_{n})=\mathsf{a}\,\rho_{n}(X)$, 3. (3)

$\rho_{n+1}(XE_{n})=\mathsf{b}\,\rho_{n}(X)$.

With this theorem and because the braid group is represented (naturally) in the bt–algebra we are ready to define an invariant for links. More precisely, denote by $\pi_{\sqrt{\mathsf{L}}}$ the (natural) representation of $B_{n}$ in $\mathcal{E}_{n}$, namely $\sigma_{i}\mapsto\sqrt{\mathsf{L}}T_{i}$. With the same procedure used to get (10) and (31), we define

[TABLE]

Definition 17**.**

For $\sigma\in B_{n}$, we define

[TABLE]

Theorem 26**.**

Let $L$ be a link obtained by closing the braid $\sigma\in B_{n}$. Then the map $L\mapsto\overline{\Delta}(\sigma)$ defines an ambient isotopy invariant for oriented links, taking values in $\mathbb{K}(\mathsf{a}\,,\sqrt{\mathsf{L}})$.

Proof.

We need only to prove that $\overline{\Delta}$ agrees with the Markov replacements. Because $\pi_{\sqrt{\mathsf{L}}}$ is an homomorphism and the properties of $\rho$ imply that $\overline{\Delta}$ agrees with M1. On the second replacement, we note that it is a routine to check $\overline{\Delta}(\sigma)=\overline{\Delta}(\sigma\sigma_{n})$. Thus it remains only to check $\overline{\Delta}(\sigma)=\overline{\Delta}(\sigma\sigma_{n}^{-1})$, for every $\sigma\in B_{n}$ . Now, put $\alpha=\pi_{\sqrt{\mathsf{L}}}(\sigma)$, then:

[TABLE]

Therefore, $\overline{\Delta}(\sigma\sigma_{n}^{-1})=\overline{\Delta}(\sigma)$, for all $\alpha\in B_{n}$. ∎

Now, in the way we define $\Theta_{m}$ we can define $\bar{\Theta}$, that is, taking out its definition by using now the presentation of the bt–algebra $\mathcal{E}_{n}(\mathsf{v}\,)$ in the Jones recipe. Having in mind what we did above, (35) and Definition 15, we define $\mathsf{L}^{\prime}$ and $\overline{\Theta}$ as follows:

[TABLE]

[TABLE]

Theorem 27**.**

Let $L$ be a link obtained by closing the braid $\alpha\in B_{n}$. Then the map $L\mapsto\overline{\Theta}(\alpha)$ defines an invariant of ambient isotopy for oriented links, which take values in $\mathbb{K}(\mathsf{a}\,,\sqrt{\mathsf{L}^{\prime}})$.

Proof.

Same proof as Theorem 26. ∎

Remark 14**.**

The invariant $\overline{\Delta}$ contains the invariant $\Delta_{m}$, for every $m\in\mathbb{N}$; that is specializing the variable $\mathsf{b}\,$ to $1/m$ we get $\Delta_{m}$. With the same specialization, the invariant $\overline{\Theta}$ contains the invariants $\Theta_{m}$, for every $m$.

12. Tied links

In this section we introduce the tied links. This class of knotted-like objects contains the classical link, so invariants of tied links yield invariants of classical links. We start the section recalling the definition of tied links and the monoid of braid tied links. Also, the respective Alexander and Markov theorems are exhibited.

12.1.

Tied links were introduced in [2] and roughly correspond to links whose components may be connected by ties; thus ties are connecting pairs of points of two components or of the same component.

The ties in the picture of the tied links are drawn as springs or dashed lines, to outline that they can be contracted and extended, letting their extremes to slide along the components.

Notation 8**.**

We will use the notation $C_{i}\leftrightsquigarrow C_{j}$ to indicate that either there is a tie between the components $C_{i}$ and $C_{j}$ of a link, or $C_{i}$ and $C_{j}$ are the extremes of a chain of $m>2$ components $C_{1},\dots,C_{m}$, such that there is a tie between $C_{i}$ and $C_{i+1}$, for $i=1,\dots,m-1$.

Definition 18** ([2, Definition 1.1]).**

Every 1–link is by definition a tied 1–link. For $k>1$, a tied $k$–link is a link whose set of components $\{C_{1},\ldots,C_{k}\}$ is partitioned into parts according to: two components $C_{i}$ and $C_{j}$ belong to the same part if $C_{i}\leftrightsquigarrow C_{i+1}$.

Notation 9**.**

We denote by $\widetilde{\mathfrak{L}}$ the set of oriented tied links.

In Fig. 21 two tied links with four components; moreover, if $C_{1}$ is the blue component, $C_{2}$ the red component, $C_{3}$ the yellow components and $C_{4}$ the green component, then the partition associated to the first tied link is $\{\{C_{1},C_{2}\},\{C_{3},C_{4}\}\}$ and the partitions associated to the second tied link is $\{\{C_{1}\},\{C_{2},C_{3}\},\{C_{4}\}\}$.

Notice that a tied $k$–link $L$, with components’ set $C_{L}=\{C_{1},\ldots,C_{k}\}$, determines a pair $(L,I(C_{L}))$ in $\mathfrak{L}_{k}\times\mathsf{P}_{k}$, where $i$ and $j$ belong to the same block of $I(C_{L})$, if $C_{i}\leftrightsquigarrow C_{j}$.

Example 12**.**

For instance in Fig. 21, the set partition determined by the first tied link is $\{\{1,2\},\{3,4\}\}$ and the set partition determined by the second tied link is $\{\{1\},\{4\},\{2,3\}\}$. The set partition determined by the tied link of Fig. 20 is $\{\{1,2,3,4\}\}$.

Definition 19** (Cf. [2, Definition 1.6]).**

A tie of a tied link is said essential if it cannot be removed without modifying the partition $I(C_{L})$, otherwise the tie is said unessential.

Notice that between the $c$ components indexed by the same block of the set partition, the number of essential ties is $c-1$; for instance, in the tied link of Fig. 21, left, among the three ties connecting the first three components, only two are essential. The number of unessential ties is arbitrary. Ties connecting one component with itself are unessential.

Definition 20**.**

The $k$–tied link $L$ and the $k^{\prime}$–tied link $L^{\prime}$ with, respectively, components $C=\{C_{1},\ldots,C_{k}\}$ and $C^{\prime}=\{C^{\prime}_{1},\ldots,C^{\prime}_{k^{\prime}}\}$, are t–isotopic if:

1. (1)

The links $L$ and $L^{\prime}$ are ambient isotopic (hence $k=k^{\prime})$. 2. (2)

The set partitions $I(C)$ and $I(C^{\prime})$ satisfy $I(C^{\prime})=w_{L,L^{\prime}}(I(C))$, where $w_{L,L^{\prime}}$ is the bijection from $C(L)$ to $C(L^{\prime})$ induced by the isotopy.

Example 13**.**

In the Fig. 22, we have that $L1$ and $L2$ are not t–isotopic. Indeed, the set partition, respectively, of $L1$ and $L2$ are $I_{1}=\{\{1,2,3,4\}\}$ and $I_{2}=\{\{1,3,4\},\{2\}\}$ and $w_{L1,L2}=(1,2,4,3)$, but $I_{2}\not=w_{L1,L2}(I_{1})$.

Now, $L3$ has associated the set partition $I_{3}=\{\{1,2,3\},\{4\}\}$ and $L4$ has associated the set partition $I_{4}=\{\{1\},\{2,3,4\}\}$ and $w_{L3,L4}=(4,1,2,3)$; thus, $I_{4}=w_{L3,L4}(I_{3})$. Then, $L3$ and $L4$ are t–isotopic

Remark 15**.**

The Definitions 18 and 20 not only say that classical links are included in tied links but also that the classical links can be identified to the set of tied links which components are all tied. Both ways to see the classical links allows to study the isotopy of links through the t–istopy of tied links. Observe that for $k$–tied links with set of components $C$, we have $I(C)$ is $\{1,2,\ldots,n\}$ if it has all components tied and is $\{\{1\},\{2\},\ldots,\{n\}\}$ if does not have ties or has only ties that are unessential.

Remark 16**.**

Everything established for tied links can be translated in terms of diagrams in the obvious way. Informally, it is enough to change ‘links’ by ‘diagrams of a link’  and so on.

12.2.

The classical theorems of Alexander and Markov in knot theory have their analogous in the world of tied links. The starting point to establish these theorems for tied links is the so–called tied braid monoid.

Definition 21**.**

[2, Definition 3.1]* The tied braid monoid $T\!B_{n}$ is the monoid generated by usual braids $\sigma_{1},\ldots,\sigma_{n-1}$ and the tied generators $\eta_{i},\ldots,\eta_{n-1}$, such the $\sigma_{i}$’s satisfy braid relations among them together with the following relations:*

[TABLE]

We denote $TB_{\infty}$ the inductive limit determined by the natural monomorphism monoid from $TB_{n}$ into $TB_{n+1}$.

Diagrammatically, as usual, $\sigma_{i}$ is represented as the usual braid and the tied generator $\eta_{i}$, as the diagram of $E_{i}$, that is, a tie connecting the $i$ with $(i+1)$–strands, see Fig. 19.

The tied braid monoid is to the bt–algebra as the braid group is to the Hecke algebra. In particular, we have the following proposition and its corollary.

Proposition 13**.**

The mapping $\eta_{i}\mapsto E_{i}$, $\sigma_{i}\mapsto T_{i}$ defines an homomorphism, denoted by $\widetilde{\pi}$, from $TB_{n}$ to $\mathcal{E}_{n}$.

Proof.

∎

Corollary 3**.**

$\mathcal{E}_{n}$* The bt–algebra is a quotient of $\mathbb{K}TB_{n}$. More precisely,*

[TABLE]

where $I$ is the two–sided ideal generated by $T_{i}^{2}-(\mathsf{u}\,-1)E_{i}(1+T_{i})$, for $1\leq i\leq n-1$.

$TB_{n}$ has a decomposition like semidirect product of groups, this decomposition allows a study purely algebraic–combinatorics of the tied link, see [5], and will be used below. To establish this decomposition, we start by noting that the action of $\mathtt{S}_{n}$ on $\mathsf{P}_{n}$, see (36), together with the natural projection $\mathsf{p}$ of (4), define an action of $B_{n}$ on $\mathsf{P}_{n}$. We denote by $\sigma(I)$, the action of the braid $\sigma$ on $I\in\mathsf{P}_{n}$, that is, $\sigma(I)$ is the result of the application of the permutation $\mathsf{p}(\sigma)$ to the set partition $I$. Define now the following product in $\mathsf{P}_{n}\times B_{n}$:

[TABLE]

$\mathsf{P}_{n}\times B_{n}$ with this product is a monoid, which is denoted by $\mathsf{P}_{n}\rtimes B_{n}$. We shall denote $I\sigma$ instead $(I,\sigma)$.

Theorem 28** ([5, Theorem 9]).**

The monoid $TB_{n}$ and $\mathsf{P}_{n}\rtimes B_{n}$ are isomorphic.

Now, as for braid, we can define the closure of a tied braid in the same way of the closure of the braids. Evidently, the closure of a tied braid is a tied link. Moreover, in [2, Theorem 3.5] we have proved the Alexander theorem for tied links; namely.

Theorem 29** ([2, Theorem 3.5]).**

Every oriented tied link can be obtained as closure of a tied braid.

Before establishing the Markov theorem for tied links, notice that according to Theorem 28, every element $a$ in $TB_{n}$ can be written uniquely in the form $a=I\sigma$, where $I\in\mathsf{P}_{n}$ and $\sigma\in B_{n}$. We use this fact in the definition of Markov moves for tied links and also we use the notation of the $\mu_{i,j}$’s as in (40).

Definition 22**.**

We say that $a,b\in TB_{\infty}$ are t–Markov equivalents, denoted $a\sim_{tM}b$, if $b$ can be obtained from $a$ by using a finite sequence of the following replacements:

1. tM1.

t–Stabilization: for all $a=I\sigma\in TB_{n}$, we can replace $a$ by $a\mu_{i,j}$, if $i,j$ belong to the same cycle of $\mathsf{p}(\sigma)$. 2. tM2.

Commuting in $TB_{n}$. For all $a,b\in TB_{n}$, we can replace $ab$ by $ba$, 3. tM3.

Stabilizations: for all $a\in TB_{n}$, we can replace $a$ by $a\sigma_{n}$ or $a\sigma_{n}^{-1}.$

The relation $\sim_{tM}$ is an equivalence relation on $TB_{\infty}$ and in [2, Theorem 3.7], cf. [5, Theorem 5], the following theorem was proved.

Theorem 30**.**

Two tied links define t–isotopic tied links if and only if they are t–Markov equivalents.

Remark 17**.**

The Markov replacements $tM2$ and $tM3$ are the classical Markov moves but including now the ties, so their geometrical meaning is clear. The replacement tM1 says that the ties provided by $\mu_{i,j}$ in the closure of $a\mu_{i,j}$ become unessential ties, so the closure of $a$ and $a\mu_{i,j}$ defines the same set partition.

13. The invariant $\mathcal{F}$

We will define the invariant $\mathcal{F}$ for tied links. This invariant is of type Homflypt since evaluated on classical links coincide with the Homflypt polynomial. We start by defining it by skein relations and later their definition by the Jones recipe.

13.1.

To define $\mathcal{F}$ by skein relations we need, as in (13), to introduce the following notation: $L_{+},L_{-},L_{0},L_{+,\sim},L_{-,\sim}$ and $L_{0,\sim}$ denote oriented tied links which have, respectively, oriented diagrams of tied links $D_{+},D_{-},D_{0},D_{+,\sim},D_{-,\sim}$ and $D_{0,\sim}$, that are identical outside a small disk into which enter two strands, whereas inside the disk the strands look, respectively, as Fig. 23 shows.

We shall call $L_{+,\sim},L_{-,\sim}$ and $L_{0,\sim}$ a tied Conway triple.

Set $\mathsf{w}\,$ a variable with unique condition that commutes with $\mathsf{u}\,$ and $\mathsf{a}\,$.

Theorem 31** ([2, Theorem 2.1]).**

There is a unique function $\mathcal{F}:\widetilde{\mathfrak{L}}\longrightarrow\mathbb{K}(\mathsf{a}\,,\mathsf{w}\,)$, defined by the rules:

1. (1)

$\mathcal{F}(\bigcirc)=1$, 2. (2)

For all tied link $L$,

[TABLE] 3. (3)

The Skein rule:

[TABLE]

Proposition 14**.**

The defining skein relation of $\mathcal{F}$ is equivalent to the following skein rules:

1. (1)

[TABLE] 2. (2)

[TABLE] 3. (3)

[TABLE]

Proof.

∎

Define $\widetilde{\mathfrak{L}}_{0}$ the set of tied links with all components tied.

Proposition 15**.**

The restriction of $\mathcal{F}$ to $\widetilde{\mathfrak{L}}_{0}$, is determined uniquely by the rules:

1. (1)

$\mathcal{F}(\bigcirc)=1$, 2. (2)

Skein relation on tied Conway triple $L_{+,\sim}$, $L_{-,\sim}$ and $L_{0,\sim}$,

[TABLE]

where $\mathsf{r}\,=\mathsf{w}\,\sqrt{u}$ and $\mathsf{s}\,=\sqrt{\mathsf{u}\,}-\sqrt{\mathsf{u}\,}^{-1}$.

Proof.

The computation of $\mathcal{F}$ on a such tied link can be realized by rules (1) Theorem 31 and the skein relation (1) Proposition 14. Multiplying this skein relations by $\sqrt{\mathsf{u}\,}$, we get the values of $\mathsf{r}\,$ and $\mathsf{s}\,$. Hence the proof is concluded. ∎

Remark 18**.**

According to Remark 15, $\widetilde{\mathfrak{L}}_{0}$ is identified to classical links, so, the above proposition and Theorem(12)) say that $\mathcal{F}$ and Homflypt polynomial coincide on classical links. For this reason we say that $\mathcal{F}$ is invariant of type Homflypt.

Proposition 16**.**

The restriction of $\mathcal{F}$ to classical links is more powerful than the Homflypt polynomial.

Proof.

∎

13.2.

In order to define $\mathcal{F}$ through the Jones recipe, we extend first the domain $\pi_{\sqrt{\mathsf{L}}}$ to $TB_{n}$; we denote this extension by $\widetilde{\pi}_{\sqrt{\mathsf{L}}}$. Secondly, we define $\widetilde{\Delta}$ by

[TABLE]

Theorem 32**.**

Let $L$ be a tied link obtained as the closure of the tied braid $\eta$, then

[TABLE]

The relation among, $\mathsf{u}\,$, $\mathsf{w}\,$ and parameters trace $\mathsf{a}\,$ and $\mathsf{b}\,$ of $\rho$, is given by the equation:

[TABLE]