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Getting back now to knots and links, a quick comparison between our main results so far, namely Theorem 8.8 due to Reidemeister, and then Theorem 8.9 due to Alexander, suggests the following question, whose answer will certainly advance us: \begin{question} What is the analogue of the Reidemeister theorem, in the context of braids? That is, when do two braids produce, via closing, the same link? \end{question} And this is, and we insist, a very good question, because assuming that we have an answer to it, no need afterwards to bother with plane projections, decorated graphs, Reidemeister moves, and amateurish topology in general, it will be all about groups and algebra. Which groups and algebra questions, you guessed right, we will eat them raw.

In answer now, we have the following theorem, due to Markov:

Theorem

Two elements of the full braid group, obtained as the increasing union of the various braid groups, with embeddings given by [math]\beta\to\beta\,|[/math],

[[math]] B_\infty=\bigsqcup_{k=1}^\infty B_k [[/math]]

produce the same link, via closing, when one can pass from one to another via:

• Conjugation: [math]\beta\to\alpha\beta\alpha^{-1}[/math].
• Markov move: [math]\beta\to g_k^{\pm1}\beta[/math].

Show Proof

This is a version of the Reidemeister theorem, the idea being as follows:

(1) To start with, it is clear that conjugating a braid, [math]\beta\to\alpha\beta\alpha^{-1}[/math], will produce the same link after closing, because we can pull the [math]\alpha,\alpha^{-1}[/math] to the right, in the obvious way, and there on the right, these [math]\alpha,\alpha^{-1}[/math] will annihilate, according to [math]\alpha\alpha^{-1}=1[/math].

(2) Regarding now the Markov move from the statement, with [math]\beta\in B_k\subset B_{k+1}[/math] and with [math]g_1,\ldots,g_k\in B_{k+1}[/math] being the standard Artin generators, from Theorem 8.10 and its proof, this is the tricky move, which is worth a proof. Taking [math]k=3[/math] for an illustration, and representing [math]\beta\in B_3[/math] by a box, the link obtained by closing [math]g_4\beta[/math] is as follows, which is obviously the same link as the one obtained by closing [math]\beta[/math], and the same goes for [math]g_4^{-1}\beta[/math]:

[[math]] \xymatrix@C=9pt@R=4pt{ &\ar@{-}[dddd]\ar@{-}[rrrrrrrrrrr]&&&&&&&&&&&\ar@{-}[ddddddddddddd]\\ &&\ar@{-}[ddd]\ar@{-}[rrrrrrrrr]&&&&&&&&&\ar@{-}[ddddddddddd]\\ &&&\ar@{-}[dd]\ar@{-}[rrrrrrr]&&&&&&&\ar@{-}[ddddddddd]\\ &&&&&&&\ar@{-}[rr]\ar@{-}[dddd]&&\ar@{-}[ddddddd]\\ \ar@{-}[rrrr]\ar@{-}[dd]&&&&\ar@{-}[dd]\\ \\ \ar@{-}[rrrr]&\ar@{-}[ddddddd]&\ar@{-}[dddddd]&\ar@{-}[d]&&&\\ &&&\ar@{-}[drr]&&&&\ar@{-}[ddllll]\\ &&&&&\ar@{-}[drr]&\\ &&&\ar@{-}[dd]&&&&\ar@{-}[d]&\\ &&&&&&&\ar@{-}[rr]&&\\ &&&\ar@{-}[rrrrrrr]&&&&&&&\\ &&\ar@{-}[rrrrrrrrr]&&&&&&&&&\\ &\ar@{-}[rrrrrrrrrrr]&&&&&&&&&&& } [[/math]]

(3) Thus, the links produced by braids are indeed invariant under the two moves in the statement. As for the proof of the converse, this comes from the Reidemeister theorem, applied in the context of the Alexander theorem, or perhaps simpler, by reasoning directly, a bit as in the proof of the Reidemeister theorem. We will leave this as an exercise.

■

As explained before, the above kind of theorem is exactly what we need, in order to reformulate everything in terms of groups and algebra. To be more precise, looking now more in detail at what Theorem 8.12 exactly says, we are led to the following strategy: \begin{strategy} In order to construct numeric invariants for knots and links:

• We must map [math]B_\infty[/math] somewhere, and then apply the trace.
• And if the trace is preserved by Markov moves, it's a win.

\end{strategy} You get the point with all this, if we are do (1) then, by using the trace property [math]tr(ab)=tr(ba)[/math] of the trace, we will have [math]tr(\beta)=tr(\alpha\beta\alpha^{-1})[/math], in agreement with what Theorem 8.12 (1) requires. And if we do (2) too, whatever that condition exactly means, and more on this in a moment, we will have as well [math]tr(\beta)=tr(g_k^{\pm1}\beta)[/math], in agreement with what Theorem 8.12 (2) requires, so we will have our invariant for knots and links.

This sound very good, but before getting into details, let us be a bit megalomaniac, and add two more ambitious points to our war plan, as follows: \begin{addendum} Our victory will be total, with a highly reliable invariant, if:

• The representation and trace are faithful as possible.
• And they depend, if possible, on several parameters.

\end{addendum} Here (1) and (2) are obviously related, because the more parameters we have in (2), the more chances for our constructions in (1) to be faithful will be. In short, what we are wishing here for is an invariant which distinguishes well between various knots and links, and this can only come via a mixture of faithfulness, and parameters involved.

So long for the plan, and in practice now, getting back to what Strategy 8.13 says, we are faced right away with a problem, coming from the fact that [math]B_\infty[/math] is not that easy to represent. You might actually already know this, if you have struggled a bit with the exercise that I left for you, at the end of the previous section. So, we are led to: \begin{question} How to represent the braid group [math]B_\infty[/math]? \end{question} So, this was the question that Reidemeister, Alexander, Markov, Artin and the others were fighting with, a long time ago, in the first half of the 20th century. Quite surprisingly, the answer to it came very late, in the 80s, from Jones , with inspiration from operator algebras, and more specifically, from his previous paper about subfactors.

Retrospectively looking at all this, what really matters in Jones' answer to Question 8.15 is the algebra constructed by Temperley and Lieb in , in the context of questions from statistical mechanics. But then, by looking even more retrospectively at all this, we can even say that the answer to Question 8.15 comes from nothing at all, meaning basic category theory. So, this will be our approach in what follows, with our answer being: \begin{answer} Thinking well, [math]B_\infty[/math] is self-represented, without help from the outside. \end{answer} So, ready for some category theory? We first need objects, and our set of objects will be the good old [math]\mathbb N[/math]. As for the arrows, somehow in relation with topology and braids, we will choose something very simple too, with our definition being as follows:

Definition

The Temperley-Lieb category [math]TL_N[/math] has the positive integers [math]\mathbb N[/math] as objects, with the space of arrows [math]k\to l[/math] being the formal span

[[math]] TL_N(k,l)=span(NC_2(k,l)) [[/math]]

of noncrossing pairings between an upper row of [math]k[/math] points, and a lower row of [math]l[/math] points

[[math]] \xymatrix@R=10pt@C=5pt{ &&1\ar@{-}[dd]&2\ar@{-}[d]&3\ar@{-}[d]&4&5\ar@{-}[ddd]\\ &&&\ar@{-}[r]&&&&&\\ &&\ar@{-}[rr]&&&&&&\\ \ar@{-}[rrr]&&&&&&\ar@{-}[rr]&&\\ &\ar@{-}[r]&&&&&\ar@{-}[r]&&\\ 1\ar@{-}[uu]&2\ar@{-}[u]&3\ar@{-}[u]&4\ar@{-}[uu]&5\ar@{-}[uuu]&6\ar@{-}[uuuuu]&7\ar@{-}[u]&8\ar@{-}[u]&9\ar@{-}[uu] } [[/math]]

and with the composition of arrows appearing by composing the pairings, in the obvious way, with the rule [math]\bigcirc=N[/math], for the closed loops that might appear.

This definition is something quite subtle, hiding several non-trivial things, and is worth a detailed discussion, our comments about it being as follows:

(1) First of all, our scalars in this chapter will be complex numbers, [math]\lambda\in\mathbb C[/math], and the “formal span” in the above must be understood in this sense, namely abstract complex vector space spanned by the elements of [math]NC_2(k,l)[/math]. Of course it is possible to use an arbitrary field, at least at this stage of things, but remember that we are interested in quantum mechanics, and related mathematics, where the field of scalars is [math]\mathbb C[/math].

(2) Regarding the composition of arrows, this is by vertical concatenation, with our usual convention that things go “from up to down”. And with this coming from care for our planet, and for entropy at the galactic level, I mean why pushing things from left to right, when we can have gravity work for us, pulling them from up to down:

[[math]] \xymatrix@R=40pt@C=5pt{ up\ar[d]\\ down } [[/math]]

(3) Less poetically, this “from up to down” convention is also useful for purely mathematical purposes, because the left-right direction will be reserved for the intervention of sums [math]\Sigma[/math] and scalars [math]\lambda\in\mathbb C[/math], while the up-down direction will be reserved for “action”.

(4) Let us discuss now what happens with the closed circles, when concatenating. As an example, let us consider a full capping of noncrossing pairings, also called meander:

[[math]] \xymatrix@R=10pt@C=5pt{ \ar@{-}[rrrrrrrrrrrrrrr]&&&&&&&&&&&&&&&\\ &\ar@{-}[rrrrrrr]&&&&&&&&\ar@{-}[rrrrr]&&&&&&\\ &&\ar@{-}[rrrrr]&&&&&&&&\ar@{-}[rrr]&&&&&\\ &&&\ar@{-}[r]&&\ar@{-}[r]&&&&&&\ar@{-}[r]&&&&\\ 1\ar@{-}[uuuu]&2\ar@{-}[uuu]&3\ar@{-}[uu]&4\ar@{-}[u]&5\ar@{-}[u]&6\ar@{-}[u]&7\ar@{-}[u]&8\ar@{-}[uu]&9\ar@{-}[uuu]&10\ar@{-}[uuu]&11\ar@{-}[uu]&12\ar@{-}[u]&13\ar@{-}[u]&14\ar@{-}[uu]&15\ar@{-}[uuu]&16\ar@{-}[uuuu]\\ \ar@{-}[u]\ar@{-}[r]&\ar@{-}[u]&&&\ar@{-}[u]\ar@{-}[r]&\ar@{-}[u]&&&\ar@{-}[u]\ar@{-}[r]&\ar@{-}[u]&&&&\ar@{-}[u]\ar@{-}[r]&\ar@{-}[u]&\\ &&&\ar@{-}[uu]\ar@{-}[rrr]&&&\ar@{-}[uu]&\ar@{-}[uu]\ar@{-}[rrr]&&&\ar@{-}[uu]&&\ar@{-}[uu]\ar@{-}[rrr]&&&\ar@{-}[uu]\\ &&\ar@{-}[uuu]\ar@{-}[rrrrrrrrr]&&&&&&&&&\ar@{-}[uuu]&&&& } [[/math]]

According to our conventions, this meander appears as the product [math]\pi\sigma\in NC_2(0,0)[/math] between the upper pairing [math]\sigma\in NC_2(0,16)[/math] and the lower pairing [math]\pi\in NC_2(16,0)[/math]. But, what is the value of this product? We have two loops appearing, namely:

[[math]] 1-2-9-10-15-14-11-8-3-12-13-16 [[/math]]

[[math]] 4-5-6-7 [[/math]]

Thus, according to Definition 8.17, the value of this meander is [math]N^2[/math], with one [math]N[/math] for each of the above loops, and with these two values of [math]N[/math] multiplying each other.

(5) The same discussion applies to an arbitrary composition [math]\pi\sigma\in NC_2(k,m)[/math] between an upper pairing [math]\sigma\in NC_2(k,l)[/math] and a lower pairing [math]\pi\in NC_2(l,m)[/math], with a certain number of loops appearing in this way, each contributing with a multiplicative factor [math]N[/math].

(6) Finally, in Definition 8.17 the value of the circle [math]N=\bigcirc[/math] can be pretty much anything, but due to some positivity reasons to become clear later, we will assume in what follows [math]N\in[1,\infty)[/math]. Also, we will call this parameter [math]N[/math] the “index”, with the precise reasons for calling this index to become clear later, too, as this book develops.

With all this discussed, what is next? More category theory I guess, and matter of having a theorem formulated too, instead of definitions only, let us formulate:

Theorem

The Temperley-Lieb category [math]TL_N[/math] is a tensor [math]*[/math]-category, with:

• Composition of arrows: by vertical concatenation.
• Tensoring of arrows: by horizontal concatenation.
• Star operation: by turning the arrows upside-down.

Show Proof

This is more of a definition, disguised as a theorem. To be more precise, we already know about (1), from Definition 8.17, and we can talk as well about (2) and (3), constructed as above, with (2) using of course multiplicativity with respect to the scalars, and with (3) using antimultiplicativity with respect to the scalars:

[[math]] \left(\sum_i\lambda_i\pi_i\right)\otimes\left(\sum_j\mu_j\sigma_j\right)=\sum_{ij}\lambda_i\mu_j\pi_i\otimes\sigma_j [[/math]]

[[math]] \left(\sum_i\lambda_i\pi_i\right)^*=\sum_i\bar{\lambda}_i\pi_i^* [[/math]]

And the point now is that our three operations are compatible with each other via all sorts of compatibility formulae, which are all clear from definitions, with the conclusion being that what we have a tensor [math]*[/math]-category, as stated. We will leave the details here, basically amounting in figuring out what a tensor [math]*[/math]-category exactly is, as an exercise.

■

In order to further understand the category [math]TL_N[/math], let us focus on its diagonal part, formed by the End spaces of various objects. With the convention that these End spaces embed into each other by adding bars at right, this is a graded algebra, as follows:

[[math]] \Delta TL_N=\bigcup_{k\geq0}TL_N(k,k) [[/math]]

Moreover, for further fine-tuning our study, let us actually focus on the individual components of this graded algebra. These components will play a key role in what follows, and they are worth a dedicated definition, and new notation and name, as follows:

Definition

The Temperley-Lieb algebra [math]TL_N(k)[/math] is the formal span

[[math]] TL_N(k)=span(NC_2(k,k)) [[/math]]

with multiplication coming by concatenating, with the rule [math]\bigcirc=N[/math].

In other words, [math]TL_N(k)[/math] appears as the formal span of the noncrossing pairings between an upper row of [math]k[/math] points, and a lower row of [math]k[/math] points, with multiplication coming by concatenating, with [math]\bigcirc=N[/math]. As an example, here is a basis element of [math]TL_N(8)[/math]:

[[math]] \xymatrix@R=10pt@C=5pt{ 1\ar@{-}[d]&2\ar@{-}[d]&3\ar@{-}[dd]&4\ar@{-}[d]&5\ar@{-}[d]&6\ar@{-}[ddd]&7\ar@{-}[d]&8\ar@{-}[d]\\ \ar@{-}[r]&&&\ar@{-}[r]&&&\ar@{-}[r]&\\ &&\ar@{-}[rr]&&&&&\\ \ar@{-}[rrr]&&&&&\ar@{-}[rr]&&\\ &\ar@{-}[r]&&&&\ar@{-}[r]&&\\ 1\ar@{-}[uu]&2\ar@{-}[u]&3\ar@{-}[u]&4\ar@{-}[uu]&5\ar@{-}[uuu]&6\ar@{-}[u]&7\ar@{-}[u]&8\ar@{-}[uu] } [[/math]]

Getting back now to what we know about [math]TL_N[/math], from Theorem 8.18, the tensor product operation makes sense in the context of the diagonal algebra [math]\Delta TL_N[/math], but does not apply to its individual components [math]TL_N(k)[/math]. However, the involution is useful, and we have:

Proposition

The Temperley-Lieb algebra [math]TL_N(k)[/math] is a [math]*[/math]-algebra, with involution coming by turning the diagrams upside-down.

Show Proof

This is something trivial, which follows from Theorem 8.18, and can be verified as well directly, and we will leave this as an instructive exercise.

■

Getting back now to knots and links, we first have to make the connection between braids and Temperley-Lieb diagrams. But this can be done as follows:

Theorem

The following happen:

• We have a braid group representation [math]B_k\to TL_N(k)[/math], mapping standard generators to standard generators.
• We have a trace [math]tr:TL_N(k)\to\mathbb C[/math], obtained by closing the diagrams, which is positive, and has a suitable Markov invariance property.

Show Proof

Again, this is something quite intuitive, with the generators in (1) being by definition the standard ones, on both sides, and with the closing operation in (2) being similar to the one for braids, from Theorem 8.9. To be more precise:

(1) The idea here is to map the Artin generators of the braid group to suitable modifications of the following Temperley-Lieb diagrams, called Jones projections:

[[math]] \xymatrix@C=3pt@R=7pt{ &\circ\ar@/_/@{-}[rr]&&\circ&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]\\ e_1=&&&&&&\ldots&\\ &\circ\ar@/^/@{-}[rr]&&\circ&\circ&\circ&&\circ&\circ} [[/math]]

[[math]] \xymatrix@C=3pt@R=7pt{ &\circ\ar@{-}[dd]&\circ\ar@/_/@{-}[rr]&&\circ&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]\\ e_2=&&&&&&\ldots&\\ &\circ&\circ\ar@/^/@{-}[rr]&&\circ&\circ&&\circ&\circ} [[/math]]

[[math]] \ \ \ \ \vdots [[/math]]

[[math]] \xymatrix@C=3pt@R=7pt{ &\circ\ar@{-}[dd]&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]&\circ\ar@/_/@{-}[rr]&&\circ\\ \ \ \ \ \ \ \ e_{k-1}=&&&\ldots&&&&\\ &\circ&\circ&&\circ&\circ&\circ\ar@/^/@{-}[rr]&&\circ&&&&} [[/math]]

As a first observation, these diagrams satisfy [math]e_i^2=Ne_i[/math], with [math]N=\bigcirc[/math] being as usual the value of the circle, so it is rather the rescaled versions [math]f_i=e_i/N[/math] which are projections, but we will not bother with this, and use our terminology above. Next, our Jones projections certainly satisfy the Artin relations [math]e_ie_j=e_je_i[/math], for [math]|i-j|\geq2[/math]. Our claim now is that is that we have as well the formula [math]e_ie_{i\pm1}e_i=e_i[/math]. Indeed, by translation it is enough to check [math]e_ie_{i+1}e_i=e_i[/math] at [math]i=1[/math], and this follows from the following computation:

[[math]] \xymatrix@C=3pt@R=7pt{ &\circ\ar@/_/@{-}[rr]&&\circ&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]\\ &&&&&&&\ldots&\\ &\circ\ar@{-}[dd]\ar@/^/@{-}[rr]&&\circ\ar@/_/@{-}[rr]&&\circ&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]\\ e_1e_2e_1=&&&&&&&\ldots&&&=e_1\\ &\circ\ar@/_/@{-}[rr]&&\circ\ar@/^/@{-}[rr]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]\\ &&&&&&&\ldots&\\ &\circ\ar@/^/@{-}[rr]&&\circ&&\circ&\circ&&\circ&\circ} [[/math]]

As for the verification of the relation [math]e_2e_1e_2=e_2[/math], this is similar, as follows:

[[math]] \xymatrix@C=3pt@R=7pt{ &\circ\ar@{-}[dd]&&\circ\ar@/_/@{-}[rr]&&\circ&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]\\ &&&&&&&\ldots\\ &\circ\ar@/_/@{-}[rr]&&\circ\ar@/^/@{-}[rr]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]\\ e_2e_1e_2=&&&&&&&\ldots&&&=e_2\\ &\circ\ar@/^/@{-}[rr]&&\circ\ar@/_/@{-}[rr]&&&\circ\ar@{-}[dd]&&\circ\ar@{-}[dd]&\circ\ar@{-}[dd]\\ &&&&&&&\ldots\\ &\circ\ar@{-}[uu]&&\circ\ar@/^/@{-}[rr]&&\circ&\circ&&\circ&\circ} [[/math]]

Now with the relations [math]e_ie_{i\pm1}e_i=e_i[/math] in hand, let us try to reach to the Artin relations [math]g_ig_{i+1}g_i=g_{i+1}g_ig_{i+1}[/math]. For this purpose, let us set [math]g_i=te_i-1[/math]. We have then:

[[math]] \begin{eqnarray*} g_ig_{i+1}g_i &=&(te_i-1)(te_{i+1}-1)(te_i-1)\\ &=&t^3e_i-t^2(Ne_i+e_ie_{i+1}+e_{i+1}e_i)+t(2e_i+e_{i+1})-1\\ &=&t(t^2-Nt+2)e_i+te_{i+1}-t^2(e_ie_{i+1}+e_{i+1}e_i) \end{eqnarray*} [[/math]]

On the other hand, we have as well the following computation:

[[math]] \begin{eqnarray*} g_{i+1}g_ig_{i+1} &=&(te_{i+1}-1)(te_i-1)(te_{i+1}-1)\\ &=&t^3e_{i+1}-t^2(Ne_{i+1}+e_ie_{i+1}+e_{i+1}e_i)+t(2e_{i+1}+e_i)-1\\ &=&t(t^2-Nt+2)e_{i+1}+te_i-t^2(e_ie_{i+1}+e_{i+1}e_i) \end{eqnarray*} [[/math]]

Thus with [math]t^2-Nt+1=0[/math] we have a representation [math]B_k\to TL_N(k)[/math], as desired.

(2) This is something more subtle, especially in what regards the positivity properties of the trace [math]tr:TL_N(k)\to\mathbb C[/math], which requires a bit more mathematics. So, no hurry with this, and we will discuss all this, and applications, in the remainder of this chapter.

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General references

Banica, Teo (2024). "Graphs and their symmetries". arXiv:2406.03664 [math.CO].