Alagic, G., Jarret M., and Jordan S.P.

Yang-Baxter operators need quantum entanglement to distinguish knotsarXiv:1507.05979

Their result is what it says in the title. Namely, we comb the knot into a braid, and assign R-matrices to crossings. An R-matrix underlies a linear operator . The authors prove that if the this operator maps product states to product states, then it gives rise (via a certain “taking the normalized trace of the operator the braid gives” procedure) to a trivial quantum knot invariant. Thus, entanglement is an essential part of being a nontrivial quantum invariant. Very cool!

There’s a suggestive picture in my head. Entanglement is all about nonlocality, where two non-interacting objects (an overstrand and an understrand?) cannot be described as separate systems (crossing?), but are inseparably intertwined in that they share some sort of coordination. It’s the entanglement which allows the overstrand to “communicate” to the understrand that it is there, making it possible to construct a nontrivial quantum invariant.

I suspect there’s a lot more to this story. Well done Alagic, Jarret, and Jordan!

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One of the constructions of Tangle Machines, which I previously discussed HERE and HERE, is a universal set of logic gates using coloured tangles (and in fact we cheated, because our colouring wasn’t by a quandle but by a more general algebraic structure). It turns out that this idea isn’t new, and actually it’s been done better a long time ago in a different setting, and in a very nice way (thanks anonymous referee!). Boolean logic can be realized using coloured braids! And it’s even potentially useful in quantum computing! So today I’ll discuss this paper and the papers it references:

Alagic, G., Jeffery, S., and Jordan, S. Circuit Obfuscation Using Braids. In

9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)(eds. S.T. Flammia and A.W. Harrow), Vol. 27, pp. 141–160.

We start with a group in which we choose two elements, one which we call `zero’ and the other which we call `one’. Our operation is conjugation . A *computation* is a braid coloured by elements of , with strands incident to endpoints at the bottom each of which is coloured by a zero or a one called the *input* of the computation, and strands incident to endpoints at the top each of which is coloured by a zero or a one called the *output* of the computation. The braid may also have additional strands at the top and bottom (indeed it must if ). These extra strands, called *ancilla*, may be coloured by any element of . The ancilla serve as catalysts for the computation and nothing more.

How does a braid compute? Moving up through the braid from bottom to top, we encounter braid group generators describing strand crossing over strand , and their inverses. If the colour of strand at the bottom of is and the colour of strand at the bottom of is , then the colour of strand at the top is still while the colour of strand at the top is .

What computations can a braid perform? A-priori, we might suspect that maybe it can’t do all that much, because conjugations are rather special operations. In particular, it doesn’t seem like we should be able to realize an AND gate with a braid. To remind you, , pronounces ` and ‘, returns . So it returns if both and are one, and zero otherwise. The reason AND seems really hard to realize using coloured braids (try it if you don’t believe me!) is that it responds differently to ones than to zeros, so there is an `if’ concept built in; and how could a braid do that? More formally, it’s recovering a product from conjugation, which seems impossible.

It really looks like one should be able to prove that AND can’t be realized by braids, and this is what people indeed believed for a long time. This is one case in which our intuition is wrong, however. Kitaev showed that if the group is the symmetric group , then AND can indeed be realized by a -coloured braid (note: ancilla are important- this, and the fact that is `sufficiently rich’, is what comes to the rescue). The proof is pretty similar to a celebrated result of Barrington and of Krohn-Maurer-Rhodes about representing maps using conjugations. Ogburn and Preskill showed that the alternating group , which is half as large as , is enough.

Monchon constructed a Toffoli gate braid, which is explicitly drawn in Appendix A of Alagic-Jeffery-Jordan. It has 132 crossings and 14 strands (11 of which are ancilla), so it isn’t an easy construction.

To remind the reader, a Toffoli gate is a universal reversible logic gate which contains within it a universal set of gates. Thus, an AND gate is a subcomputation of a Toffoli gate. A Toffoli gate accepts 3 bits , , and , returns and , and inverts if and only if .

Explicitly, encodes zero and encodes one. You can see the construction there.

There are some obvious questions which this raises- as any binary boolean function can be realized by a coloured braid (this is what the above result shows), and because the boolean circuit model is Turing complete, we know that any computable function can be represented by an -coloured braid with ancilla. It’s pretty clear, however, that it would be tremendously suboptimal to realize individual Toffoli gates and to string them all together- the braid language can recover boolean circuits, but not in a simple or intuitive way. How could a simple braid computation be performed? What is the simples braid that would realize some given binary boolean function, and how would we set-out to find it? Is the group the best possible, or are there more reasonable groups (in whatever sense) which do the same job?

How is all of this practical? Well, an element in the braid group can be thought of as a motion of points in a disc (more accurately, as an element of the fundamental group of a configuration space on the disc). The timelines of these moving points trace out the braid- for instance, if we have three points, one of which is stationary and two of which change places, we obtain (or its inverse) in the braid group on 3 strands.

Now imagine 2-dimensional particles- anyons they are called, and they are just-about physical- which move around one another in a disc. Say they have 60 states, and when one moves around another it picks up a phase which corresponds to the conjugation operation. Again, this is just about physical. Then voilá! The above construction gives rise to a Toffoli gate -coloured braid!

There’s more to the story. We can use braid relations, AKA Reidemeister moves on braids, to obfuscate or to disguise our computation for security purposes, so an eavedropper with access to the system would have trouble figuring out which computation was being performed.

Computing seems to have a definite topological side, which is only just now slowly emerging into the light. The visionary who first saw this was Kauffman. There are many sides to the story, and I’m sure that topology appears in many different and even independent ways; a Toffoli gate braid is a particularly pretty one.

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On the converse side, there are a lot of new OA journals out there, some Green and some Gold. This is a very good thing! Submit there, and we can beat predatory publishers through market competition.

One thing I’m seeing right now is a proliferation of metrics enhancers, promoted by publishers. One which has recently caught my attention is Kudos. It seems a decent enough service and I see no reason not to sign on to it, but the picture it paints of today’s research scene is bleak. Research consists, in that picture, of the production of mountains of papers which nobody reads, with researchers having to promote papers on Twitter and Facebook and through short catchy pop-science paper summaries for anyone to actually read them. And budgets being evaluated on the basis of citation counts, h-indexes, and various altmetrics, which may depend primarily on being in many people’s peripheral vision rather that on actually advancing a research field. I like to think things are not quite that bad in mathematics.

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Akbulut-Kirby Conjecture: If 0-surgeries on two knots give the same 3-manifold, then the knots with relevant orientations are concordant.

Note that some knots are not concordant to their reverses (Livingston), but the 0-surgery of a knot and its reverse are homeomorphic, so Akbulut-Kirby had to revise their original formalism to allow for arbitrary orientations. Abe and Tagami recently showed that if the Slice-Ribbon Conjecture is true then the Akbulut-Kirby Conjecture is false. Thus Yasui has eliminated an avenue to falsify the Slice-Ribbon Conjecture.

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G. Borot and B. Eynard, All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials,

Quantum Topol.6(2015), 39-138.

In it, the authors apply the Eynard-Orantin topological recursion to conjecture an all-order asymptotic expansion of the coloured Jones polynomial of the complement of a hyperbolic knot, extending the volume conjecture.

To get an overview of Eynard-Orantin Theory, I’m looking at:

- The original paper.
- Eynard’s own overview– an expanded version of an ICM talk.
- Some superb slides on the topic by Mulase.

Eynard-Orantin Theory studies a certain universal recursion formula based on a plane analytic curve (a Riemann surface) which Eynard calls the `spectral curve’, the Cauchy differentiation kernel, and the residue calculus on it. Like quantum topology itself, the main question about Eynard-Orantin Theory is what it is exactly that it is computing.

The recursion associates to a Riemann surface with extra structure a family of differential forms called `invariants’. The initial terms and are canonically given, and remaining terms are defined by the universal recursive formula by `removing pairs of pants’.

Examples of such recursions had occured previously *e.g.* in work of Mirzakhani.

In low dimensional topology, the curve is canonically associated with the A-polynomial. The form essentially determines the hyperbolic volume of the knot complement, while corresponds to the Reidemeister torsion.

Mirror symmetry relates these quantities with quantum invariants such as coloured Jones polynomials. Dijkgraaf, Fuji, and Manabe pointed out that mirror symmetry relates not only with the coloured Jones polynomial, but also the higher terms. They required them to add some ad-hoc terms which Borot-Eynard eliminate by chosing a different and perhaps more suitable wave function.

The topological recursion lives on the B-model side of mirror symmetry. To trully understand it, however, and to prove it in the cases of interest in low dimensional topology, the goal is to find an A-model proof of it.

More and better conjectures are a sure indicator for the vibrancy of the topic. Eynard-Orantin Theory is coming into play in many places in mathematics, and now it is coming to topology. Have the famous conjectures concerning asymptotics of quantum invariants at last met their match?

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There are a number of different strands coming together, and the one that is pulling the others right now is physics, and string theory in particular. In the physicsÂ approach, knot polynomials are seen as Wilson-loop averages in some version of –dimensional Chern-Simons theory.The knot polynomials which play a particular role are the coloured Jones polynomials and their generalizations, coloured HOMFLYPT polynomials.

Ooguri and Vafa, building on work of Witten, showed how knot and link polynomials are associated to string theories. For an polynomial, certain topological strings are defined, of which wrap around and one corresponds each component of the knot, and the polynomial invariant is thought of as a amplitude of the resulting string in some sense. It turns out that the relevant thing is how the branes corresponding to the knot complement behave at infinity. Mirror symmetry relates this behaviour at infinity with geometric quantities at infinity. We can view several big conjectures in quantum topology at the moment, such as the A-J conjecture and various flavours of the Volume Conjecture. One of the recent strands of work has been on extending these conjectures to links from the physical perspective, which turns out to involve a generalization of the very notion of mirror symmetry itself! Thus, in some sense, low dimensional topology is providing an impetus for deeper physical research into mirror symmetry. This is of course one major aspect of why knot theory has proven historically important inside topology and elsewhere- it provides very nice toy models for other theories. This is the paper I’ve been looking at.

Another active direction has to do with understanding knot homology (Khovanov homology). A lot of sophisticated mathematics and physics seems to be going into this. Everyone seems to be focussing on torus knots and links right now, because the circle action simplifies things- but the concensus seems to be that the results should all generalize somehow. I’ll just mention some things I’ve looked at:

- From the physics perspective, there’s work of Aganagic and Shakirov reframing knot homology (of torus knots at least) in terms of refinements of Chern-Simons theory, rather than more exotic objects. This is again a string theory approach, and follows a conjectural picture of Gukov, Vafa, and Schwarz.
- There’s a lot of work right now on finding lots and lots of gradings (and coloured differentials) on coloured HOMFLYPT homology, in order to make it universal in an interesting way and in order to facilitate its computation. http://arxiv.org/pdf/1304.3481.pdf
- Topological recursion, which is a way of accessing the asymptotic expansion of colored Jones or HOMFLYPT polynomials by looking at the curve associated to the A-polynomial, is a hot topic right now. Various new ideas are popping up, but my general impression looking at this is that what’s going on right now in this field is mainly a lot of groping around in the dark, and that there’s a lot more work to be done in this direction.
- From a mathematical perspective, what I find most exciting are the connections to known deep mathematical structures. Perhaps most intriguing of all is the connection to rational doubly-affine Hecke algebras, AKA rational Cherednik algebras, AKA rational DAHA, which you can read about HERE. Knots and links are arising here via their diagrams, as diagrammatic formalisms for aspects of representation theory, and all kinds of number theory and theory of hypergeometric functions is entering the picture. This direction seems white-hot right now.

I can’t really say that I understand the endgame of any of this, but, on the one hand, the invariants coming out of these lines of work are among the most powerful that we know, and on the other hand, there’s hope that they will become easy (or at least easier) to compute. Beyond all of this, all of this categorified coloured HOMFLYPT theory represents a second entry of quantum knot theory into really deep mainstream mathematics.

Maybe knots and links are nothing more than nice toy objects for all sorts of serious mathematical tools. But I like to dream that maybe the algebraic structure of a knot diagram genuinely does encode some central aspect of the mathematical universe.

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A recently published paper by Gelca and Uribe, which is also the topic of a book by Gelca and some nice slides, constructs the MOO invariant from theta functions completely classically essentially without using anything quantum at all (although the representation theory behind it was originally developed for quantum mechanical purposes). Thus, like the Alexander polynomial and the linking number, MOO is seen to be quantum but also classical.

There is also a more analytic, heat-equation-based way of seeing the same thing due to Andersen, but I haven’t read Andersen’s paper and therefore I can’t say anything about that.

Amongst the most useful functions in mathematics are the trigonometric functions. The function arises as a cross-section of the complex exponential function. It is periodic with period (I’ve become a proponent of tau) and it allows you to parametrize the points on a circle.

The formula for the theta function, as formulated by Jacobi, is

Theta functions can be thought of as sort of complex analogues of trigonometric functions. They are not doubly periodic (otherwise they would have to be constant by Liouville’s Theorem), but they are as close to being doubly periodic as possible, satisfying

where and are integers, and has positive imaginary part. Note that adding an integer to the -input of a theta function leaves it unchanged. Riemann taught us that we should think of such a function as a multi-valued function from the torus, that is the complex plane quotiented by .

Personally, I really like theta functions. In an alternative life I’d like to study theta functions. In another alternate life, I’d like to study K-theory. These are all things I think are really beautiful, but I never really sank myself properly into.

Anyway, the theta function is genuinely periodic in the integer direction. If we’re adding to the -argument, which is times the meridian of the torus and times its longitude, then we can throw away, contracting the meridian to a point. Geometrically, this is like viewing the torus as the boundary of a solid torus, and this is where handlebodies and later Heegaard splittings begin to enter the picture.

Riemann generalized theta functions to higher genera, and somebody (Mumford?) generalized them to depend on a natural number (in Riemann or Jacobi’s case ) to yield *theta functions with characteristic* so that adding times the meridian to gives identity. It is these Reimann theta functions with characteristic which are related to the MOO invariant.

Gelca and Uribe’s basic idea is to think of theta functions as oriented framed multicurves in handlebodies. These basically encode the curves around which the theta function is periodic. Thus, the torus meridian is the Jacobi theta function, the Jacobi theta function in characteristic is times the meridian, and in higher genus, a theta function is just an oriented multicurve which represents some element in a lagrangian subspace of the first homology of the surface.

Theta functions are naturally acted on by two groups, the Heisenberg group and the modular group (AKA the mapping class group of the surface).

Acting by the Heisenberg group multiplies the theta function by some exponential function, which can in turn be represented as an oriented multicurve representing an element of the **other** lagrangian. This is where the framing comes in- the theta function is periodic in the contractible direction, and the multiplicative factor is captured by its framing.

Because we only really care about the homology class of the oriented framed multicurve we can quotient by a load of relations and what we really have here is a skein module called the *reduced linking number skein module*.

This whole construction is wonderfully elucidated by Example 5.6 on Pages 240-241 of Gelca’s book, but all sources of this I can find are behind paywalls, I can’t be bothered to scan and then cut and paste, and I can’t be bothered to redraw it… I apologise.

The action of the mapping class group turns out to be more or less what you’d expect, as long as you remember that the it is the class of the multicurve in the skein module, not the multicurve itself, which represents the theta function. Namely, push the framed multicurve representing the theta function to the boundary of the handlebody in all possible ways, Dehn-twist the picture, and push the result back into the handlebody, and you get the correct theta function (times whatever multiplicative factor).

Now things get a bit more interesting. There’s an identity in the theory of theta functions called the exact Egorov identity, which turns out to say that handlesliding the multicurve representing the theta function (or rather a certain theta series) over the curve representing an element of the mapping class group gives the identity. In other words, invariance under handleslides.

And now for MOO. The reduced linking number skein module modulo the action of the mapping class group turns out to be isomorphic to the field of complex numbers. Viewing the theta function multicurve as a surgery link, such that surgery on by this link gives our manifold, the image of this link is the MOO invariant. It is invariant under handleslides by the exact Egorov identity. We have thus constructed the MOO invariant without a shred of quantum field theory.

I’ll conclude with some random thoughts:

- Yoshida has a paper in Annals proposing to reformulate the Reshetikhin-Turaev invariants in terms of “higher level theta functions”. Hansen and I wrote some notes on this paper… Teleman pointed out a gap. I’m currently doubtful that that such a simple formula has any chance to hold.
- I have wondered for a long time whether and in what sense -manifolds “exist”. Surfaces certainly “exist” because they arise inevitably from other objects which “exist”, for example as closed Riemann surfaces. But it’s never been clear to me that -manifolds “exist” in the same way, and reading Poincaré’s original papers, it wasn’t clear to Poincaré either.
At first I was excited about the Gelca-Uribe paper, because it seemed they were pulling –manifolds “out of a hat”, and that their Heegaard splittings were an inevitable consequence of the theory of theta functions. I no longer feel that this is the case. Their whole theory is really about skein modules, and can be reformulated in diagrammatic algebraic terms without reference to topology.

- Continuing on from that last thought, somebody ought to reformulate their paper completely diagrammatically! Doing so might provide new ways of thinking about aspects of the theory of theta functions, and so might be useful in a wider sense (I dream). For example, the exact Egorov identity corresponds to invariance under handleslides. But we know that handleslides are unzips of embedded theta graphs. So if embedded curves correspond to theta functions, what to embedded theta graphs correspond to? What are embedded trivalent graphs in analytic language? What are unzips?
- It would be cool to explain the whole idea in 2-3 pages with no reference to fancy concepts and formulae, and with nothing quantum in sight. The core idea (which I interpret as sort of a diagrammatic calculus for theta functions) is so simple and elegant that I believe it deserves to be better known.

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- Major improvements to the link and planar diagram component, including link simplification, random links, and better documentation.
- Basic support for spun normal surfaces.
- New extra features when used inside of Sage:
- HIKMOT-style rigorous verification of hyperbolic structures,

contributed by Matthias Goerner. - Many basic knot/link invariants, contributed by Robert

Lipschitz and Jennet Dickinson. - Sage-specific functions are now more easily accessible as

methods of Manifold and better documented. - Improved number field recognition, thanks to Matthias.

- HIKMOT-style rigorous verification of hyperbolic structures,
- Better compatibility with OS X Yosemite and Windows 8.1.
- Development changes:
- Major source code reorganization/cleanup.
- Source code repository moved to Bitbucket.
- Python modules now hosted on PyPI, simplifying installation.

All available at the usual place.

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Deraux, M. & Falbel, E. 2015 Complex hyperbolic geometry of the figure-eight knot.

Geometry & Topology19, 237–293.

In it, the authors study a very different geometric structure for the figure-eight knot complement, as the manifold at infinity of a complex hyperbolic orbifold.

Geometrization contains a characterization of manifolds that admit a geometry modeled on real hyperbolic 3-space. CR-geometry, on the other hand, is about manifolds which admit a geometry that is locally modeled on the CR structure (the largest subbundle in the tangent bundle that is invariant under the complex structure) of viewed as the boundard of the unit ball . Such a structure is called *uniformizable* roughly if our manifold M has a discrete cover which sits inside , with the CR-structure on M lifting to the standard CR-structure on (I think).

Quotients of such as lens spaces give the simplest examples of manifolds with uniformizable spherical CR-structures. A nice question, which is a bit reminiscent of a part of Geometrization, is to topologically classify which manifolds admit such structures, and how many of them there are.

CR structures are very good to have around; CR-structures are the kind of structures you get on real hypersurfaces in complex manifolds. There is a whole theory around them which parallels Riemannian geometry, and there seems to be a lot of deep analysis going on (which I know next to nothing about) around trying to understand various fundamental operators and their spectra (sub-Laplacian, Kohn Laplacian…).

This paper gives a really interesting example of a manifold with a uniformizable spherical CR-structure, namely the figure-eight knot complement. This manifold played an important motivational role in the development of real hyperbolic geometry, and the hope is that here too it will provide a good motivational example to study, which is simple enough to work with “by hand” but complicated enough to exhibit somehow “generic” behaviour.

A quite fascinating research programme! Looking at the a class of manifolds as manifolds at infinity of complex hyperbolic orbifolds! I look forward to reading this paper and to learning more about this!

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V.F.R. Jones,

Some Unitary Representations of Thompson’s Groups and, arXiv:1412.7740.

Links occur as braid closures, and so links can be studied via braid theory. This is the starting point for the Jones polynomial , and it’s very nice, because braids form a group (good algebraic property) which is orderable (even biorderable) and automatic (even biautomatic). But here are some complaints I have:

- Combing a link introduces an artificial structure- a `timeline’ where strands always `move forward through time’- which a link doesn’t naturally have. By trading links with braids, maybe we’re missing out on some essential
*je ne sais quoi*that makes a link a link. - More concretely, all known quantum link invariants come from Lie (bi)algebra constructions. But not all subfactors arise this way. For example, the Haagerup subfactor does not. If we believe that quantum invariants should correspond to finite index subfactors and not just to e.g. quantum groups, then surely braids are the wrong way to go.
- Everyone and their cousin uses braid groups nowadays.

Jones’s manuscript provides something new, that is a way to obtain links from elements of Thompson groups, and elements of Thompson groups from links. His algorithm seems just as natural as combing tangles to obtain braids. And it leads directly to new polynomial link invariants.

Thompson groups may be just as nice as braid groups. At the very least, they’re new in this context. It’s an open problem whether they are orderable, automatic, or amenable- as far as I know, they certainly might be. Indeed, they might be *much nicer*, because, unlike braid groups on more than two generators, they don’t contain a subgroup isomorphic to the free group on two generators.

As far as I know, Thompson groups have only shown up in mathematics so far as counterexamples. But in Jones’s paper, they naturally pop out of the structure of planar algebras. There’s nothing artificial about the correspondence between knots and Thompson group elements.

**Edit**: Yemon Choi points out a very nice interpretation of Thompson group elements which both sense in context and also in which the Thompson group is not a counterexample. The reference is HERE. What I said about Thompson groups only being counterexamples is even more false than that- they’ve featured before in quantum topology (and I wonder what the relation is, if any, with the construction in Jones’s new preprint).

**Silly convention**: Elements of braid groups are called braids. I suggest that elements of Thompson groups be called Thompsons. Indeed, two (dependent) elements of the Thompson group are used to construct links, and so, to be precise: What is at hand is truly a case of Thompson and Thompson.

**Edit**: Scott Carter points out that “Thompson and Thompson” ought to be Thomson and Thompson. Which makes sense to me- the link is a diagrammatic inner product of one Thompson with another Thompson acted on by a representation . Perhaps it’s reasonable to call a new polynomial invariant the `Thomson and Thompson Polynomial’ after all, with the `p’ reminding us of the action of .

Jones investigates the scaling limit of tangle diagrams, by letting the number of boundary points of the tangle fill out the boundary disc of the diagram (tangles with more and more strands). It wouldn’t make much sense, at least diagrammatically, to do this all at once- what would a tangle with uncountably many endpoints look like?- but what Jones does instead is to present a directed set construction, increasing the number of tangle endpoints inductively by gluing on tangles with . This mimics the block spin renormalization procedure from physics on a diagrammatic level. Jones credits Dylan Thurston with having suggested this idea to him.

Because the only way to add endpoints to a tangle is with caps or with cups, the number of intervals between tangle endpoints is always even. Jones chooses the interval endpoints (tangle endpoints) to correspond to dyadic rationals, that is to numbers of the form for some . The group of PL homeomorphisms of whose non-differential points are the dyadic rationals and whose slopes are all powers of is Thompson’s group , and this is how Thompson groups enter the picture.

The preprint opens up vast new vistas, and the relationship between links and Thompson groups which it points out is at once so powerful and so natural (*e.g.* the analogous construction for a different planar algebra gives the Tutte polynomial of a planar graph) that it would be difficult to imagine the new polynomials which his construction defines *not* once again reshaping major parts of quantum topology.

Two challenges are presented at the end of the preprint:

- Links are obtained from Thompsons by an
*inner product*, that is by pairing a tangle obtained from the Thompson with a mirror image of another tangle obtained in a different way from the same Thompson. This sort-of reminds me of things like Heegaard splittings by the way, although of course that’s a very different idea. Anyway, the -index of a link is the smallest number of leaves of a Thompson (presented as a pair of rooted binary trees) required for this construction. Investigate bounds for the -index. - Find and prove Markov Theorem for Thompson group presentations. This has got to be a major goal. I would be totally amazed if the `Markov moves’ for Thompsons were anything other than the set of moves on pairs of trees corresponding to the Reidemeister moves, as Jones himself suggests. This makes Thompsons *much more* natural group element representations of tangles than braids are.

Another obvious question is how to compute the polynomial invariant defined in this preprint, at least for small examples, and to establish whether it satisfies a reasonable-looking skein relation.

To complain (never forget to complain!), like the Jones polynomial, the construction of this invariant also uses planarity in an essential way, and therefore it is difficult to see how its construction might extend to virtual and to welded tangles.

I thank Ian Agol for drawing my attention to this preprint.

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