A groundbreaking paper which made a deep impression on a lot of people, including me, was Cochran-Orr-Teichner’s Knot concordance, Whitney towers and signatures. This paper revealed an unexpected geometric filtration of the topological knot concordance group, which formed the basis for much of Tim Cochran’s subsequent work with collaborators, and the work of many other people.

In this post, in memory of Tim, I will say a few words about roughly what all of this is about.

It would be very nice to be able to equip the space of knots with a good algebraic structure. Somehow, the natural binary operations on knots seem to be the satelite operations, of which the connect sum may be considered a special (degenerate) case.

Unfortunately, unless is the trivial knot, there is no knot which can be `satelited’ to to obtain a trivial knot. Thus, the set of knots under connect sum, or indeed under any satelite operations, does not form a group.

But the set of knots can be quotiented by an equivalence relation called concordance, and concordance classes of knots do form a group under the connect sum. Concordance takes place in an ambient 4-dimensional space, and so it provides an avenue for knot theory to be used to study 4-dimensional topology. For most of the world, this is the ultimate motivation to study link concordance. This point of view is beautifully laid out in Freedman-Quinn’s Topology of 4-Manifolds.

The way the field has gone, every conjecture about how `good’ the structure of the link concordance group is has turned out to be wrong. Almost every paper which has come out about knot concordance in the last 20 years, as far as I know, has been a negative result. Cochran’s work has been instrumental in showing `how bad things are’. The group isn’t trivial, and its non-triviality is detected by the Casson-Gordon invariants. The next step was taken in Cochran-Orr-Teichner; Casson-Gordon invariants do not detect knots up to concordance. They’re just the first step in an infinite geometric filtration of invariants, which is non-trivial at every step.

Stavros Garoufalidis suggested a long time ago that the Cochran-Orr-Teichner filtration should be investigated through the lens of quantum topology. This was a major research interest of mine at one point, and to the best of my knowledge, nobody has yet achieved this aim. I remain convinced that this is an interesting avenue of research worthy of future investigation.

A recent paper of Tim Cochran which captured by imagination was his joint work with his mathematical daughter Shelly Harvey on The Geometry of the Knot Concordance Space. In it, Cochran and Harvey suggest viewing the topological knot concordance space in a metric space in various different ways, and suggest investigating its coarse geometry. Again, the structure isn’t neat- it isn’t quasi-isometric to a finite product of hyperbolic spaces- but it is possible to address the question of whether it is what the authors call a `fractal space’, that is roughly a space which admits a natural system of self-similarities. The conjecture that the knot concordance space is a fractal space looks intuitively highly plausible to me; and the investigation of the coarse geometry of the knot concordance space looks to me like a marvelous research project which will surely lead to many fruitful results in the future, both positive and negative.

And apart from all of his fantastic and groundbreaking ideas, Tim was an inspiring teacher, lecturer, and colleague: a true powerhouse of good mathematics.

RIP, Tim Cochran.

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Feedback is very welcome (as are “how do I…?” questions), especially for a brand new port such as this.

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The question asks whether, rather than searching for Reidemeister moves to simplify a knot diagram, we could instead search for “big Reidemeister moves” in which we view a section which passes underneath the whole knot (only undercrossing) or over the whole knot (only overcrossing) as a single unit, and we replace it by another undersection (or oversection) which has the same endpoints.

This question (or more generally, the question of how to efficiently simplify knot diagrams in practice) loosely relates to a fantasy about being able to photograph a knot with a smartphone, and for the phone to be able to identify it and to tag it with the correct knot type. Incidentally, I’d like to also draw attention to a question by Ryan Budney on the topic of computer vision identification of knots, which is topic I speculated about here:

A core question to which all of this relates is:

And perhaps more generally, are there any very hard ambient isotopies of knots?

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Gilmer, P.M. and Masbaum, G., Maslov Index, Mapping Class Groups, and TQFT, Forum Math.

25(2013), 1067-1106.

It makes me think a lot about just what the anomaly `actually means’…

I’ll start with some vague philosophical musings. I’m quite taken with the information physics idea that everything is information, and I think that Chern-Simons theory should really be all about information as well. But I’m not sure how. A google search turns up load of physics papers with keywords “anomaly”, “Chern Simons”, and “entropy” in close proximity, so I’m sure that some physicists know the whole story, but I don’t. Maybe somebody could explain it in the comments?

There’s a theme in physics which says that the `interesting’ information content of naturally occuring systems on `things with boundaries’ is contained entirely on the boundary and not in the interior. Manifestations of this theme include the holographic principle which roughly claims that the maximal entropy in a region scales like the surface area of the boundary of that region instead of like its volume (so that the entire information content of a black hole lies on its event horizon), and area laws which roughly claim that the amount of quantum entanglement between particles in a region and in its complement depends on the area of the boundary of the region and not on the volume of the region.

Because every closed oriented –manifold bounds an oriented -manifold, this physics theme suggests a way in which physics might be unreasonably effective in low dimensional topology. Namely, a physically interesting information measure on a bounded -manifold ought to give rise to a -manifold invariant. This is sort-of the meta-intuition I have for why we have Topological Quantum Field Theory (TQFT) invariants of -manifolds. My vague feeling is that because Fisher information and the Chern-Simons action both have something to do with curvature, perhaps Chern-Simons Theory and quantum -manifold invariants have a clear and legitimate information-physics interpretation (if you know what it is, please tell me!).

Be that as it may, it turns out that quantum invariants coming from -dimensional TQFTs tend not quite to give numerical topological invariants of their –dimensional boundaries. We need an integer worth of extra information from the interior of the bounded -manifold to get a numerical –manifold invariant. This `extra information from the interior’ is called the anomaly. The anomaly ought not to even exist for `physically interesting information’ according to the naive interpretation of the physicist’s theme outlined above. Maybe that’s why it’s called an *anomaly*- because a physicist would wish that it not exist. A lot of surveys and entry-level texts seem to gloss over the anomaly, maybe partly for that reason.

It seems to be only recently that anomalies are becoming respectable. Perhaps this is due to Lurie’s higher categorical formalisms which sheds some light on anomolies, and perhaps to work on TQFT’s for manifolds with boundaries and corners, and perhaps to interest in “type II superstring orientifolds” in which anomalies are both tricky and important, but in any event, there does seem to be a resurgence of interested in anomalies. It seems that an anomaly should be considered an (invertible) field theory itself. This paper is the most interesting recent paper I have seen on the subject… maybe I’ll talk about it another time.

Back on the subject of Chern-Simons TQFTs, or rather Reshetikhin-Turaev TQFTs, our setting is a closed oriented -manifold bounding an oriented -manifold. This -manifold matters only up to cobordism (i.e. two cobordant -manifolds are considered equivalent from the point of view of our TQFT, because, while there may be a wee bit of information in a dimension 4 interior, there is no relevant information in a dimension 5 interior). Cobordism classes of oriented -manifold are classified by the signature (an integer), and I think that this is the secret reason that the signature keeps popping up in all kinds of formulae for quantum invariants of -manifolds.

So really, the domain of our TQFT ought to be a pair consisting of a manifold and an integer, or a manifold with an integer-worth of extra structure. How to usefully specify that integer? There are a variety of of approaches- structures, -framings, various choices of largrangian thisses or thats, Masbaum-Roberts explicit methods… There’s an algebraic approach as well, in which we trade our -manifold for a mapping class group element. Remember how every -manifold has a Heegaard splitting? This constructs our -manifold by gluing together two genus handlebodies using an element of the mapping class group . The TQFT induces a representation of the mapping class group which is only projective but not linear because of the anomaly. Gauge-fixing/ choosing a cobordism class of a bounded -manifold/ fixing the anomaly corresponds to choosing a central extension of the mapping class group. And it turns out the has a universal central extension, and (unsurprisingly) the cohomology class of this extension is a generator of a cohomology group which is isomorphic to the integers (the most famous such generator is the Meyer cocycle, and the second most famous is its cohomological negative, the Maslov cocycle).

So the whole problem of fixing the anomaly has been algebratized, and the goal has now become to describe explicit elements of the universal central extension of , which are the algebraic objects which have now replaced “-manifold together with a cobordism class of -manifolds which it bounds”.

That’s pretty-much the goal of Gilmer-Masbaum. Some major steps which were outlined in Walker’s iconic TQFT notes are worked out explicitly. This is at long last a careful treatment of a TQFT anomaly. I know more than I knew before.

Now that we have a technically coherent and careful treatment of the anomaly in the Chern-Simons context which seems more or less amenable to concrete computation (I’m haven’t followed through the details carefully enough to strengthen the above sentence), the next thing I’d love to read would be a survey-level treatment of the anomaly, which explains all of the different approaches to fixing it, the strengths and weaknesses of each, and how they relate to one another.

I’d also really like to understand how quantum invariants measure information (entropy), and in particular what information is measured by the anomaly. And what is the conceptual reason that Chern-Simons theory violates the theme that all interesting information lies on the boundary? Or maybe it doesn’t? I wish I understood more.

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This is great! Knots looking cool in semi-mainstream media!

A completely unrelated thing I’m chuffed about is that, a week ago, I broke my 2004 record of 550km in 10 days by walking 600km, Osaka to Tokyo, in nine and a half days. The trick? Walk slowly but without resting, and sleep less.

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There’s several new features, such as:

- rigorous certification of hyperbolicity (using angle structures and linear programming);
- fast and automatic census lookup over much larger databases;
- much stronger simplification and recognition of fundamental groups;
- new constructions, operations and decompositions for triangulations;
- and more—see the Regina website for details.

You will find (1) and (2) on the Recognition tab, (3) on the Algebra tab, and (4) in the Triangulation menu.

If you work with hyperbolic manifolds then you may be happy to know that Regina now integrates more closely with SnapPy / SnapPea. In particular, if you import a SnapPea triangulation then Regina will now preserve SnapPea-specific data such as fillings and peripheral curves, and you can use this data with Regina’s own functions (e.g., for computing boundary slopes for spun-normal surfaces) as well as with the in-built SnapPea kernel (e.g., to fill cusps or view tetrahedron shapes). Try File -> Open Example -> Introductory Examples, and take a look at the figure eight knot complement or the Whitehead link complement for examples.

Finally, a note for Debian and Ubuntu users: the repositories have moved, and you will need to set them up again as per the installation instructions (follow the relevant Install link from the GNU/Linux downloads table).

Enjoy!

- Ben, on behalf of the developers.

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A binary operation is **distributive** over another operation if . If then the operation is said to be **self-distributive**. Examples of self-distributive operations include conjugation , conditioning (assume X and Y are both Gaussian so that such a binary operation makes sense, essentially as covariance intersection), and linear combinations with (say), and elements of a real vector space.

Two nice survey papers about self-distributivity are:

- J. Przytycki, Distributivity versus associativity in the homology theory of algebraic structures. arXiv:1109.4850.
- M. Elhamdadi, Distributivity in Quandles and Quasigroups. arXiv:1209.6518

I won’t survey these paper today- instead I’ll relate some abstract philosphical musings on the topic of associativity vs. distributivity.

Algebraic topology detects information not only about associative structures like groups, but also about self-distributive structures like quandles. I wonder to what extent distributivity can stand in for associativity. Might our associative age give way to a distributive age? Will future science will make essential use of distributive structures like quandles, racks, and their generalizations? At the moment, such structures appear prominently only in low dimensional topology.

I think that there is a philosophical difference between an *associative world* and a *distributive world*. The associative world is a geometric world; a world in which space and time are important and fundamental concepts. The distributive world seems different to me. I think that it is a quantum world without space and time, in which only information exists.

Analogous to mass being a manifestation of energy via , so energy may be viewed as a manifestation of information via Shannon/Boltzman entropy. From a physics perspective, there exists the `future physics’ idea that space and time might be emergent, and that the only true fundamental physical quantity is information. Vendral has written a book expounding this point of view. If this idea takes hold, then future fundamental physics will include information physics, and I believe that its underlying mathematics will belong not to the associative world, but rather to the distributive world. I speculate that information physics will some day make essential use of quandles, racks, and related structures.

The associative world is more familiar, so I’ll begin with a survey of the history of the distributive world, followed by a brief survey of both worlds. Then I’d like to compare and contrast them.

But perhaps there is more in heaven and earth than is dreamt of in associative philosophy. The person credited with this observation is the great American logician C.S. Pierce when in 1880 he concluded:

These are other cases of the distributive principle… These formulae, which have hitherto escaped notice, are not without interest.

For the next century or so, like stray ants who don’t follow paths to establish food sources, there were occasional bursts of realization that distributivity might be fundamental. Notable among the mavericks is M. Takasaki. Alone and isolated as a fresh Japanese math PhD in Harbin during wartime, Takasaki defined an involutive quandle in 1942 as an abstraction of the geometric idea of a symmetric linear transformation. Takasaki envisioned his self-distributive `keis’ as alternatives to groups, but his dream is still largely unrealized. In 1959 another group of mavericks, John Conway and Gavin Wraithe, discovered quandles and racks whose operations were abstractions of the conjugation operation in group theory. But it was only in 1982, with the work of Joyce, and another great independent discoverer Matveev, that quandles and racks entered the mathematical consciousness. Other independent thinkers who discovered or rediscovered such structures (racks, in this case) include Brieskorn and Kauffman. There were ideas about using quandles in the context of geometry (Takasaki), singularity theory (Breiskorn), and symmetric spaces (Joyce), but I think that quandles and suchlike only really ever took hold in low dimensional topology.

From the knot theorist’s perspective, quandles and racks were popularized by Fenn, Rourke, and Sanderson’s 1992 discovery of rack cohomology (the quandle version is due to Carter et.al., and the history is explained in his survey). It turns out that algebraic topology works just fine when associativity is replaced by distributivity, and quandle cocycles yield computable knot invariants. Algebraic topology of quandles and racks has become a bit of a subfield inside low dimensional topology, and this is more or less the only quasi-popular use of quandles of which I am aware.

Note: quandles and racks are only part of the mathematical consciousness of low-dimensional topologists! Physicists, biologists, chemists, computer scientists, engineers, and the rest of humanity don’t really know what a quandle is. I think that we’re a few steps ahead of the pack.

Viewed broadly enough, I think that every associative operation is an abstraction of one or more of the following archetypes:

**Addition**: The archetypal geometric picture for addition is concatenation of segments of specified lengths. To add natural numbers and , start with a number line, represent the number by the segment , mark a second point at distance from point in the positive direction representing as , and concatenate the two line segments to represent by the concatenated directed segment . Associativity is seen in the geometry (the space), in that , and both are represented by the same directed segment .**Multiplication**: The archetypal geometric picture for multiplication is to fill a cycle by a cell. To multiply natural numbers and , represent by the directed segment along the x-axis and represent by the directed segment along the y-axis, and form the rectangle . The product is visualized as the area of the rectangle (the 2-cell) in the upper right quadrant whose boundary is the above rectangle. Associativity is seen from the fact that both measure the area of the same cube in Euclidean 3-space.

In the associative world, it makes sense to represent objects by 0-cells and maps by 1-cells. Data structures can sensibily be represented using labeled graphs. A composition of maps from an object represented by a vertex to an object represented by a vertex on a graph is represented by a path on the graph between and . It makes sense to represent a composition of maps in this way thanks to associativity- there is no need for brackets along the path. Maps between maps can be represented by directed higher cells, sort of like our geometric picture for multiplication. Again, this makes sense thanks to associativity.

The claim that I am making is that formalisms such as category theory and graph theory are native to the associative world. So too classical probability theory. Probabilities are added and multiplied, and they are always between and . So too, the theory of computable functions relies on associative compositions.

Let’s consider the following archetypes for distributive operations:

**Convex combination**: Our first archetype is with elements of a real vector space, and .**Conjugation**: The second archetype is .

Neither of these operations are associative in general. For example,

.

Both operations have natural archetypes in the world of information (their best-known archetypes are in low dimensional topology of course). One archetype for convex combination is from Bayesian statistics. I estimate the mean of data based on a sample, and I obtain a number . But I have a prior belief that the mean should actually be . Based on external information (*e.g.* the number of elements in the sample and my choice of standard of `absolute credibility’), I compute a constant , and my updated estimate becomes . Fusion operations satisfy .

I can view convex combination as `mixing'; I mix units of with units of .

An archetype for conjugation might quantum interference, where quantum evolution of density operator conjugates it by a unitary operator. So `interaction’ is convex combination, and `evolution’ is conjugation…

It doesn’t make much sense to represent words in **D**istibutive **N**on-**A**ssociative (DNA) structures using concatenated edges in labeled graphs, because concatenating edges would not correspond to a well-defined composition of operations (because of non-associativity). There are still notions of Cayley graphs for quandles and racks (e.g. Chapter 4 of Winker’s thesis); I don’t feel qualified to comment on these.

The natural way to represent words in DNA structures, I would think, would be to walk along (modified) tangle diagrams. A Reidemeister III move on tangle diagrams coloured by distributive structures makes sense, because :

One idea behind tangle machines is to make use of this fact to do distributive algebra on tangles. So, while for an associative operation one might diagrammatically represent in some way like this:

In a distributive world we might represent maybe like this:

Is there a DNA (**D**istributive **N**on-**A**ssociative) analogue to category theory, where morphisms distribute but don’t have associative composition? I wonder… I also wonder whether quantum probability, suitably formulated using convex combination and conjugation operations, would be a valid DNA analogue to probability. If we take Reidemeister 2 seriously, and apply it to the DNA structure of Gaussian distributions whose operation is conditioning, we have to define `unconditioning’ X by Y, and the resulting probability might be negative. Classically this makes no sense, but from a quantum perspective it’s fine, and even natural; it feeds my confirmation bias for the philosophical thesis we are considering. Consider the following quote by Feynmann:

The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative.

Most quantum topology of tangles is actually associative, in that we speak of the *category of tangles*, whose operation is stacking. Objects are tangles with tops and bottoms:

Stacking is an associative operation. Via a TQFT formalism, braided monoidal bla bla bla categories give rise to tangle invariants and to knot invariants.

Dror Bar-Natan suggested that this might not really be the right way to think about tangles. Tangles should not have `tops’ and `bottoms’- such information certainly does not exist topologically. Instead, endpoints of tangles should be marked points around a disc (more generally a disjoint union of spheres with holes):

Surprisingly, this disc, which (partially following Bar-Natan) I think we should call the `firmament’, is quite important: See Dror’s “cosmic coincidences” talk.

You then concatenate by connecting two endpoints, and extending the firmament appropriately. This way of thinking is behind Dror’s Khovanov homology work, and current work on various w-knotted objects by him and collaborators.

A major difference between the “stacking” worldview and the “circuit algebra” worldview is that the former views a tangle as a morphism from data stored in the “boundary points at the bottom” to data stored in the `boundary points at the top”. So a tangle encodes an operator (reference: Chapter 3 of Ohtsuki’s book Quantum Invariants). But in the latter worldview, a tangle just encodes some relationship between a bunch of data at endpoints. In this worldview, a tangle cannot encode a mapping in any meaningful sense- this worldview does not support the idea of operator invariants of tangles. This worldview isn’t imposing any non-topological artificial structure on tangles. All it has are the Reidemiester moves, including Reidemeister III. So tangles in this sense are a distributive-world structure.

As an example, let’s consider a single crossing. When tangles express morphisms to be stacked, this `represents’ an R-matrix representing a linear transformation from a vector space to itself. Bottom happens before top, and there’s an implicit time axis. But with no up-down information, it represents a transition from one undercrossing arc to the other by way of an overcrossing arc, . No braided monoidal categories anywhere it sight.

Having tops and bottoms to tangles is nice because associative structures tend to be more amenable to explicit computation. Computing in a quandle is usually very hard, perhaps **because** the Turing machine formalism itself belongs to the associative world. My vague thought is that we can probably do a lot better in the future using different sorts of (probabilistic?) tools… but that’s a speculation for another day. I also think that distributed and parallel computing could provide better ways to compute in distributive structures, and may in turn have distributive algebraic models (Marius Buliga has some work in this direction: e.g. Chemlambda, joint with Louis Kauffman).

Although people have began looking at the distributive world only quite recently, it’s already rife with terminology. The more this world is explored, the more terminology there will be, so I’d just like to point out some parallels. Recalling some axioms, consider the following axioms on a set with a set of binary operations :

**Idepotence**: for all $a\in Q$ and for all .**Injectivity**: If for some and , then .**Distributivity**: for all and .

If contains only one element , and assuming that is also surjective for all , we have the following cases.

- If is both distributive and idempotent then you’re looking at a
*spindle*. - If is distributive and injective then you’re looking at a
*rack*. - If all three, then you’re looking at a
*quandle*. - Only distributive and you’ve got yourself a
*shelf*.

Lots of operations and you might add words like *multi-*, so you have multiracks, multiquandles, multishelfs… or maybe G-families of quandles, or irq’s, or whatever.

Staring at these DNA structures though, they look quite parallel to familiar associative structures. Injectivity parallels invertibility of elements (*i.e.* it tells us that is left-invertible) and distributivity parallels associativity. I’m not sure what the parallel associative concept to idempotence is (idempotence involves both the element and the operation ), but I think it might be orthogonality; because reminds me of in orthogonal groups. Also, conjugation distributes over convex combination, but not the opposite. We might therefore think of convex combination as being parallel to addition, and conjugation as parallel to multiplication. So, using the adjective `DNA’ for `distributive non-associative’, a quandle might be a `DNA orthogonal group’, a rack might be a `DNA group’, if you have both conjugation and convex combination, maybe you have a `DNA near-field‘.

Why would you use a structure like that? Well, as an example of how it might be useful, here’s an AND Gate without trivalent vertices, where and stand in for the digits and correspondingly. The operations are convex combinations, and is conjugation.

It seems to be very natural to consider structures where has lots of elements- it doesn’t inhibit their algebraic topology, it occurs naturally in our archetypes (in the Bayesian probability archetype, to expect all `new’ information to have the same credibility is unnatural; see also Buliga’s work on irq’s, emergent algebras, and related structures- all DNA structures HERE and HERE), and it allows us to construct various topological invariants such as invariants of knotted handlebodies (“A G-family of quandles and handlebody-knots” by A.Ishii, M. Iwakiri, Y. Jang, and K. Oshiro).

The term `DNA’ suggests that distributive non-associative structure are in some way fundamental (like DNA is fundamental to cells in living organisms), and I think that they are. There are some simple transforms between the associative world and the distributive world too: Given a group, you can look at it’s associated conjugation quandle. Conversely, automorphisms of a quandle form a group. In another direction, you can represent a tangle diagram by a graph for example by representing each arc as a vertex, drawing edges from the vertex representing the overcrossing to the two vertices representing undercrossings, and drawing an edge between the undercrossings. By doing this you’ve thrown away all your symmetries- graphs are rigid and there are no Reidemeister moves on graphs. This construction is also partially reversible.

I think there’s a whole distributive world waiting to be discovered, and we’re just looking at the tip of the iceberg. I can’t wait to see these distributive structures play a role outside low dimensional topology, in other parts of mathematics and in other sciences!

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One major problem with this story, and with similar stories, is that the knot diagrams have to be photographed (and thus identified) by hand. The pictures are not always easy to interpret (e.g. distinguishing overcrossings from undercrossings):

Also resolution might be low, objects might be in the way…

This is a computer vision problem as opposed to a math problem- but wouldn’t it be nice if a computer could recognise a knot type from a suboptimal picture? If you could snap a picture of yourself standing in front of an knot making bunny ears behind it, and your computer would automatically tag it with the correct knot type? Furthermore, wouldn’t it be nice if a computer could recognise your knot on the basis of many noisy pictures, perhaps taken from different angles?

In computer vision, there is a concept of a geon. A geon is a fundamental shape, such as a sphere or a cube, which a computer or the human brain can recognise from any angle even if the resolution is low and even if there are other objects in the way. The Recognition by components (RBC) theory asserts that vision is a bottom-up process which works by combining geons.

Geons have always been defined geometrically. A. Carmi suggested to me that topological geons should also exist. Indeed- a human can recognise a trefoil in any “reasonable” (i.e. fairly close to “minimal energy”) configuration, from any angle, even at low resolution and even if there are objects in the way. A computer ought to be able to do the same thing; and actually much more.

Computer vision is the most intensively researched field in applied computer science. It contains a huge body of research; all geometric and analytical as far as I know. Would it help to introduce some low-dimensional topology? Could topological geons such as knots and links help computers to see the world better? This would be a further manifestation of a “low-dimensional topology of information”!

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Here is a lovely, simple theorem.

Given a non-trivial link in the 3-sphere with all pairwise linking numbers equal to zero, it is impossible to put that link into a position where every component is a round circle.

Definition: A link in S^3 is “round” if every component is the intersection of an affine-linear 2-dimensional subspace of R^4 with S^3.

The idea for the proof is that if all the components of a link are round the linking number of components would either be 0 or +-1, depending on whether or not the affine-linear 2-discs they bound in D^4 intersect or not. If the pairwise linking numbers were zero, the discs do not intersect, so shrinking the radius of the sphere produces an animation where the link component radii go to zero, and the link components remain disjoint.

A corollary of this observation is that the Borromean rings (and the Whitehead link, etc) can not be put into a position where every component is round — this holds true in R^3 as well as S^3, since stereographic projection preserves round circles.

Although the Borromean rings can not be realized by round circles in R^3, they can be realized by ellipses. Haefliger used a higher-dimensional version of the ellipsoidal Borromean rings to construct his exotic smooth embedding of S^3 in S^6, so this is an idea that “has legs.”

Here is one elliptical embedding of the Borromean rings in R^3:

x^2 + 2y^2 = 1, z=0

y^2 + 2z^2 = 1, x=0

z^2 + 2x^2 = 1, y=0

You might ask “what does all this have to do with spaces of knots?” It’s about time we got to that.

Much time has been spent in geometric topology on relatively foundational problems, like classification problems. Manifolds up to diffeomorphism. Rigid hyperbolic structures. Various cobordism relationships between manifolds, surgery relationships, and so on. These are relatively discrete-ish problems. There are times when that’s less of the case. Cerf theory, sweep outs, singularity theory, open book decompositions and Teichmuller theory all have aspects of the spaces-of-things philosophy, where one studies families.

In spaces of knots, the objects of study tend to be things like the space of all C^1-smooth embeddings S^1 –> S^3 with the C^1-metric topology. That’s the topology where one takes as a distance between two smooth embeddings f,g : S^1 –> S^3 the maximum of |f(z)-g(z)| + |f'(z)-g'(z)| where z is in the circle S^1, it is sometimes called the Whitney Topology. So in this topology two such embeddings are close only when there is a “small” isotopy from one to the other.

One of the natural reasons to study spaces of knots comes not from foundational 3-manifold theory questions, but from mechanical engineering (considered broadly!). Specifically, continuum mechanics: subjects like elasticity and plasticity. These subjects study materials and their behaviours under different stresses and conditions. The connection to spaces of knots is the idea of thinking of a physical process as a dynamical system on a state space, a space of all possible configurations. Knots are one of the most basic examples of infinite-dimensional state spaces that allow for deformable objects. A more typical continuum mechanics problem would be 2-dimensional continuua, like the study of how a plastic shopping bag deforms when its carrying groceries, or the dynamics of human flesh, or the dynamics of a big canvas tent. On the extreme end, general relativity is very close to continuum mechanics. On the more pragmatic end, high-dimensional state spaces are increasingly important in subjects like robotics where one has to plan the motion of a complex object. In that sense, spaces of knots could be viewed as a “baby” case of a much wider collection of problems.

A `physical’ dynamical systems on spaces of knots is the electrostatic potential. The idea would be to imagine a knot as being an elastic band embedded in S^3, and one places a uniform electric charge along that elastic band. The elastic band is made of rubber, so the charges do not move relative to the rubber. One can write down differential equations such as this and construct various potential functions on the space of knots Emb(S^1,S^3), see for example the work of Jun O’Hara at Tokyo Metropolitan University. Knowledge of the homotopy-type of spaces of knots tells you about what kind of critical points your potential function must have (and conversely), via traditional subjects such as Morse Theory.

Here is one of the most direct connections with low-dimensional interests. An open problem in knot theory is whether or not there is an efficient algorithm to determine if a knot is trivial, say, starting from a knot diagram. The Haken algorithm has nice implementations in Regina, but it’s exponential run-time. And although it gives one access to the isotopy to the trivial knot provided it verifies the knot is trivial, it isn’t the most convenient access one could hope for.

Consider the subspace UK of Emb(S^1,S^3) consisting of knots that are isotopic to the trivial knot. We know via Allen Hatcher’s work in the 1980’s that UK has the homotopy-type of the subspace of parametrized great circles, i.e. UK has the homotopy-type of S^3 x S^2. From this we can conclude that there exists a smooth, real-valued function UK –> R where the only critical points are the global minima, that being the great circles. At present we only know such a “potential function” exists in the weak Zermelo-Frankel sense. Due to the nature of Hatcher’s proof, we do not know the *form* of such a function. If the potential function had a nice geometric or physical interpretation (something like an electrostatic potential, for example) then perhaps the gradient-flow could be turned into an efficient mechanism to recognise trivial knots. By and large the issue of finding critical points on physically-defined potential functions Emb(S^1,S^3) –> R is an open problem. But as Hatcher shows in the final section of his paper (linked above), if you had such a potential function, you could give a new proof of the Smale Conjecture. The electrostatic potential is not the only potential function that could potentially be used in a new proof of the Smale Conjecture, the Menger curvature is another seemingly-reasonable candidate, and has its own appeal.

The person that really got the study of spaces of knots off the ground and into peoples’ imaginations is Victor Vassiliev. Vassiliev had been studying singularity theory with Arnold, in the spirit of how Arnold used singularity theory to describe the (co)homology of configuration spaces. One can think of a configuration space of points in the plane as the space of embeddings of a finite set into the plane, Emb({1,2,…,n}, R^2). That embedding space sits in the space of all maps Maps({1,2,…,n}, R^2), which is just R^{2n}. So the configuration space is the complement of a “discriminant” space, sometimes also called the “diagonal” where the points in R^2 are required to have some collisions. Similarly, the embedding space Emb(S^1, S^3) is a subspace of the mapping space Map(S^1,S^3) whose homotopy-type is known, this is S^3 x \Omega S^3. So if one is content to study (co)homology of Emb(S^1,S^3) one can study it via Spanier-Whitehead duality. This turns the relatively tricky problem of studying the (co)homology of Emb(S^1,S^3) into the somewhat more tractible problem of studying the singular maps S^1 –> S^3. The singular maps are “more tractible” precisely because they form a stratified space. You can count the double points, triple points, etc, similarly you can count the places where the derivative is zero, giving a filtration. This gives you a non-homogeneous object to work with, and suddenly there are details to study. Vassiliev went quite far with this perspective, giving a spectral sequence that converges to the (co)homology of Emb(S^1,S^n) for n at least 4. In the 3-dimensional case, it’s unclear precisely what the Vassiliev spectral sequence says about the homotopy-type of Emb(S^1,S^3), and that is an open problem. The invariants of H_0(Emb(S^1,S^3)) that it produces are known as “Vassiliev invariants” or “finite type invariants”. It remains an open problem whether or not one can distinguish knots via Vassiliev invariants. Due to the nature of their definition in terms of double points, one might expect that the key property of Vassiliev invariants is how they depend on crossing changes. You would be right!

There are some wonderful connections, though. For example, the first non-trivial finite-type invariant of knots is called “the type two invariant”. It has many interpretations, my favourite being a signed count of the number of families of “satanic circles” intersecting the knot, these are the round circles that intersect the knot in 5 points making a pentagram. See Daniel’s write-up, linked, for details. This interpretation also “has legs”. The type-2 invariant of knots, from the perspective of Vassiliev, is a cohomology class defined in H^{2n-6}(Emb(S^1,S^n)) for all n>2. So it is an isotopy invariant in dimension 3, but it is also a non-trivial cohomology class in all higher dimensions as well, having a fundamental interpretation. 2n-6 is the dimension of the first non-trivial homotopy class in Emb(S^1,S^n) that does not come from the homotopy of the free loop space on S^n. Moreover, the type-2 invariant faithfully detects this homotopy/homology class. Just as in dimension 3, it is a signed count of the number of “satanic circles” on the knot. This result appears rather tersely, here. If you want to work out the proof you’ll have to understand the relation with the long-knot space, outlined in the linked paper, first.

I’m starting to hope questions such as “do Vassiliev invariants distinguish knots” are perhaps answerable in the near future. There are a variety of ways to attack this problem but I’m increasingly drawn to a relatively formal perspective. I don’t want to bore you with too much operads verbiage, but let me tell you about the geometric-topology input to this perspective. The homotopy-type of the space of smooth embeddings Emb(S^1, S^3) has a rather beautiful description in the language of operads (operads are something like topological monoids, and are a general language for operations on spaces). The most immediate analogy I can think of would be to consider the subgroups of braid groups that preserve a system of closed curves in the punctured disc. They clearly have semi-direct product descriptions. The space of knots is comparable to that, with the key ingredient being an operad that codifies satellite operations. I call it the Splicing Operad, in reference to Larry Siebenmann’s work on JSJ-decompositions of homology 3-spheres. A key theorem that allows one to compute the homotopy-type of the splicing operad (and Emb(S^1,S^3)) is:

Given an (n+1)-component hyperbolic link L in S^3, with the components denoted (L_0,L_1,…,L_n)=L, if we know the n-component sublink (L_1,…,L_n) is the trivial link, then one can isotope L into a position in S^3 so that each of the components L_1 through L_n are round circles, and if we let G be the group of orientation-preserving isometries of S^3 that restrict to homeomorphisms of L, and which restrict to homeomorphisms also of L_0, then we can ensure that the restriction of G to S^3 \setminus L is the full group of orientation-preserving hyperbolic isometries on the exterior which preserve the L_0 cusp (and which extend to continuous functions on S^3).

So this theorem is something of a partial converse to the stated theorem at the beginning of this article. While one can’t put the Borromean rings into a position where all 3 components are round circles, one can equivariantly put the Borromean rings into a position where two of the three components are round.

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K. Okazaki, The state sum invariant of 3–manifolds constructed from the linear skein.

Algebraic & Geometric Topology13(2013) 3469–3536.

It’s a wonderful piece of diagrammatic algebra, and I’d like to tell you a bit about it!

The two main constructions of 3-dimensional topological quantum field theories are:

**Reshetikhin-Turaev invariants**: These are computed from surgery presentations of 3-manifolds.**Turaev-Viro invariants**: These are based on triangulations of 3-manifolds.

Turaev-Viro invariants are defined using -symbols coming from representations of quantum groups. When everything is `nice’ enough, the Turaev-Viro invariant equals to the square of the absolute value of a corresponding Reshetikhin-Turaev invariant, and its computation reduces to a Reshetikhin-Turaev computation. But there’s a natural extension of Turaev-Viro invariants due to Ocneanu which uses other types of 6j-symbols, such as 6j symbols of subfactors. In particular, the 6j-symbol of the subfactor does not come from any Reshetikhin-Turaev invariant, and so it much be computed directly. Quantum closed 3-manifold invariants associated to 6j-symbols of the subfactor are true state-sum invariant land!!

The study of subfactors, and also of knots, challenges the classical paradigm of algebra as the science of manipulating strings of symbols. Namely, relevant algebras are algebras of diagrams drawn on the plane. To veer off on a philosophical tangent for a moment:

Before `algebra of strings’, if you wanted to solve something like , you had to write something monstrous like:

If some one say: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.” Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts

This is from Al-Khwarizmi’s Compendious Book on Calculation by Completion and Balancing. Without `algebra of strings’ itself, you couldn’t even do that. Conceptual advances which make algebra effective include appropriate notation (credit to Al-Qalasadi in the fifteenth century), thinking in terms of algebraic structures, and completing them. For example, to `balance’ terms from one side of an equation to another, you need to have zero and negative numbers (so that having five apples and giving you two is the same as having minus two apples recieving five), and you need to have fractions… even if the final answer is known to be a positive integer and if only positive integers make sense in context! As an aside, I think that concepts such as negative probability and negative information can be understood analogously.

But then came the idea, whose origins are discussed in this mathoverflow question and which was popularized in topology by Kauffman HERE, that one should really be able to concatenate algebraic symbols not only on the left and right, but also from above and below and indeed from any direction. That algebra should be done not “along a line”, but rather in the whole plane. For “higher algebra” you might need even more dimensions! And diagrammatic algebra was born.

So how can you use diagrammatic algebra to compute an invariant? You compute a diagrammatic quantity for a presentation of your object. Local moves on your presentations, such as Pachner moves on triangulations, induce local moves on your diagrams. Your goal is now to prove that, using the local moves, you can reduce your diagram to some sort of “normal form”. And then that “normal form” is your invariant! This plan fits into the Kuperberg programme for understanding state-sum invariants, which is:

- Find a presentation for your skein module (your diagrammatic algebra of diagrams modulo your moves) in terms of generators and relations.
- Use this presentation to prove properties of your invariant (and to compute it!).

Bigelow had already found a presentation for the relevant planar algebra here:

Bigelow, S., Skein theory for the ADE planar algebras.

Journal of Pure and Applied Algebra214(5) (2010), 658-666.

Okazaki modifies Bigelow’s presentation, and using his modified presentation, he shows that the planar algebra in question is -dimensional, so that any diagram reduces to a scalar multiple of the empty diagram (**update**: Okazaki just posted a simplified version of this proof HERE). This means that the state sum invariant (Turaev-Viro-Ocneanu Invariant) can be computed recursively by writing down the diagram associated to 6j-symbols of the subfactor for the triangulated closed -manifold in question, and recursively applying local moves until an empty diagram is obtained.

Given that the linear skein is a non-trivial diagrammatic algebraic object, Okazaki’s paper might represent the most archetypal piece of diagrammatic algebra I’ve ever seen. It’s 57 pages full of computations some of which look a bit like this:

At the end of the paper, he computes the invariant for some lens spaces, and he’s done many more computations since. But anyway, it’s all just a beautiful testament to the power of diagrammatic algebra- a celebration of diagrammatic algebra. I believe that diagrammatic algebra will continue to expand and will soon enter all of the sciences… What would Pierce, who envisioned a diagrammatic algebra in the 1880’s as his “chef d’oeuvre”, an outline of the mathematics of the future (see HERE), have made of all the wonderful work on skein modules that we see today? What would he have made of this paper of Okazaki?

A casual question to all of you- what’s the most aesthetically pleasing diagrammatic algebraic computation you know?

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