## March 23, 2009

### The Borromean Link Configuration

#### Posted by David Corfield

I’ve been asked by Alan Calvatti, Kuolab for Imaging Informatics, UCSD Radiology, to post here a question readers may be able to answer.

I’m looking for a categorical formalization of a concept that abstractly may be captured by the following expression [Borromean Link Configuration or BLC]

$[BLC]: a P b \wedge b P c \wedge a P c \wedge \not R a b c,$

where $R$ is a ternary relation and $P$ is a binary relation that intuitively represents the “boundary of $R$”, for some objects $a$, $b$, $c$.

Concrete examples:

• The Borromean link: $a$, $b$, $c$ are unknots, and $x S y$ iff the linking number of $x$ and $y$ is 0. Then $a S b$, $b S c$, $a S c$. $R a b c$ “represents the unlink.” [BLC] models the result that Borromean link is not isotopic to the unlink.

Note: the classical linking number is defined only pairwise, though there may be invariants that can distinguish the Borromean link from the unlink, but I’m not aware of formalizations.

• Statistical independence: $a$, $b$, $c$ are random variables (not necessarily over the same measure space). $P$ and $R$ represent pairwise and mutual independence resp. [BLC] models the result that pairwise independence does not imply mutual independence, see, e.g., this.

Note: this is a problem because typically, inferences on mutual independence of stochastic processes is based on pairwise estimates. For example, in econometrics, one can form a matrix of pairwise correlation estimates for a set of timeseries (eg, asset prices) in a portfolio. Then the Borromean Link is a problem in the sense that even if a nonlinear estimate of association is substituted for the linear correlation (eg, mutual information), such a matrix cannot detect mutual dependence in this sense.

• In transactional database theory, $a$, $b$, $c$ are processes. $P$ and $R$ represent pairwise and mutual serializable schedules. [BLC] models the result that mutual interaction of 3 pairwise serializable schedules are not mutually serializable.

Note: practical examples are given in D. Taniar et al “High-performance parallel database processing and grid databases”, 2008.

• Given a graph with 3 nodes $a$, $b$, $c$, $R a b c$ can be modeled by the interior of the simplex $\{a, b, c\}$, i.e. as a hypergraph over $a$, $b$, $c$. The relations $P$ as boundary of $R$ are the graph edges $a b$, $b c$, $a c$. Then [BLC] means that this hypergraph has only edges, no simplicial interior.
• Any bridges to Ramsey combinatorics?… Quadratic reciprocity as David suggested?

Additional details and visualizations will be presented shortly on my infotainment web site and my technical site.

[I had mentioned to Alan that Denis-Charles had written some interesting remarks about Milnor numbers as quantities measuring higher linking, and their parallel in number theory. The question is whether there is a general categorical formulation which takes in statistical independence as well – DC.]

Posted at March 23, 2009 1:03 PM UTC

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## 49 Comments & 0 Trackbacks

### Re: The Borromean Link Configuration

I see John gave more references to the number theory–knot theory connection in TWF 257.

I can’t get hold of Morishita’s papers. What happens with Borromean triples of primes? And before that, what is special about pairs of primes which aren’t linked?

Posted by: David Corfield on March 23, 2009 1:28 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

$a$, $b$, $c$ are random variables (not necessarily over the same measure space). $P$ and $R$ represent pairwise and mutual independence resp.

Are you sure you really mean this? The standard definitions of independence require the random variables to be defined on a common measure space (since it’s a property of their joint distribution).

Posted by: Mark Meckes on March 23, 2009 4:37 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Presumably Alan meant that we can have our two random variables take values in different sets, like a coin toss and a die roll. Then we might say coin tosses are independent of die rolls. But you may prefer to think of the joint distribution over $\{head, tail\} \times \{1, 2, 3, 4, 5, 6\}$, and the event ‘head’ as the subset $\{head\} \times \{1, 2, 3, 4, 5, 6\}$.

So what happens with our old friend the Giry monad?

A distibution over $X \times Y$ is an arrow in the Kleisli category for this monad: $1 \to X \times Y$, (so $1 \to P(X \times Y)$).

Then there are ‘projections’, e.g., the map $X \times Y \to X$, which must be a conditional distribution $P(x' | x, y)$. Looks like the delta function $\delta_{x' x}$.

Then variables $f: 1 \to X$ and $g: 1 \to Y$ are independent if they factor through $(X \times Y)$.

Posted by: David Corfield on March 23, 2009 5:18 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

* What are some good sources on the categorical description of stochastic systems?

* With respect to the Borromean problem, perhaps the key pragmatic question is, given a real-world dataset (and I here I am focusing on complex systems, such as bioinformatics assays, econometric time series,…) could a categorical formalization of independence tell us how to estimate higher order statistical dependence not captured by pairwise estimators (eg a matrix of pairwise correlations or mutual information…)

Posted by: Alan Calvitti on March 23, 2009 10:18 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Doberkat has a draft available of ‘Stochastic Coalgebraic Logic’.

Posted by: David Corfield on March 24, 2009 10:05 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

I see, I misinterpreted the word “over”. I was thinking in terms of “a random variable $a$ over a (probability) measure space $\Omega$ is a measurable function $a:\Omega \to X$, where $X$ is some measurable space”, but you were writing in terms of “a random variable $a$ over a measurable space $X$ is a measurable function $a:\Omega \to X$, where $\Omega$ is some probability space.” I can’t make up my mind which of these uses of “over” is closer in sense to its other uses in mathematics.

Actually the most common way to talk about such an object is probably to say $a$ is a random variable on $\Omega$ with values in $\X$.

Either way, the definition of independence requires all the random variables we’re talking about to have the same domain, but whether the codomains are equal or not is immaterial.

Unfortunately the Giry monad is not yet my old friend since I haven’t read those old threads, but I will now - they look intriguing.

Posted by: Mark Meckes on March 23, 2009 6:32 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

I’m not entirely sure what’s being asked here, but the Borromean rings are a classic example of a Massey product on cohomology – in this case, the cohomology of the complement of the rings. Massey products are related to A_oo algebra structures that arise on the cohomology of dg-algebras. In particular, if you just look at the product on the cohomology of a dg-algebra, that contains much less information than the original dg-algebra. The theorem (due to Kadeishvili, I think) is that you can supplement the product with a full A_oo structure so that you have all the information of the original dg-algebra. In the case of singular cohomology, that captures the entire rational homotopy type.

Posted by: Aaron Bergman on March 23, 2009 4:57 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

The intuition is that there’s some structure similarity going on here between random variables and links. So that something similar occurs when, on the one hand, you form a joint distribution over three variables, and the projection to any two of them yields independent variables, and, on the other hand, from a Borromean link of three circles you project to two unlinked circles (by killing off one of the links?)

Posted by: David Corfield on March 23, 2009 5:29 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Nitpicking, since I didn’t think of writing about this case myself:

The A-infinity structure that Kadeishvili’s theorem deals with is not on the cohomology of a dg-algebra - it’s on the homology of a dg-algebra. The connection to cohomology is that cochains can be made into a dg-algebra, with the prototype of the cup product as the multiplication, and that the homology of THAT dg-algebra really is cohomology.

And retention of all information in the dg-algebra is not, as far as I can tell, contained in Kadeishvili’s minimality theorem. He might have something along those lines later, but at least in non-topological settings (Ext algebras of appropriate algebraic structures), the retention of the full original algebra structure is actually kinda tricky to get to work.

Posted by: Mikael Vejdemo Johansson on March 23, 2009 6:16 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Yeah – I never keep my homologies and cohomologies straight :).

FWIW, Keller in his reviews attributes to Kadeishvili the result that there exists an A_oo structure on the homology quasi-iso to the original dg-algebra, but he alsoincludes a long list of “see also” references.

Posted by: Aaron Bergman on March 23, 2009 6:37 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

The essence of Kadeishvili’s result needs a notion of homotopy. How categoriczl is that?

FWIW = ? (I don’t text msg)

which Keller paper?

does see also’ include the commutative and Lie cases also?

Markl (Homotopy algebras are homotopy algebra) has an operadic generalization of Kadeishvili’s results.

Posted by: jim stasheff on March 24, 2009 1:18 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

FWIW = “For what it’s worth”

(I think from usenet rather than text messages; the same source uses other abbreviated hedges such as

IMHO = “In my humble opinion”
AFAIK = “As far as I know”

and a bunch of other similar things.)

Posted by: Tim Silverman on March 24, 2009 1:59 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Jim: Aaron is talking about the same two Keller survey papers I have been talking about as well. I know I’ve fed you links to the papers at least twice already - if you want the links again, please drop me an email. :-)

Posted by: Mikael Vejdemo Johansson on March 24, 2009 3:20 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Yup, that’s the Minimality theorem you’re talking about. And it does not retain all information of the dg-algebra. It does, however, retain all information of the homology of the dg-algebra - which isn’t all that surprising. :-)

To get massive information gain from the A-infinity algebra, you’re going to want to restrict the kind of algebras you start with. Keller and Lu-Palmieri-Wu-Zhang have results that lets you reconstruct A from Ext_A together with an A-infinity structure. But to do that, A needs to be from one of two specific classes of algebras.

Posted by: Mikael Vejdemo Johansson on March 24, 2009 3:18 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Yup, that’s the Minimality theorem you’re talking about. And it does not retain all information of the dg-algebra.

You get a quasi-isomorphic A_oo algebra out of it which is pretty good for a dg-algebra. I know the Keller, LPWZ (and Segal) results, but they’re of a different flavor, reconstructing an algebra from the A_oo structure on the self-Exts of the ground ring, rather than just on the homology of the dg-algebra.

Posted by: Aaron Bergman on March 24, 2009 4:57 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

It’s pretty good, I agree with that. My main objection was to your original phrasing of “so that you have all the information of the original dg-algebra”, which I think a quasi-isomorphic dg-algebra does not necessarily have.

Posted by: Mikael Vejdemo Johansson on March 24, 2009 10:03 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

You can make sense of this sentence:

$a$ and $b$ are independent, $b$ and $c$ are independent, $c$ and $a$ are independent, but $a,b,c$ are not independent”

in any matroid. A matroid is a set equipped with a well-behaved way to decide when any subset is ‘independent’.

I don’t know if random variables and statistical independence gives an example of a matroid!

You seem to be aiming for an analogy where ‘independence’ is analogous to ‘linking’. How strong is this analogy? How many formal similarities can you find between ‘independence’ and ‘linking’?

Here’s one:

I guess you can say a collection of embedded circles is ‘unlinked’ if you can wiggle them around and obtain unlinked unknots. Then if a collection $S$ is contained in a collection $T$,

$T is unlinked \implies S is unlinked$

This law also holds for collections of random variables, with ‘statistically independent’ playing the role of ‘unlinked’. It also holds for subsets any matroid, with ‘independent’ playing this role. But a matroid also satisfies other axioms.

Any more formal similarities?

Posted by: John Baez on March 23, 2009 6:03 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Just looking at your formulation with the collections S and T, this looks very much like the kind of property formulation that gives you a simplicial complex situation for free: unlinkedness is closed under taking subsets, so the property of being unlinked can be made to generate a simplicial complex, the analysis of which might be interesting.

We had a colloquium speaker in Jena last summer who did similar things to give interesting non-constructive arguments in complexity theory. And I know Kozlov et.al. do similar things to generate complexes for graph colorings.

Posted by: Mikael Vejdemo Johansson on March 23, 2009 6:19 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

> But a matroid also satisfies other axioms.

Surely each the examples in my original post satisfy additinal axioms not shared by the other examples?

Eg, in database concurrency control, there are axioms that capture the fact that “read” operations commute while “writes” do not. These axioms are not relevant to random variables or topological links. The underlying binary relation P (which represents pairwise interaction), and ternary relation R (which represents mutual interaction) may thus have distinct properties across application domains and such details may in fact play a role in the categorical formalization, I have no intuition on that.

Nevertheless, I am primarily interested in systems where “mutual interaction” is not reducible to the study of “pairwise interaction,” ie, “independence” represents “lack of interaction,” or at least give conditions where it suffices to study pairwise interactions - what do those systems have in common? Surely there are many more examples of mathematical properties that hold pairwise but not mutually?

The problem with matroids (that’s a great example!) is that they are set-theoretical constructs, whereas I require the category’s objects to be general dynamical systems (eg the software processes in the case of transactional databases, or stochastic processes in the case of random variables) and my concept of interaction likewise should be quite abstract (eg RNA-interference)

Posted by: Alan Calvitti on March 23, 2009 9:40 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

I’m a bit late to the party, but have only recently come across this stuff through discussions with Pierre Baudot. Maybe having an answer to one of John’s excellent questions will still be of interest to future readers:

I don’t know if random variables and statistical independence gives an example of a matroid!

There’s a sense in which the answer is “yes, but not quite”. First, random variables and statistical independence does not give you a matroid; you can see this by noting that the matroid axioms guarantee that all maximal independent subsets have the same cardinality, but the same property certainly fails in general for subsets of random variables and their independence. (Example: take 2+3 random variables of which the first two and the latter three are independent, but such that you have weak correlations between each one of the first two and each one of the latter three. I think that this can also happen for links and their “independence”, right? Taking 1+2 instead of 2+3 is even easier, actually.)

So, what did I say “yes, but not quite” on whether random variables form a matroid? Because they form a polymatroid! This is defined in terms of a rank function like a matroid, but you generalize the notion of rank by allowing it to be an arbitrary nonnegative number. For a bunch of random variables, you get a polymatroid if you define the “rank” of any subset of these variables to be their joint entropy, et voilà! Two variables $a$ and $b$ are independent if this rank function satisfies $r(a,b) = r(a) + r(b)$, and the “Borromean link” situation therefore corresponds to

$r(a,b) = r(a) + r(b), \qquad r(b,c) = r(b) + r(c), \qquad r(c,a) = r(c) + r(a),$

but

$r(a,b,c) \lt r(a) + r(b) + r(c).$

Posted by: Tobias Fritz on October 26, 2014 3:12 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

I’m a bit late to the party, but have only recently come across this stuff through discussions with Pierre Baudot. Maybe having an answer to one of John’s excellent questions will still be of interest to future readers:

I don’t know if random variables and statistical independence gives an example of a matroid!

There’s a sense in which the answer is “yes, but not quite”. First, random variables and statistical independence does not give you a matroid; you can see this by noting that the matroid axioms guarantee that all maximal independent subsets have the same cardinality, but the same property certainly fails in general for subsets of random variables and their independence. (Example: take 2+3 random variables of which the first two and the latter three are independent, but such that you have weak correlations between each one of the first two and each one of the latter three. I think that this can also happen for links and their “independence”, right? Taking 1+2 instead of 2+3 is even easier, actually.)

So, why did I say “yes, but not quite” on whether random variables form a matroid? Because they form a polymatroid! This is defined in terms of a rank function like a matroid, but you generalize the notion of rank by allowing it to be an arbitrary nonnegative number. For a bunch of random variables, you get a polymatroid if you define the “rank” of any subset of these variables to be their joint entropy, et voilà! Two variables $a$ and $b$ are independent if this rank function satisfies $r(a,b) = r(a) + r(b)$, and the “Borromean link” situation therefore corresponds to

$r(a,b) = r(a) + r(b), \qquad r(b,c) = r(b) + r(c), \qquad r(c,a) = r(c) + r(a),$

but

$r(a,b,c) \lt r(a) + r(b) + r(c).$

Posted by: Tobias Fritz on October 26, 2014 3:14 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

I don’t know myself how to formalize BLC, but it seems that René Guitart made an attempt to define the structure of a “Borromean object in a category”. Some slides from his talks are available here.

Posted by: Denis-Charles Cisinski on March 23, 2009 10:18 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

John mentioned monad cohomology here. Does it apply to any monad? If so, what would it give us for Giry’s probability monad?

Posted by: David Corfield on March 24, 2009 10:17 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Folks, I appreciate all the helpful comments and links, some of which seem quite informative - even if I don’t have the background to comprehend the algebra and topology.

I am hoping that someone will draw me a commutative diagram to model logical relations that “hold pairwise but not mutually among a set of objects.”

Perhaps that’s what R Guitard’s “borromean object” models, I don’t understand enough there - but is it really necessary to understand the machinery of A_oo structures and dg-algebras to explain this quite generic paradigm (“pairwise versus mutual interaction”)

Posted by: Alan Calvitti on March 25, 2009 12:32 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

I certainly don’t think that A-infinity algebras and dg-algebras are necessary to formalize the aspects of the Borromean configuration you alluded to above. We just derailed onto them since Massey products show up to capture the exact way in which the appropriate cohomology ring STILL captures the linkedness of the Borromean rings in the knot theoretic context.

Posted by: Mikael Vejdemo Johansson on March 25, 2009 2:30 AM | Permalink | Reply to this

### Connected hypergraphs; Re: The Borromean Link Configuration

These are connected hypergraphs with no graph edges.

A hypergraph is a graph in which generalized edges (called hyperedges) may connect more than two nodes (vertices).

A hyperedge is a connection between two or more vertices of a hypergraph. A hyperedge connecting just two vertices is simply a usual graph edge.

If there are no “usual graph edges” but only triples of nodes connected by hyperedges, we have the Borromean Link configurations.

Am I missing something? As Einstein said, a theory should be as simple as possible, but no simpler.

Posted by: Jonathan Vos Post on March 25, 2009 2:13 AM | Permalink | Reply to this

### non-trivial Massey product; Re: Connected hypergraphs; Re: The Borromean Link Configuration

I mean, at one level of abstraction higher.

“These are connected hypergraphs with no graph edges.” Meaning that the the Borromean rings, consisting of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings, can be mapped one circle to one point, such that the three circles have been mapped to three nodes, and the same transformation maps the topological linkage of two circles to a regular edge, and the Brunnian link of 3 circles to a hyperedge.

We know that that the cohomology of the complement of the Borromean rings supports a non-trivial Massey product.

Equivalently, the hyperedge is a mapping from a hyperbolic link: the complement of the Borromean rings in the 3-sphere admits a complete hyperbolic metric of finite volume.

If this is true to someone who can draw the diagram, we are mostly there. Aren’t we?

Posted by: Jonathan Vos Post on March 25, 2009 3:09 AM | Permalink | Reply to this

### Re: non-trivial Massey product; Re: Connected hypergraphs; Re: The Borromean Link Configuration

I agree that (afaik) Massey products seem to provide the appropriate generalization of linking number that is otherwise defined only for pairs of links.

More precisely, if a “Massey product” is defined for n-component links for any n>1, then it serves as the characteristic function for both the binary and n-ary relation described in the original posting.

it should serve as “warning” that the classical linking number is only defined pairwise, whereas quantification of the concept of “prime link” (although defined conceptually) requires the construction of Massey product.

This is in contrast to the example of statistical independence, where the relation is given as a factorization condition on n-random variables directly for any n>1. (However, in practice, the estimation of statistical dependence is often limited to consideration of pairwise dependence - eg random matrix theory.)

That is an interesting situation philosphically - that the pairwise relation (eg graph edge) and the mutual relation (eg hypergraph simplex) are somehow distinct until a novel theory is constructed to bring them under the same umbrella.

Posted by: Alan Calvitti on March 25, 2009 3:44 AM | Permalink | Reply to this

### Re: Connected hypergraphs; Re: The Borromean Link Configuration

yes, I listed the hypergraph as an example in my original post.

However, I require a categorical treatment where the objects are quite general (eg dynamical systems)

Similarly, pairwise and mutual relations also can quite general interactions (eg, feynman diagrams or conditional default probabilities in finance, whatever… - do not attempt to limit the concept of “interaction”, which is completely open-ended. hypergraphs on the other hand are set-theoretical, not categorical)

Posted by: Alan Calvitti on March 25, 2009 3:21 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

The “Borromean Configuration” popped up in discussion recently about using classifications of representations of quivers for persistent homology applications.

There, we end up using the term to refer to a configuration, where, say three copies of a 1-dim vector space inject into a 2-dim vector space such that the 2-dim space is a direct sum of any two of images, but not of all three.

So, I guess, this particular model could be translated into some sort of diagram with coproducts and stuff to form equivalences of the direct products in the vector space situation… Or something…

Posted by: Mikael Vejdemo Johansson on March 25, 2009 2:37 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

I’m resisting the metaphorical aspects of this, such as psychoanalyst Jacques Lacan who found inspiration in the Borromean rings as a model for his “topology of human subjectivity” with each ring representing a fundamental Lacanian component of reality (the “real”, the “imaginary”, and the “symbolic”). His use of the words “real”, “imaginary”, and “symbolic” was so utterly alien to any mathematician. I like a Constructive proof; not a Deconstructive analogy.

Think I’ll pour myself a Ballantine beer.

Posted by: Jonathan Vos Post on March 25, 2009 3:15 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Good one. Yes, I was perplexed by the “real” “imaginary” “symbolic” triangle in R Guitard’s slide show on the “borromean object.” Not to generalize but the french have a way of melding some of the deepest thoughts with abstract aesthetic gestures.

I assure you that my motivation is pragmatic. For example, what is the meaning of “information.” When John Daugman, who studies iris biometrics states that the the combinatorial complexity of a certain wavelet transform generates 3.2 bits/mm^2, how much of that is due to the intrinsic structure of the iris, and how much of that is due to the interaction between the iris and the measurement protocol and coding algorithm. - these are issues that pervade modern machine learning in the “far heterogeneous domain” with no easy answers. I want to know, what is intrinsic to a “pairwise measurement of interaction” versus a measurement that includes “mutual interaction.”

Related: what is “independent and identically distributed” random variable? - is that a constructive concept?

Posted by: Alan Calvitti on March 25, 2009 3:57 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Since the notion of interrelation occurs here, there is a possible link with identities among relations for presentations of groups. The Borromean link occurs on p. 188 of
R. Brown and J. Huebschmann, Identities among relations’, in “Low dimensional topology” LMS Lecture Notes No 48 (1982) CUP. (Now available under POD.)

If you search on Borromean Rings in www.popmath.org.uk you will find several sculptural versions by John Robinson. One of these he called “Creation”, as it represents the idea that the whole can be more than the sum of its parts, so showing a key feature of evolution, the emergence of new stable structures.

Posted by: Ronnie Brown on March 25, 2009 6:05 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

popmath.org.uk
said `authorization required’
??

Posted by: jim stasheff on March 26, 2009 2:07 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Jim,

I got the same answer! I did get there via

http://www.popmath.org.uk/centre/index.html

however.

Posted by: Tim Porter on March 26, 2009 3:27 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

one of the pages is at

http://www.popmath.org.uk/sculpmath/pagesm/borings.html

Posted by: Tim Porter on March 26, 2009 3:29 PM | Permalink | Reply to this

### GHZ = Borromean Entanglement; Re: The Borromean Link Configuration

In quantum information theory, a GHZ Greenberger-Horne-Zeilinger state is a certain type of entangled quantum state which involves at least three subsystems (particles), no two of which are entangled with each other. It was first studied by D. Greenberger, M.A. Horne and Anton Zeilinger in 1989; who subsequently (together with A. Shimony, upon a suggestion by N. D. Mermin) applied their arguments to certain measurements involving three observers. They have noticed the extremely non-classical properties of the state.

N. David Mermin, Quantum mysteries revisited, Am. J. Phys. 58 (8), 731 (1990)

Posted by: Jonathan Vos Post on March 26, 2009 3:49 AM | Permalink | Reply to this

### Re: GHZ = Borromean Entanglement

Actually, the plain trivalent graph dual to the trefoil (Pauli matrix quandle) is the same as the GHZ graph, and the (zero 3-way entanglement) W state graph is the Borromean link. The point is that the trefoil links up the qubits into a single strand while the Borromean link keeps them separate.

Posted by: Kea on March 28, 2009 5:30 PM | Permalink | Reply to this

### Re: GHZ = Borromean Entanglement

I stand corrected. Thank you for correctly explaining this to Alan Calvatti. My hunches sometimes point vaguely in the right direction, but are never to be relied upon…

Posted by: Jonathan Vos Post on March 30, 2009 12:51 AM | Permalink | Reply to this

### Re: GHZ = Borromean Entanglement

I came up with this independently about ten years ago, and I also associated the GHZ state with the Borromean rings. The idea was that measuring a qubit is cutting a link, so in the state

(1)$|000\rangle + |111\rangle$

if you measure any one of the qubits, you know the values of the other two. On the other hand, in

(2)$|000\rangle + |011\rangle + |101\rangle + |110\rangle$

if you measure any one qubit, the other two remain entangled.

You can form a chain if you have more than two states for inner links:

(3)$|000\rangle + |011\rangle + |120\rangle + |131\rangle$

measuring the first or last bit leaves the other two particles entangled, but measuring the middle one uniquely specifies the other two.

If this picture is wrong, can you explain in a little more detail why it is wrong, and what the right picture is?

Posted by: Mike Stay on March 31, 2009 9:41 PM | Permalink | Reply to this

### Re: GHZ = Borromean Entanglement

Doh! Yeah, GHZ is the Borromean - I mixed up some zeroes and ones. That is, taking dual graphs of a 2-coloured knot diagram (so triangle for trefoil, triangle with legs for Borromean) and then taking their incidence matrices, the set of subdeterminants is a set of 6 of the amplitudes for the states. So, for the triangle (W state) one gets a 3x3 matrix with all ones, which is |1>+|2>+|3>. I’m working on this stuff, because it’s all about black holes.

Posted by: Kea on April 1, 2009 7:52 AM | Permalink | Reply to this

### Lev Bishop, at Yale; Re: GHZ = Borromean Entanglement

Ivan Deutsch, Secretary-Treasurer of the APS GQI topical group about the winners of the best student paper awards:

We are pleased to announce the Best Student Paper awards for the 2009 APS March Meeting….

For the best theoretical paper, the winner is

Lev Bishop, Yale University

for his paper V17.9, “Towards proving non-classicality with a 3-qubit GHZ state in circuit QED”.

Posted by: Jonathan Vos Post on April 4, 2009 6:17 AM | Permalink | Reply to this

### Eric Emtander. On hypergraph algebras.; Re: The Borromean Link Configuration

A deeper way to get from the Hypergraph view of Borroemeanism is:

On hypergraph algebras
Filosofie licentiatavhandling
Eric Emtander, Att presenteras den 21 april 2008, kl. 14.00, rum 306, hus 6,
Matematiska institutionen, Stockholms universitet, Kr¨aftriket.

This thesis consists of two papers, Betti numbers of hypergraphs and A class of hypergraphs that generalizes chordal graphs.

Abstract

In this paper we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge, are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.

Posted by: Jonathan Vos Post on March 27, 2009 9:21 PM | Permalink | Reply to this

### Re: Eric Emtander. On hypergraph algebras.; Re: The Borromean Link Configuration

Oh, so THAT’s what Eric Emtander is up to! I’m looking forward to getting back to Stockholm quite a bit now!

(Eric started his PhD studies just as I left Stockholm - and is working on things similar to what I used to work on for my Masters…)

Posted by: Mikael Vejdemo Johansson on March 28, 2009 12:07 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Did you mean for xSy to be the stronger condition “x and y form an unlink” rather than “have linking number 0”? The Whitehead link is an example of a nontrivial link on two components with linking number 0.

Obviously, any subcollection of components of a trivial unlink is also a trivial unlink, but I suspect that this is the only restriction. In that case, the unlinked sets of a finite collection of components can be modeled by an arbitrary (abstract) simplicial complex. The simplicial complex corresponding to the Borromean rings is the boundary of a triangle, which is the smallest non-contractible simplicial complex.

Posted by: Tracy Hall on April 4, 2009 12:08 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

> Did you mean for xSy to be the stronger condition x and y form an unlink; rather than have linking number 0?

From a cursory read, I believed that for a 2-component link, the two statements were equivalent. Can you give a counterexample?

In any case, you are correct that abstract simplicial complex yields examples of Borromean configuration as you described - but this was already noted in my original posting (modulo your observation that such a triangle boundary is the smallest noncontractile. The logical complement of this configuration is the interior of the triangle, which captures the concept of Borromean link directly.

However, I am seeking a categorical treatment of this general phenomenon so that the vertices (objects) may represent, eg, dynamical systems, edges may represent, eg, pairwise process interactions, “triangles” may represent, eg 3-way process interactions, etc.

the motivation for my original posting comes from pattern recogntition and related sciences. Namely, what can be said about higher-order interaction when only pairwise interactions are measured (eg “mutual information” is actually only pairwise interaction. As Mumford pointed out, you cannot use Venn-type diagram and inclusion-exclusion arguments to relate pairwise mutual information with the overall interaction)

Posted by: Alan Calvitti on April 4, 2009 7:04 PM | Permalink | Reply to this

### Re: The Borromean Link Configuration

From a cursory read, I believed that for a 2-component link, the two statements were equivalent. Can you give a counterexample?

She did. The Whitehead Link.

Posted by: John Armstrong on April 5, 2009 1:00 AM | Permalink | Reply to this

### Re: The Borromean Link Configuration

Weisstein, Eric W. “Whitehead Link.” From MathWorld–A Wolfram Web Resource.

http://mathworld.wolfram.com/WhiteheadLink.html

Posted by: Jonathan Vos Post on April 5, 2009 2:19 AM | Permalink | Reply to this

### Efimov one-upped by Nägerl; Re: The Borromean Link Configuration

Apr 7, 2009 Atomic quartets spotted in ultracold gas

“…Physicists in Austria have confirmed that four atoms can be coaxed into forming bound states in an ultracold gas — even though pairs of the same atoms do not bind together….>

“…nearly 40 years ago the Russian physicist Vitali Efimov was able to calculate that three atoms should, in principle, be able to form quantum states that are loosely bound together — despite the absence of bound states of any two pairs of atoms in the system….”

Nägerl and colleagues have repeated their [2006] experiment — and have found both quartets just where von Stecher predicted.”

And what higher order analogues are there?

Posted by: Jonathan Vos Post on April 15, 2009 4:41 AM | Permalink | Reply to this

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