### 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.]

## 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?