### Coalgebraic Tangles

#### Posted by David Corfield

I’m sinking in a sea of administrative duties at the moment, so for a bit of sanity I thought I’d jot down the glimmer of a thought I had.

We spoke back here about the term model for a set of ground terms, $X$, and a set of term constructors, $\Sigma$, as the initial algebra for the functor $F: Y \to X + \Sigma(Y)$. For example, the set of finite sequences of 1s is the term algebra for $F: Y \to 1 + Y$. More interesting examples involve trees.

Dually, there are coterms, which correspond to the behaviours of a system which unpacks an element into a term-constructor and collection of elements. These form the terminal coalgebra for a functor. So in the case of $F: Y \to 1 + Y$, coterms will be possibly infinite streams of 1s, each of which is unpacked, if not empty, into a 1 and another stream. Elements correspond to the extended natural numbers, i.e., the natural numbers with the infinite stream adjoined. In the case of trees, we find potentially infinitely deep trees, such as the trace trees of computations.

Now, the new thought. Tangles are like higher-dimensional pieces of syntax, as shown by their use, for example, in calculating with representations of quantum groups. This works through their forming the free braided monoidal category on a dualizable object. So, question: could there not be dually a system of ‘coterm’ tangles with possibly infinitely many subtangles?

One small hint that there’s something to this: a nice example of an initial algebra/terminal coalgebra pair was given by Tom here. It’s the dyadic rationals in the interval $[0, 1]$, and the real interval $[0, 1]$, as devised by Peter Freyd. There’s a completion process going on.

Now there’s a link from rationals to tangles in the form of Conway’s *rational tangles*, week 228 and week 229, where each two-stranded tangle may be assigned a rational in an isotopy-invariant way. Is there a ‘completion’? Well, Louis Kauffman and Sofia Lambropoulou tell us in On the classification of rational tangles that

each non-rational real number (algebraic or transcendental) can be associated to an infinite tangle…all the approximants of which are rational tangles. (p. 24)

There it is. Now I really must get back to my admin.

Posted at January 25, 2011 11:41 AM UTC
## Re: Coalgebraic Tangles

I may be speaking to myself, but to continue with some more loosely connected thoughts, Kauffman and Lambropoulou’s account of rational tangles works through continued fractions. There’s a

continued fraction coalgebrastudied in The continuum as a final coalgebra by Pavlovic and Pratt.Rutten treats the stream behaviour of an infinite multivariate weighted automaton as a continued fraction in sec. 17 of Elements of Stream Calculus (An Extensive Exercise in Coinduction). Does this mean there’s a tangle-shaped form of these stream behaviours?

Given that “Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems”, as Alexandra Silva claims, it may be interesting that there are connections between knots and dynamics, e.g., Ghrist’s “Chaotic” knots and “wild” dynamics, where

And dynamically-defined wild knots are studied by Hinojosa.

Finally, nothing escapes the Café. I see John Armstrong was asking back here

Coalgebraically?