In praise of composition: knot groups and cospans

I have been thinking about knots recently. The context is our work (with Sabadini, Katis, Rosebrugh etc) since 1997 ([1]) in using the tensor and composition in categories of spans and cospans to describe "evolving nets of automata" - the most recent paper being [4]. After these thoughts I noticed that the main idea of this post, seen somewhat differently, has already been discovered by John Armstrong (see below). It could easily be also in other works, for example Yetter's book on Functorial Knot Theory (I don't have a copy).

The idea is to see a knot as a system composed from parts. Let's look at the trefoil:

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Hard sciences

Here in Como the hard sciences are said to be physics, chemistry and mathematics, but not computer science.

I have just been given a program by a student and I would like to know that it is correct. This is more difficult than writing a new program myself.

In mathematics instead it is usually much easier to be convinced of a proof than to construct a proof. Part of the reason is that mathematicians usually don't check a proof completely but try to find clues from the description and construct a proof themselves.

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Cultural differences

Italian children are brought up with the saying "the end justifies the means". I was brought up in Australia with the opposite exhortation.

University news

Reform The grand university reform of Mariastella Gelmini has passed into law by a whisker - immediately after a no-confidence motion passed by three votes. We are now waiting to see what effect this will have. Already many courses have closed including ours. The position of researcher has a significantly changed status and many are working to rule, putting courses in difficulty.


The fractal nature of politics I don't really mean fractal. I am referring to the fact that behaviour at a high level is reflected at a lower level.
I have rather studiously avoided talking about Italian politics even if it is one of the topics most discussed here. However the premier himself often refers to the fact that he is under siege, with the opposition including former allies.

At the University level, a president of a faculty, who pressured a previous president to resign, has publically asked the rector to resign. Now I hear rumours that he himself is under siege, with the opposition including former allies.

The good old days

This phrase (or something like it) occurred in some discussions on the Categories List, in connection with a paper I wrote with Ross Street in 1978 (Yoneda structures). The comprehensive factorization of 1973 was also mentioned recently. I have also been asked if I have an electronic version of my PhD thesis of 1970. I had already decided to scan my unpublished MSc (1967) and PhD theses and put them on line.
Today I finished scanning the master's thesis and it is available at this address. It was written at the University of Queensland in the good old days, before anyone had completed a doctorate there, and research publication by staff members was a special event.

20 January 2011: I have now finished scanning my PhD thesis, and it is available here.