Hi,

I just set up this blog to record my thoughts on maths, science, social social, politics, and anything else that strikes my fancy. Why did I call it Sachi’s hyperbolic space? Well I’m a mathematician who’s studied 3-manifold invariants (hence the “hyperbolic”) and “Sachi” is my nickname.

I’m interested in all sorts of things including applying evidence-based enquiry to problems in society (eg to politics, education and economics), ecological economics, whole earth system studies, physics, pure mathematics, applied mathematics (eg how does turbulence arise from the Navier-Stokes equations, or does it even ???) and I’m attracted to the idea of studying *m*-dimensional objects in *n*-dimensional spaces (where m is less than n-1), just because it seems beautiful.

This blog might end up being just a forum for me, or it might be part of the “blogging community” – who knows – and perhaps **B********may even write a post.

Here’s a taste of the mathematics I’m currently interested in. The Hecke algebra *H(n,q)* (also known as the *Iwahori-Hecke algebra of type A(n-1)*) taken over the complex field, is generated by the invertible elements *{1, g(1), g(2), …, g(n-1)}*, subject to the following relations (where *q* is a complex number):

*g(i)g(i+1)g(i) = g(i+1)g(i)g(i+1),* for i=1, 2, …, n-2;
*g(i)g(j) = g(j)g(i),* if |i-j| > 1;
*(g(i))^{2} = (q-1)g(i) + q,* for i=1, 2, … n-1.

Note that relations 1. and 2. are just the Braid group relations, so any representation of *H(n,q)* is automatically a representation of the Braid group *B(n)* on *n* strings.

Suppose now that you have a representation of the Braid group given by the algebra homomorphism:

*p: g(i) -> R(i),*

where each *R(i)* satisfies the equation

*(R-a)(R-b)=0;*

(suppressing the index *i*) for some non-zero complex scalars *a, b* where *a* and *b* are not equal*.* Then the algebra homomorphism defined by

*g(i) -> -R(i)/a*

yields a representation of *H(n, -b/a)* in the algebra generated by *{R(i), i=1, 2, …, n-1}* taken over the complex field.

This is *very cool*, because I think that the Jones Polynomial (of knots and links) can be obtained from the Hecke algebra (or some representation of it) – then, by being sufficiently clever (making sure you have a Markov trace etc), you can obtain a link polynomial from your representation of *H(n, -b/a),* which, after some suitable reparametrisation, is just the Jones polynomial.

I have to check this, but I think it’s on the right track. Something similar happens with the Birman-Wenzl-Murakami algebra. If you have any representation of this algebra, you automatically have a link polynomial. Very nice.