Matrix elements of Thiemann's Hamiltonian constraint in loop quantum gravity

We present an explicit computation of matrix elements of the Hamiltonian constraint operator in non-perturbative quantum gravity. In particular, we consider the Euclidean term of Thiemann's version of the constraint and compute its action on trivalent states, for all its natural orderings. The calculation is performed using graphical techniques from the recoupling theory of coloured knots and links. We exhibit the matrix elements of the Hamiltonian constraint operator in the spin-network basis in compact algebraic form.