  Grothendieck's theory of dessins provides a bridge between algebraic numbers
and combinatorics. This paper adds a new concept, called 'bias', to the bridge.
This produces: (i) from a biased plane tree the construction of a sequence of
algebraic numbers, and (ii) a Galois invariant lattice structure on the set of
biased dessins. Bias brings these benefits by (i) using individual polynomials
instead of equivalence classes of polynomials, and (ii) applying properties of
covering spaces and the fundamental group. The new features give new
opportunities.
  At the 2014 SIGMAP conference the author spoke [1] on 'The decorated lattice
of biased dessins'. This decorated lattice $\mathcal{L}$ is combinatorially
defined, and its automorphism group contains the absolute Galois group
$\Gamma$, perhaps as an index 6 subgroup.
  This paper defines new families of invariants of dessins, although they
require further work to be understood and useful. For this, $\mathcal{L}$ is
vital. This paper relies on the the existing, unbiased, theory. Also, it only
sketches the construction of $\mathcal{L}$. In [2], [3] the author will remove
this dependency, develop the biased theory further, with a focus on $\Gamma$,
and make the theory more accessible.
