  We address the question of different representation of Bloch states for
lattices with a basis, with a focus on topological systems. The representations
differ in the relative phase of the Wannier functions corresponding to the
diffferent basis members. We show that the phase can be chosen in such a way
that the Wannier functions for the different sites in the basis both become
eigenstates of the position operator in a particular band. A key step in
showing this is the extension of the Brillouin zone. When the distance between
sites within a unit cell is a rational number, $p/q$, the Brillouin extends by
a factor of $q$. For irrational numbers, the Brillouin zone extends to
infinity. In the case of rational distance, $p/q$, the Berry phase "lives" on a
cyclic curve in the parameter space of the Hamiltonian, on the Brillouin zone
extended by a factor of $q$. For irrational distances the most stable way to
calculate the polarization is to approximate the distance as a rational
sequence, and use the formulas derived here for rational numbers. The use of
different bases are related to unitary transformations of the Hamiltonian, as
such, the phase diagrams of topological systems are not altered, but each phase
can acquire different topological characteristics when the basis is changed. In
the example we use, an extended Su-Schrieffer-Heeger model, the use of the
diagonal basis leads to toroidal knots in the Hamiltonian space, whose winding
numbers give the polarization.
