  We study the topological nature of a class of lattice systems, whose Bloch
vector can be expressed as the difference of two independent periodic vector
functions (knots) in an auxiliary space. We show exactly that each loop as a
degeneracy line generates a polarization field, obeying the Biot-Savart law:
The degeneracy line acts as a current-carrying wire, while the polarization
field corresponds to the generated magnetic field. Applying the Ampere's
circuital law on a nontrivial topological system, we find that two Bloch knots
entangle with each other, forming a link with the linking number being the
value of Chern number of the energy band. In addition, two lattice models, an
extended QWZ model and a quasi-1D model with magnetic flux, are proposed to
exemplify the application of our approach. In the aid of the Biot-Savart law,
the pumping charge as a dynamic measure of Chern number is obtained numerically
from quasi-adiabatic processes.
