  We proposed an entangled multi-knot lattice model to explore the exotic
statistics of anyon. This knot lattice model bears abelian and non-abelian
anyons as well as integral and fractional filling states that is similar to
quantum Hall system. The fusion rules of anyon are explicitly demonstrated by
braiding on crossing states of the multi-knot lattice. The statistical
character of anyon is quantified by topological linking number of multi-knot
link. Long-range coupling interaction is a fundamental character of this knot
lattice model. The short range coupling models, such as Ising model, fermion
paring model, Kitaev honeycomb lattice model and so on, appears as the short
range coupling case of the knot lattice model. We introduced link lattice
pattern as geometric representation of the eigenstate of quantum many body
model to explore the topological nature of quantum eigen-states. For example, a
convection flow loop is introduced into the well-know BCS fermions pairing
model to show the Pseudo-gap state in unconventional super-conducting state.
The integral and fractional filling numbers in quantum Hall system is directly
quantized by topological linking number. The quantum phase transition between
different quantum states in quantum spin model is also directly quantified by
the change of topological linking number, which revealed topological character
of phase transition. This multi-knot lattice has a promising physical
implementation by circularized photons in optical firbre network. It may also
provide another different path to topological quantum computation.
