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A knot theory for two-dimensional square lattice is proposed, which shedslight on design of new two-dimensional material with high topological numbers.We consider a two-band model, focusing on the Hall conductance {\sigma}xy =e^2/hbar*P, where P is a topological number, the so-called Pontrjagin index. Byre-interpreting the periodic momentum components kx and ky as the stringparameters of two entangled knots, we discover that P becomes the Gauss linkingnumber between the knots. This leads to a successful re-derivation of thetypical P-evaluations in literature: P = 0;{\pm}1. Furthermore, with the aid ofthis explicit knot theoretical picture we modify the two-band model to achievehigher topological numbers, P = 0;{\pm}1;{\pm}2.

Knot theory for two-band model of two-dimensional square lattice with high topological numbers