(a) Two anyons are exchanged within | ± 1 〉 states with three crossings. (b) The equivalent solenoid helix scheme for the same braiding operation with respect to panel (a). (c) An odd number of many writhing loops generated on square lattice. (d) One writhing loop generated on one arc. (e) The twisted free arcs generated by one braiding. (f) The output of one braiding on the crossing point of the writhing loop leads to one untwisted free arc and one free loop.