Mathematical Logic of the Jones and Homfly Polynomials of Knotted Trivalent Networks

Two rational functions are defined logically for special type of knotted trivalent networks as state models of planar trivalent networks. The restriction of these two rational functions reduce to the Jones and Hom y polynomials for non oriented links. Also, these two models are used to define two invariants for this special type of knotted trivalent networks embedded in R3. Finally, we study some congruences of these two polynomials for periodic knotted trivalent networks this generalize the work of periodicity of the Jones and Hom y polynomials on knots to these two rational functions of knotted trivalent networks.