Measures of entanglement in systems employing Periodic Boundary Conditions

We introduce a definition for the Jones polynomial of open or

closed curves in systems employing Periodic Boundary Conditions (PBC). This is a one

variable Laurent polynomial of a finite link in 3-space. For closed curves, this gives a

topological invariant that captures the grain of entanglement in this infinite periodic

system. In fact, we show that for systems of closed chains in 1 PBC, the periodic

Jones polynomial is a repetitive factor of the Jones polynomial of the infinite

component link. For open curves, this gives a polynomial with real coefficients which

are continuous functions of the chain coordinates. We show with some illustrative

examples that the periodic Jones polynomial is a useful tool for measuring knotting in

periodic systems.