Applied Knot Theory
Physical filaments can have many architectures: linear, ring, branched etc. Traditional tools from knot theory refer to simple closed curves. New mathematical measures of topological/geometrical complexity are needed to study entanglement of open curves.
Knot polynomials of open chains
A new measure of entanglement of open chains in 3-space, stronger than the Gauss linking integral, is obtained using the Jones polynomial.
The Jones polynomial of open chains in 3-space is a continuous function of the chain coordinates that coincides with the classical Jones polynomial of a knot as the endpoints of the chain come together.
This study lead us to define Vassiliev measures of open curves in 3-space.