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.
Entanglement in systems employing Periodic Boundary Conditions
Polymer melts are simulated using Periodic Boundary Conditions (PBC). generate an infinite collection of chains in 3-space (see (a), (b) and (c) in Figure). The Periodic Linking Number is a mathematical tool that can be used to measure entanglement for closed, open and infinite chains in PBC. Using the Periodic Linking Number one can also define the Periodic Linking Matrix, a measure that captures all the pairwise linking in the system.
Under certain conditions, polymer chains can be seen as random coils. By studying the entanglement complexity of random walks one can gain insight on the behavior of physical filaments. What is the expected value of a topological/geometrical measure of entanglement for a random walk of fixed length/stiffness? or under confinement?
The Kauffman Bracket polynomial of open chains
A new measure of entanglement of open chains in 3-space, stronger than the Gauss linking integral, is obtained using the Kauffman bracket polynomial.
The Kauffman bracket polynomial of open chains in 3-space is a continuous function of the chain coordinates that coincides with the classical Kauffman bracket polynomial of a knot as the endpoints of the chain come together