  We discuss the appropriate techniques for modelling the geometry of open
ended elastic polymer molecules. The molecule is assumed to have fixed
endpoints on a boundary surface. In particular we discuss the concept of the
winding number, a directional measure of the linking of two curves, which can
be shown to be invariant to the set of continuous deformations vanishing at the
polymer's end-point and which forbid it from passing through itself. This
measure is shown to be the appropriate constraint required to evaluate the
geometrical properties of a constrained DNA molecule. Using the net winding
measure we define a model of an open ended constrained DNA molecule which
combines the necessary constraint of self-avoidance with being analytically
tractable. This model builds upon the local models of Bouchiat and Mezard
(2000). In particular, we present a new derivation of the polar writhe
expression, which detects both the local winding of the curve and non local
winding between different sections of the curve. We then show that this
expression correctly tracks the net twisting of a DNA molecule subject to
rotation at the endpoints, unlike other definitions used in the literature.
