  We develop topological methods for characterizing the relationship between
polymer chain entanglement and bulk viscoelastic responses. We introduce
generalized Linking Number and Writhe characteristics that are applicable to
open linear chains. We investigate the rheology of polymeric chains entangled
into weaves with varying topologies and levels of chain density. To investigate
viscoelastic responses, we perform non-equilibrium molecular simulations over a
range of frequencies using sheared Lees-Edwards boundary conditions. We show
how our topological characteristics can be used to capture key features of the
polymer entanglements related to the viscoelastic responses. We find there is a
linear relation over a significant range of frequencies between the mean
absolute Writhe $Wr$ and the Loss Tangent $\tan(\delta)$. We also find an
approximate inverse linear relationship between the mean absolute Periodic
Linking Number $LK_P$ and the Loss Tangent $\tan(\delta)$. Our results show
some of the ways topological methods can be used to characterize chain
entanglements to better understand the origins of mechanical responses in
polymeric materials.
