Abstract: Consider a link $L$ in the trivial circle bundle $N$ over a surface $S$ of negative Euler characteristic. If the fiber-wise projection of $L$ to $S$ is a collection $C$ of closed curves in minimal position, then $N \ L$ is hyperbolic if and only if $C$ is filling and $N \ L$ is acylindrical. We would like to understand the behavior of $vol(N \ L)$ in terms of the topology of $C$. When $C$ is a composed of simple closed curves and $L$ is stratified, as we will define, we show that $vol(N \ L)$ is quasi-isometric to expressions involving distances in the pants graph. When $S$ is a punctured torus or a four punctured sphere and $N = PT^1(S)$, we show that the tangent field lift of $C$ is always stratified and that the volume is quasi-isometric to curve complex distance. In this talk, we will outline our proofs and give several natural examples of stratified links. This is joint work with T. Cremaschi and J. A. Rodriguez-Migueles.
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