We will also talk about how circular orderability behavior with respect to cyclic branched covers compares to left-orderability. This is a joint work with Adam Clay.

This is joint work with Cameron Gordon and Ying Hu.

This is part of joint work with Rylee Lyman.

The problem falls naturally into three cases, where the exterior of $L$ is Seifert fibered, hyperbolic, or toroidal. In this talk we will discuss some background to the ADE Conjecture and the fact that it is true in the Seifert fibered case.

This is joint work with Steve Boyer and Ying Hu.

(a) Let $M_0 = M - Int(B^{3})$, where $B^{3}$ is a $3$-ball in $M$. We show that there exists a Reebless co-orientable foliation $\mathcal{F}$ in $M_0$, whose leaves may be transverse to $\partial M_0$ or tangent to $\partial M_0$ at their intersections with $\partial M_0$, such that $\mathcal{F}$ has a transverse $(\pi_1(M_0),\mathbb{R})$ structure and that $\mathcal{F}$ is analogous to taut foliations (in closed $3$-manifolds) in the following sense: there exists a compact $1$-manifold (i.e. a finite union of properly embedded arcs and/or simple closed curves) transverse to $\mathcal{F}$ that intersects every leaf of $\mathcal{F}$.

(b) We have a process to produce a foliation $\mathcal{F}$ as given in (a), which deponds on the choice of a left-invariant order of $\pi_1(M)$ and certain fundamental domain $\Gamma$ of $M$. If $M$ admits a taut foliation that has a transverse $(\pi_1(M),\mathbb{R})$ structure, then some resulting foliation of our process can extend to a taut foliation in $M$ that has a transverse $(\pi_1(M),\mathbb{R})$ structure. If $M$ admits an $\mathbb{R}$-covered foliation, then some resulting foliation of our process can extend to an $\mathbb{R}$-covered foliation in $M$. Furthermore, we conjecture that every resulting foliation of our process can extend to a taut foliation in $M$ that has a transverse $(\pi_1(M),\mathbb{R})$ structure.