3 problems

Let $P$ be a branched standard spine embedded as a faithful flow-spine for a contact structure $cxi$ on a closed $3$-manifold $\widehat{M}$, and let $\mathcal{F}_{\xi}(P)$ denote i…

Let $B$ be a branched surface in a compact triangulated atoroidal irreducible $3$-manifold $N$. A branched surface is quasi-hyperbolic if it does not carry any sphere or torus but…

Let $N$ be a compact atoroidal 3-manifold, and let $B$ be a branched surface in $N$. A branched surface is quasi-hyperbolic if it carries no sphere or torus but fully carries a sur…