Templates are branched 2–manifolds with semi-flows used to model “chaotic” hyperbolic invariant sets of flows on 3–manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be bounded. We prove a special case when the template is positive; the general case is now known to be false.
"Knots on a positive template have a bounded number of prime factors." Algebr. Geom. Topol. 5 (2) 563 - 576, 2005. https://doi.org/10.2140/agt.2005.5.563