The non semi-simple TQFT of the sphere with four punctures

In this work, we compute the representation of the mapping class group of the sphere with $4$ punctures arising from the non semi-simple TQFT (constructed by Blanchet–Costantino–Geer–Patureau). We show that it is faithful. Lastly, we compare quantum representations of punctured spheres in general with those of braid groups.