  Following an idea of Fr\'ed\'eric le Roux, we define in this paper a family
of Hofer-type pseudonorms on braid groups, computing the minimal energy of a
Hamiltonian diffeomorphism which fixes a Lagrangian configuration of circles on
the unit disc and realises that braid type. We prove that in the case of braids
with two strands we have in fact a norm, and we give lower estimates for braids
with more strands. The main tool is Link Floer Homology, recently defined by D.
Cristofaro-Gardiner, V. Humili\`ere, C.-Y. Mak, S. Seyfaddini and I. Smith,
which we use to construct a family of quasimorphisms on the group of compactly
supported Hamiltonian diffeomorphisms which is sensitive to the linking number
of diffeomorphisms fixing Lagrangian links.
