Let π:M→B be an elliptic fibration over B=D2 or B=S2
Nearby in the stack
with
n
nodal fibers over
Δ⊆B
. We study the universal liftable braids for
π
: those braids that admit a fiber-preserving lift to
M
for all choices of coordinates on
(B,Δ)
. When
B=S2
, we show that nontrivial universal braids do not exist by proving a Zariski-density theorem on the
SL2
-character variety for
(S2,Δ)
. When
B=D2
we classify when the subgroup of universal braids has finite index in the braid group
Bn=Mod(D2,Δ)
, and relate these examples to Coxeter's factor groups of braid groups, which in turn are related to the platonic solids. Finally, we generalize the results derived in the finite-index cases by considering a canonical family of branched covers of the base
B
associated to any elliptic fibration. The generalization naturally connects the universal braids to the integral Burau representation reduced modulo 3.