Braid groups have an infinite cyclic group center, generated by the square of the fundamental braid.
Geometrically, the fundamental braid has the property that any two strands cross positively exactly once.
For a braid group of about four strands it is easy to show that the square of the fundamental braid commutes with the generators, using the braid relations (I have used the Artin presentation). How can this be generalized? Is there an intuitive reasoning of why this property holds for all braid groups? Maybe an inductive proof or appeal to the geometric nature of every brand crossing twice in the square of the fundamental braid?
Also, is there a way of showing that there are not any other word that commutes with every generator?