Constructing Thompson representatives via pointed links

We extend Jones' construction to obtain a surjective map from the Brown-Thompson group $F_3$ to the set of pointed links up to pointed isotopy. We then introduce an operation on $F_3$, and use it to define a new monoid $(F_3, \diamond)$, called the central monoid. Using the extended version of Jones' construction, we obtain a surjective monoid homomorphism from the central monoid to the monoid of pointed links with connected sum. This allows us to introduce a standard form for connected sum representatives in $F_3$, and we extend this construction to a certain family of links by defining disjoint union and linking moves on $F_3$.