Pointed Racks and Their Applications to Braid Theory

We define a new algebraic structure called a \emph{pointed rack} and use it to construct ambient isotopy invariants of $n$-braids. We first introduce an integer-valued invariant of braids using pointed racks. This is then strengthened by defining a matrix-valued invariant using racks. Moreover, our invariant determines the rack coloring invariant previously defined for the closure of the braid. Finally, we include examples of braids that are distinguished by these new invariants.