Surfaces in the 4-ball constructed using generators of knits and their graphical description

We introduce a new construction of surfaces in $D^2 \times B^2$, called knitted surfaces or BMW surfaces, which are described as the trace of deformations of knits. Here, knits are tangles obtained from classical braids from splicing at some crossings. Knitted surfaces are a generalization of braided surfaces. Further, we generalize charts of braided surfaces to BMW charts of knitted surfaces, which are finite graphs in $B^2$, and we show that a knitted surface has a BMW chart description. We show that every compact surface with non-empty boundaries properly embedded in $D^2 \times B^2$ is ambiently isotopic to some knitted surface: so such surfaces are described by BMW charts.