Consider the braid group with $n$ strands $B\_n$. Each braid can be drawn (say) *from bottom to top* as $n$-intertwining strictly monotonic strands. Moreover, the group $B\_n$ is generated by $n-1$ generators $\sigma\_i$. Geometrically, the strands of $\sigma\_i$ are trivial except for the $i^{\text{th}}$ strand which passes *over* the ${i+1}^{\text{th}}$ strand. Similarly, the pure braid group is generated by $\binom{n}{2}$ elements. Explicitly these are $A\_{ij}=\sigma\_{j-1}...\sigma\_{i+1}\sigma\_i^2\sigma\_{i+1}^{-1}...\sigma\_{j-1}^{-1}$, for $1 \leq i <j \leq n$.


Any element of the braid group $B\_n$ can be represented as (the isotopy class of) a diffeomorphism of the $n$-punctured $2$-disc (the diffeomorphim permutes the punctures and fixes the boundary of the disc). If I am not mistaken, $\sigma\_i $ is represented by the half Dehn twist around the loop encircling the $i^{\text{th}}$ and ${i+1}^{\text{th}}$ punctures.


Is it correct that the $A\_{ij}$ are represented by full Dehn twists around the curves encircling the $i^{\text{th}}$ and ${j}^{\text{th}}$ punctures while avoiding the ones in between?



Braids, pure braids and Dehn twists
Your description of $A\_{ij}$ isn't quite correct in that it does not uniquely specify the curve. There are infinitely many non-isotopic curves such that the disc they bound in the plane contains only the $i$-th and $j$-th punctures. To specify the curves uniquely, one convention would be to put all the puncture points on a common line, and to demand your curves stay below the line except in a small neighbourhood of the puncture points. 


That would be one way to be precise about which curves you're talking about. 


