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?


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.


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