Simple expressions for the holed torus relations

In the mapping class group of a $k$-holed torus with $0 \leq k \leq 9$, one can factorize the boundary multi-twist (or the identity when $k=0$) as the product of twelve right-handed Dehn twists. Such factorizations were explicitly given by Korkmaz and Ozbagci for each $k \leq 9$ and an alternative one for $k=8$ by Tanaka. In this note, we simplify their expressions for the $k$-holed torus relations.

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