Presentations of Braid Groups of Type A Arising from \((m+2)\)-angulations of Regular Polygons

Coloured quiver mutation, introduced by Buan, A.B., Thomas, H (Adv. Math. 222(3), 971–995 2009), gives a combinatorial interpretation of tilting in higher cluster categories. In type A work of Baur, K., Marsh, B. (Trans. Am. Math. Soc. 360(11), 5789-5803 2008) shows that m-coloured quivers and m-coloured quiver mutations have a nice geometrical description, given in terms of \((m+2)\) $(m+2)$ -angulations of a regular polygon, and rotations of an m-diagonal. In this paper, using such correspondence, we describe presentations of braid groups of type A arising from coloured quivers of mutation type A.