On the construction of foldings of branched covers along knots

In this talk, we study braiding and folding of branched covers of the 3-sphere along knots, focusing on constructions derived from quandle colorings of knots and quipu diagrams for finite groups. These techniques were developed in earlier work. We present a detailed folding of the dihedral cover of \(S^3\) branched along the torus knot \(T(2,5)\), and describe a related example for the trefoil knot using the alternating group \(A_4\)​. These constructions provide explicit geometric models for branched covers and suggest potential extensions to surface-knot branch sets in \(S^4\).