Loop Fundamental Quandles: A Geometric Generalization of the Knot Quandle

The knot quandle, introduced independently by Joyce and Matveev, admits a geometric description in terms of nooses. Motivated by this description, we introduce the \textit{loop fundamental quandle} $LQ(X,A,\ast)$ of a pair $(X,A)$ with a fixed base point $\ast\in X$. Its elements are represented by