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 $\mathit{\text{loop}}\mathit{\text{fundamental}}\mathit{\text{quandle}}$ $LQ(X,A,\ast )$ of a pair $(X,A)$ with a fixed base point $\ast \in X$. Its elements are represented by pairs consisting of a loop in $A$ and a path joining the base point of the loop to the fixed base point $\ast$. More generally, for a loop condition $P$ on $A$, invariant under free homotopy, we define the loop fundamental quandle $LQ(X,A,\ast ;P)$. We establish functoriality and homotopy invariance of the loop fundamental quandle and show that it admits a natural augmented quandle structure over ${\pi }_1(X,\ast )$. For a knot $K$ in $S^3$, applying the construction to its exterior gives the $\mathit{\text{knot}}\mathit{\text{loop}}\mathit{\text{quandle}}$ $LQ(K,\ast ;P)$. Appropriate choices of the loop condition recover the knot quandle, its $n$-th power quandles, and the full knot quandle. The construction is inspired by Joyce’s noose-theoretic approach, but allows loop data that need not arise as the boundary of a disk satisfying the noose condition.