Year: 2026
Type: preprint
Abstract: 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 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 $π_1(X,\ast)$. For a knot $K$ in $S^3$, applying the construction to its exterior gives the \textit{knot loop 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.
Source: arXiv (Cornell University)
Author: Kamada, Seiichi
Institutions: Osaka University of Economics, The University of Osaka
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221763462