Loop Fundamental Quandles: A Geometric Generalization of the Knot Quandle

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,∗) of a pair (X,A) with a fixed base point ∗∈ 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 ∗. More generally, for a loop condition P on A, invariant under free homotopy, we define the loop fundamental quandle LQ(X,A,∗;P). We establish functoriality and homotopy invariance of the loop fundamental quandle and show that it admits a natural augmented quandle structure over π₁(X,∗). For a knot K in S³, applying the construction to its exterior gives the \textit{knot loop quandle} LQ(K,∗;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.

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BibTeX RIS

BibTeX

@misc{kamada2026loop,
  title = {{Loop Fundamental Quandles: A Geometric Generalization of the Knot Quandle}},
  author = {Seiichi Kamada},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16056},
  eprint = {2610.16056},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16056},
}

RIS

TY  - UNPB
TI  - Loop Fundamental Quandles: A Geometric Generalization of the Knot Quandle
AU  - Seiichi Kamada
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16056
UR  - https://arxiv.org/abs/2610.16056
AB  - 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,∗) of a pair (X,A) with a fixed base point ∗∈ 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 ∗. More generally, for a loop condition P on A, invariant under free homotopy, we define the loop fundamental quandle LQ(X,A,∗;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,∗). For a knot K in S^3, applying the construction to its exterior gives the \textit{knot loop quandle} LQ(K,∗;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.
ER  -

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