Multivariate Quandles as Groupoid Invariants
Abstract
We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid cal G, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings ℤ[ℤ], the multivariate version involves modules over groupoid rings ℤ[cal G]. A manifestly categorical description is given in terms of ℤ-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between ℤ-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.
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BibTeX
@misc{arsiwalla2026multivariate,
title = {{Multivariate Quandles as Groupoid Invariants}},
author = {Xerxes D. Arsiwalla and Louis H. Kauffman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30262},
eprint = {2609.30262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30262},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multivariate Quandles as Groupoid Invariants
AU - Xerxes D. Arsiwalla
AU - Louis H. Kauffman
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30262
UR - https://arxiv.org/abs/2609.30262
AB - We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid cal G, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings ℤ[ℤ], the multivariate version involves modules over groupoid rings ℤ[cal G]. A manifestly categorical description is given in terms of ℤ-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between ℤ-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.
ER -