arXiv subjects

Luc Ta

Publications and source records attributed to Luc Ta.

13 recordsLinked to original sources

arXiv2026

Fourier Analysis and Idempotents in Quandle Algebras

In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].

math.GT↗

arXiv2026

Idempotents, automorphism groups, and commutator widths of quandle algebras

The paper develops the theory of quandle algebras. We show that over integral domains of characteristic other than 2, quandle algebras of ordered commutative quandles have no nontrivial idempotents. We also compute the automorphism groups of quandle algebras of trivial quandles and dihedral quandles of odd orders over arbitrary commutative rings, with partial results for dihedral quandles of even orders. Finally, a computer search provides the first examples of quandle algebras of commutator width 2.

math.RA↗

arXiv2026

On medial Latin quandles and affine modules

In this note, we show that the category of Latin (resp. commutative) medial quandles is equivalent to the category of affine modules over a certain Laurent polynomial ring (resp. the dyadic rationals). As applications, we describe free objects in these categories and obtain a structure theorem for finitely generated medial commutative quandles. We also characterize racks whose duals are commutative. Collectively, this solves two open problems of Bardakov and Elhamdadi (arXiv:2601.07057).

math.GR↗

arXiv2025

From affine algebraic racks to Leibniz algebras and Yang-Baxter operators

We introduce analogues of algebraic groups called algebraic racks, which are pointed rack objects in the category of schemes over a ground field. Addressing a problem of Loday, we construct functors assigning left and right Leibniz algebras to affine algebraic racks. These functors are compatible with closed subracks and ideals, and they recover the Lie algebras of linear algebraic groups (via conjugation quandles) and the Leibniz algebras of algebraic Lie racks. We also study properties of coordinate algebras and Leibniz algebras of affine algebraic racks. Finally, we use rack schemes to functorially construct (co-)nondegenerate Yang-Baxter operators in various categories.

math.AG↗

arXiv2025

Distinguishing Power of 4-Legendrian Permutation Racks

We study 4-Legendrian racks and their effectiveness at distinguishing Legendrian knots. We prove that permutation racks with 4-Legendrian rack structures cannot distinguish Legendrian knots that share the same knot type, Thurston-Bennequin number, and rotation number. However, they also recover these three classical invariants.

math.GT↗

arXiv2025

Groups versus quandle-like invariants of 3-manifolds

Risandles are nonassociative algebraic structures recently introduced to construct invariants of 3-manifolds. In this note, we show that the categories of groups and nonempty, faithful risandles are equivalent. In analogy to knot quandles, we also introduce fundamental risandles of 3-manifolds, which categorify the risandle coloring invariants of Ishii, Nakamura, and Saito. For infinitely many 3-manifolds, the equivalence of categories recovers the fundamental group from the fundamental risandle and vice versa.

math.GT↗

arXiv2025

Good involutions of twisted conjugation subquandles and Alexander quandles

We completely describe good involutions of free quandles and subquandles of twisted conjugation quandles of groups, including all Alexander quandles. As an application, we enumerate good involutions of linear quandles, and we provide explicit mappings for those up to order 23 via a computer search. Along the way, we completely characterize connected, involutory Alexander quandles, which may be of independent interest.

math.GT↗

arXiv2025

Enumeration of virtual quandles up to isomorphism

Virtual racks and virtual quandles are nonassociative algebraic structures based on the Reidemeister moves of virtual knots. In this note, we enumerate virtual dihedral quandles and several families of virtual permutation racks and virtual conjugation quandles up to isomorphism. We also classify virtual racks and virtual quandles up to order 8 using a computer search. These classifications are based on the conjugacy class structures of rack automorphism groups. In particular, we compute class numbers of holomorphs of finite cyclic groups, which may be of independent interest.

math.GT↗

arXiv2025

Graph quandles: Generalized Cayley graphs of racks and right quasigroups

This article lays the foundations for an analogue of geometric group theory that studies actions on graphs by right quasigroups, including racks and quandles. We study markings of graphs that realize racks, and we introduce (di)graph invariants based on such markings. We show that all right quasigroups are realizable by edgeless graphs and complete (di)graphs. Using Schreier (di)graphs, we also characterize Cayley (di)graphs of right quasigroups Q that realize Q. In particular, all racks are realizable by their full Cayley (di)graphs. This solves two problems of Valeriy Bardakov. Finally, we give graph-theoretic characterizations of labeled Cayley digraphs of right-cancellative magmas, right-divisible magmas, right quasigroups, racks, quandles, involutory racks, and kei.

math.GT↗

arXiv2025

Good involutions of conjugation subquandles

Posed by Taniguchi, the classification of quandles with good involutions is a difficult question with applications to surface-knot theory. We address this question for subquandles of conjugation quandles, including all core quandles. We also study good involutions of faithful racks. In particular, we obtain sharp bounds on the number of good involutions of racks in these families. As an application of our results, we implement group-theoretic algorithms that compute all good involutions of conjugation quandles and core quandles; we provide data for those up to order 23. As another application, we construct infinite families of connected, noninvolutory symmetric quandles. We also classify symmetric and Legendrian racks, quandles, and kei up to order 8 using a computer search. Finally, we exhibit an equivalence of categories between racks and Legendrian racks that induces an equivalence between involutory racks, Legendrian kei, and symmetric kei.

math.GT↗

arXiv2025

Classification and structure of generalized Legendrian racks

This paper develops the structure theory of generalized Legendrian racks, which are algebraic structures used to study Legendrian knots. First, we give a group-theoretic characterization of GL-racks. As applications, we classify GL-racks up to order 8 up to isomorphism via an exhaustive search algorithm, and we classify certain infinite families of GL-racks. Then we compute the center of the category of GL-racks and construct an equivalence of categories between racks and GL-quandles. Finally, we study tensor products of racks and GL-racks coming from universal algebra.

math.GT↗

arXiv2025

Constructions of and Bounds on the Toric Mosaic Number

Knot mosaics were introduced by Kauffman and Lomonaco in the context of quantum knots, but have since been studied for their own right. A classical knot mosaic is formed on a square grid. In this work, we identify opposite edges of the square to form mosaics on the surface of a torus. We provide two algorithms for efficiently constructing toric mosaics of torus knots, providing upper bounds for the toric mosaic number. Using these results and a computer search, we provide a census of known toric mosaic numbers.

math.GT↗

arXiv2024

Bounds on the mosaic number of Legendrian Knots

Mosaic tiles were first introduced by Lomonaco and Kauffman in 2008 to describe quantum knots, and have since been studied for their own right. Using a modified set of tiles, front projections of Legendrian knots can be built from mosaics as well. In this work, we compute lower bounds on the mosaic number of Legendrian knots in terms of their classical invariants. We also provide a class of examples that imply sharpness of these bounds in certain cases. An additional construction of Legendrian unknots provides an upper bound on the mosaic number of Legendrian unknots. We also adapt a result of Oh, Hong, Lee, and Lee to give an algorithm to compute the number of Legendrian link mosaics of any given size. Finally, we use a computer search to provide an updated census of known mosaic numbers for Legendrian knots, including all Legendrian knots whose mosaic number is 6 or less.

math.GT↗