共查询到20条相似文献,搜索用时 500 毫秒
1.
Monatshefte für Mathematik - In the paper we describe the class of principal quandles and we show that connected quandles can be decomposed as a disjoint union of principal quandles. We also... 相似文献
2.
This paper gives the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles. 相似文献
3.
We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles. 相似文献
4.
Takuro Mochizuki 《Journal of Pure and Applied Algebra》2003,179(3):287-330
We calculate some quandle cohomology groups; the rational cohomology groups of any finite Alexander quandles, the second cohomology groups with a finite field coefficient of any finite Alexander quandles over a finite fields, and the third cohomology groups of the finite Alexander quandles of the form . 相似文献
5.
J. Scott Carter Daniel Jelsovsky Seiichi Kamada Laurel Langford Masahico Saito 《Transactions of the American Mathematical Society》2003,355(10):3947-3989
The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids).
A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in -space and knotted surfaces in -space.
Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.
6.
We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove several results on the strong representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the Pontryagin dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations. 相似文献
7.
Valérian Even 《Applied Categorical Structures》2014,22(5-6):817-831
The purpose of this article is to clarify the relationship between the algebraic notion of quandle covering introduced by M. Eisermann and the categorical notion of covering arising from Galois theory. A crucial role is played by the adjunction between the variety of quandles and its subvariety of trivial quandles. 相似文献
8.
Kanako Oshiro 《Topology and its Applications》2012,159(4):1092-1105
We introduce the notion of pallets of quandles and define coloring invariants for spatial graphs which give a generalization of Fox colorings studied in Ishii and Yasuhara (1997) [4]. All pallets for dihedral quandles are obtained from the quotient sets of the universal pallets under a certain equivalence relation. We study the quotient sets and classify their elements. 相似文献
9.
《Journal of Pure and Applied Algebra》2022,226(7):106936
Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures. 相似文献
10.
Chuichiro Hayashi 《代数通讯》2013,41(9):3340-3349
We introduce a notion of natural orderings of elements of connected finite quandles. Let Q be such a quandle of order n. Any automophism on Q is a natural ordering when the elements are already ordered naturally. Suppose that the permutation *q is a cycle of length n ? 1. Then, the operation tables for such orderings coincide, which leads to the classification of automorphisms on Q. Moreover, every row and column of the operation table contains all the elements of such a quandle, which is due to K. Oshiro. We also consider the general case of finite connected quandles. 相似文献
11.
Markus Szymik 《代数通讯》2018,46(1):230-240
Racks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and their relation to the theory of permutations. In particular, we compute the centers of the categories and describe power operations on them, thereby revealing free extra structure that is not apparent from the definitions. This also leads to precise characterizations of these theories in the form of universal properties. 相似文献
12.
T. S. R. Fuad J. D. Phillips Xiaorong Shen 《Southeast Asian Bulletin of Mathematics》2000,24(2):217-224
The right universal multiplication group of a quandle in the variety of all quandles is constructable from the right Cayley graph of that quandle.AMS Subject Classification (2000), 20N 相似文献
13.
Peter Ulrickson 《代数通讯》2018,46(7):2964-2967
We show that the only endofunctors of the category of quandles commuting with the forgetful functor to sets are the power operations. We also give a similar statement for racks. 相似文献
14.
Applied Categorical Structures - This article is the second part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in... 相似文献
15.
We revisit finite racks and quandles using a perspective based on permutations which can aid in the understanding of the structure. As a consequence we recover old results and prove new ones. We also present and analyze several examples. Communicated by M. Dixon. 相似文献
16.
D. A. Fedoseev 《Moscow University Mathematics Bulletin》2011,66(6):239-243
Virtual quandles with two operations and the corresponding invariants of long virtual knots are discussed. A certain knot
invariant is constructed and an example of proof that two knots are not equivalent in terms of this invariant is presented. 相似文献
17.
We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex volume of a hyperbolic link only from its diagram. 相似文献
18.
Eric Ramos 《Journal of Pure and Applied Algebra》2018,222(12):3858-3876
A quandle is an algebraic structure which attempts to generalize group conjugation. These structures have been studied extensively due to their connections with knot theory, algebraic combinatorics, and other fields. In this work, we approach the study of quandles from the perspective of the representation theory of categories. Namely, we look at collections of conjugacy classes of the symmetric groups and the finite general linear groups, and prove that they carry the structure of FI-quandles (resp. -quandles). As applications, we prove statements about the homology of these quandles, and construct FI-module and -module invariants of links. 相似文献
19.
Vassily O. Manturov 《Acta Appl Math》2004,83(3):221-233
In the present work, we construct an invariant of virtual knots valued in (infinite-dimensional) Lie Algebras and establish some properties of it. This leads to some heuristic ideas how to construct quandles and extract (virtual) link invariants. 相似文献
20.
Takefumi Nosaka 《Topology and its Applications》2011,158(8):996-1011
For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995) [13], and the quandle homotopy invariant of links is defined in Z[π2(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13]. It is known that the cocycle invariants introduced in Carter et al. (2005) [3], Carter et al. (2003) [5], Carter et al. (2001) [6] can be derived from the quandle homotopy invariant.In this paper, we show that, for a finite quandle X, π2(BX) is finitely generated, and that, for a connected finite quandle X, π2(BX) is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate π2(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links. 相似文献