https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00018708_v178_n2_p177_Andruskiewitsch.pdf

From racks to pointed Hopf algebras

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

3-Manifold Groups

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces (CX, c^q), where C is the field of complex numbers, X is a rack and q is a 2-cocycle on X with values in C^*. Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks contaninig properly the existing ones. We introduce a "Fourier transform" on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.

Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation

This article surveys many aspects of the theory of quandles which algebraically encode the Reidemeister moves. In addition to knot theory, quandles have found applications in other areas which are only mentioned in passing here. The main purpose is to give a short introduction to the subject and a guide to the applications that have been found thus far for quandle cocycle invariants.

A survey of quandle ideas

Knots and links Algebraic structures Quandles Quandles and groups Generalizations of quandles Enhancements Generalizd knots and links Bibliography Index

Quandles: An Introduction to the Algebra of Knots

Wedescribeawayofrepresentingfinitebiquandleswithnelements as 2n ◊ 2n block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting invariants of non-quandle biquandles can reveal information not present in the knot quandle, such as the nontrivialityofthevirtualtrefoilandvariousKishinoknots.Wealso exhibitanorientedvirtualknotwhichisdistinguishedfromboth its obverse and its reverse by a finite biquandle counting invariant.Weclassifybiquandlesoforder2,3and4andprovideaURL for our Maple programs for computing with finite biquandles.

https://intlpress.com/site/pub/files/_fulltext/journals/hha/2006/0008/0002/HHA-2006-0008-0002-a003.pdf

Matrices and finite biquandles

Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander quandles of the form Z_n[t,t^-1]/(t-a) where gcd(n,a)=1 (called linear quandles) are isomorphic, as well as specific conditions on when two linear quandles are dual and which linear quandles are connected. We apply this result to obtain a procedure for classifying Alexander quandles of any finite order and as an application we list the numbers of distinct and connected Alexander quandles with up to fifteen elements.

Classification of Finite Alexander Quandles

Finite quandles with n elements can be represented as n-by-n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group of each finite quandle. As an application, we classify finite quandles with up to 5 elements and compute the automorphism group for each quandle.

Matrices and Finite Quandles

We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.

https://www.ams.org/tran/2009-361-10/S0002-9947-09-04751-5/S0002-9947-09-04751-5.pdf

Sam Nelson

Papers

Quandles: An Introduction to the Algebra of Knots

Matrices and finite biquandles

Classification of Finite Alexander Quandles

Matrices and Finite Quandles

Virtual Yang-Baxter cocycle invariants