Identities of the kauffman monoid K4 and of the Jones Monoid J4

Kauffman monoids Kn and Jones monoids Jn, n=2,3,…, are two families of monoids relevant in knot theory. We prove a somewhat counterintuitive result that the Kauffman monoids K3 and K4 satisfy exactly the same identities. This leads to a polynomial time algorithm to check whether a given identity holds in K4. As a byproduct, we also find a polynomial time algorithm for checking identities in the Jones monoid J4. © Springer Nature Switzerland AG 2020.M. V. Volkov—Supported by Ural Mathematical Center under agreement No. 075-02-2020-1537/1 with the Ministry of Science and Higher Education of the Russian Federation

Identities in twisted Brauer monoids

We show that it is co-NP-hard to check whether a given semigroup identity holds in the twisted Brauer monoid $\mathcal{B}^\tau_n$ with $n\ge5$.Comment: 20 pages, 5 figures. In v2, several typos and inaccuracies are corrected, and a few suggestions by the anonymous referee are implemente

Identities and bases in plactic, hypoplactic, sylvester, and related monoids

The ubiquitous plactic monoid, also known as the monoid of Young tableaux, has deep connections to several areas of mathematics, in particular, to the theory of symmetric functions. An active research topic is the identities satisfied by the plactic monoids of finite rank. It is known that there is no “global" identity satisfied by the plactic monoid of every rank. In contrast, monoids related to the plactic monoid, such as the hypoplactic monoid (the monoid of quasi-ribbon tableaux), sylvester monoid (the monoid of binary search trees) and Baxter monoid (the monoid of pairs of twin binary search trees), satisfy global identities, and the shortest identities have been characterized. In this thesis, we present new results on the identities satisfied by the hypoplactic, sylvester, #-sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, thus showing that the identity checking problems of these monoids are in the complexity class P, and prove that the varieties generated by these monoids have finite axiomatic rank, by giving a finite basis for them. We also give a subdirect representation ofmultihomogeneous monoids by finite subdirectly irreducible Rees factor monoids, thus showing that they are residually finite.O ubíquo monóide plático, também conhecido como o monóide dos diagramas de Young, tem ligações profundas a várias áreas de Matemática, em particular à teoria das funções simétricas. Um tópico de pesquisa ativo é o das identidades satisfeitas pelos monóides pláticos de característica finita. Sabe-se que não existe nenhuma identidade “global” satisfeita pelos monóides pláticos de cada característica. Em contraste, sabe-se que monóides ligados ao monóide plático, como o monóide hipoplático (o monóide dos diagramas quasifita), o monóide silvestre (o monóide de árvores de busca binárias) e o monóide de Baxter (o monóide de pares de árvores de busca binária gémeas), satisfazem identidades globais, e as identidades mais curtas já foram caracterizadas. Nesta tese, apresentamos novos resultados acerca das identidades satisfeitas pelos monóides hipopláticos, silvestres, silvestres-# e de Baxter. Mostramos como mergulhar estes monóides, de característica estritamente maior que 2, num produto direto de cópias do monóide correspondente de característica 2. Confirmamos assim que todos os monóides da mesma família, de característica maior ou igual a 2, satisfazem exatamente as mesmas identidades. A seguir, damos uma caracterização completa dessas identidades, mostrando assim que os problemas de verificação de identidades destes monóides estão na classe de complexidade P, e provamos que as variedades geradas por estes monóides têm característica axiomática finita, ao apresentar uma base finita para elas. Também damos uma representação subdireta de monóides multihomogéneos por monóides fatores de Rees finitos e subdiretamente irredutíveis, mostrando assim que são residualmente finitos

Introduction to Vassiliev Knot Invariants

This book is a detailed introduction to the theory of finite type (Vassiliev) knot invariants, with a stress on its combinatorial aspects. It is intended to serve both as a textbook for readers with no or little background in this area, and as a guide to some of the more advanced material. Our aim is to lead the reader to understanding by means of pictures and calculations, and for this reason we often prefer to convey the idea of the proof on an instructive example rather than give a complete argument. While we have made an effort to make the text reasonably self-contained, an advanced reader is sometimes referred to the original papers for the technical details of the proofs. Version 3: some typos and inaccuracies are corrected.Comment: 512 pages, thousands picture

Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout