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We define invariants of oriented surface-links by enhancing the biquandle counting invariant using \textit{biquandle modules}, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that bead colorings of marked graph diagrams are preserved by Yoshikawa moves and hence define enhancements of the biquandle counting invariant for surface links. We provide examples illustrating the computation of the invariant and demonstrate that these invariants are not determined by the first and second Alexander elementary ideals and characteristic polynomials., Comment: 13 pages; version 2 includes typo corrections. To appear in Proc. of the AMS

We introduce a new algebraic structure called \textit{local biquandles} and show how colorings of oriented classical link diagrams and of broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles and show that it is isomorphic to Niebrzydowski's tribracket (co)homology. This implies that Niebrzydowski's (co)homology theory can be interpreted similary as biqandle (co)homology theory. Moreover through the isomorphism between two cohomology groups, we show that Niebrzydowski's cocycle invariants and local biquandle cocycle invariants are the same., 41 pages. Version 2 includes changes suggested by referee. To appear in Topology and Its Applications

Biquandles and multiple conjugation biquandles are algebras which are related to links and handlebody-links in $3$-space. Cocycles of them can be used to construct state-sum type invariants of links and handlebody-links. In this paper we discuss cocycles of a certain class of biquandles and multiple conjugation biquandles, which we call $G$-Alexander biquandles and $G$-Alexander multiple conjugation biquandles, with a relationship with group cocycles. We give a method to obtain a (biquandle or multiple conjugation biquandle) cocycle of them from a group cocycle., Comment: 31 pages

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., 38 pages, 4 figures

In their paper entitled “Quantum Enhancements and Biquandle Brackets”, Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. We prove herein that if a biquandle bracket [Formula: see text] is the pointwise product of the pair of functions [Formula: see text] with a function [Formula: see text], then [Formula: see text] is also a biquandle bracket if and only if [Formula: see text] is a biquandle 2-cocycle (up to a constant multiple). As an application, we show that a new invariant introduced by Yang factors in this way, which allows us to show that the new invariant is in fact equivalent to the Jones polynomial on knots. Additionally, we provide a few new results about the structure of biquandle brackets and their relationship with biquandle 2-cocycles.

We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures. We also continue the study of biquandle homomorphisms into a medial biquandle begun by the second author et al., finding biquandle analogs of results about Hom-quandles. We describe the biquandle structure of the Hom-biquandle, and consider the relationship between the Hom-quandle and Hom-biquandle., 15 pages, 2 figures

We enhance the tribracket counting invariant with \textit{tribracket brackets}, skein invariants of tribracket-colored oriented knots and links analogously to biquandle brackets. This infinite family of invariants includes the classical quantum invariants and tribracket cocycle invariants as special cases, as well as new invariants. We provide explicit examples as well as questions for future work., 12 pages. Version 2 includes typo corrections and suggestion from referee

In this short survey article we collect the current state of the art in the nascent field of quantum enhancements, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes classical skein invariants and quandle and biquandle cocycle invariants as well as new invariants.

This paper explores the structure groups $G_{(X,r)}$ of finite non-degenerate set-theoretic solutions $(X,r)$ to the Yang-Baxter equation. Namely, we construct a finite quotient $\overline{G}_{(X,r)}$ of $G_{(X,r)}$, generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if $X$ injects into $G_{(X,r)}$, then it also injects into $\overline{G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of $G_{(X,r)}$. We show that multipermutation solutions are the only involutive solutions with diffuse structure group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free., 32 pages. Final version. Accepted for publication in Proc. Edinburgh Math. Soc

We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant and are not determined by the Jablan polynomial.

In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant [Formula: see text] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams [Formula: see text], the following formula holds: [Formula: see text], where [Formula: see text] is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any positive integer $n$., 20 pages

To every oriented link $L$, we associate a topologically defined biquandle $\widehat{\mathcal{B}}_{L}$, which we call the topological biquandle of $L$. The construction of $\widehat{\mathcal{B}}_{L}$ is similar to the topological description of the fundamental quandle given by Matveev. We find a presentation of the topological biquandle and explain how it is related to the fundamental biquandle of the link., 14 pages, 12 figures

In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the shadow biquandle cocycle invariants for oriented surface-links., 34 pages, 28 figures. arXiv admin note: text overlap with arXiv:1502.01450

Given a shadow biquandle $(B,X)$ composed of a biquandle $B$ and a strongly connected $B$-set $X$, we have a local biquandle structure on $X$. The (co)homology groups of such shadow biquandles are isomorphic to those of the corresponding local biquandles. Moreover, cocycle invariants, of oriented links and oriented surface-links, using such shadow biquandles coincide with those using the corresponding local biquandles. These results imply that for some cases, the Niebrzydowski's theory in [14, 15, 16] for knot-theoretic ternary quasigroups is the same as shadow biquandle theory. We also show that some local biquandle $2$- or $3$-cocycles and some $1$- or $2$-cocycles of the Niebrzydowski's (co)homology theory can be induced from Mochizuki's cocycles., Comment: arXiv admin note: text overlap with arXiv:1809.09442

Title: Algebraic Structures for Knot Coloring Author: Martina Vaváčková Department: Department of Algebra Supervisor: doc. RNDr. David Stanovský, Ph.D., Department of Algebra Abstract: This thesis is devoted to the study of the algebraic structures providing coloring invariants for knots and links. The main focus is on the relationship between these invariants. First of all, we characterize the binary algebras for arc and semiarc coloring. We give an example that the quandle coloring invariant is strictly stronger than the involutory quandle coloring invariant, and we show the connection between the two definitions of a biquandle, arising from different approaches to semiarc coloring. We use the relationship between links and braids to conclude that quandles and biquandles yield the same coloring invariants. Keywords: knot, coloring invariant, quandle, biquandle iii

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for $S^1$-oriented handlebody-links using $2$-cocycles. When a multiple conjugation biquandle $X\times\mathbb{Z}_{\operatorname{type}X_Y}$ is obtained from a biquandle $X$ using $n$-parallel operations, we provide a $2$-cocycle (or $3$-cocycle) of the multiple conjugation biquandle $X\times\mathbb{Z}_{\operatorname{type}X_Y}$ from a $2$-cocycle (or $3$-cocycle) of the biquandle $X$ equipped with an $X$-set $Y$., Comment: arXiv admin note: text overlap with arXiv:1702.01363; text overlap with arXiv:1704.07792 by other authors

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Ayumu Inoue's article arXiv:1205.5891., 14 pages. Version 3 includes some corrections and typo fixes

We introduce a multiple conjugation biquandle, and show that it is the universal algebra for defining a semi-arc coloring invariant for handlebody-links. A multiple conjugation biquandle is a generalization of a multiple conjugation quandle. We extend the notion of $n$-parallel biquandle operations for any integer $n$, and show that any biquandle gives a multiple conjugation biquandle with them.

We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving strands over and under traces and identify a new stop condition for the recursive expansion. In the case of monochromatic crossings we show that biquandle brackets satisfy a Homflypt-style skein relation and we identify algebraic conditions on the biquandle bracket coefficients to allow pass-through trace moves., Comment: 22 pages; version 2 includes typo fixes. To appear in Int'l J. Math

A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, $(t,s)$-racks, Alexander biquandles and Silver-Williams switches, known as $(\tau,\sigma,\rho)$-biracks. We consider enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials., Comment: 14 pages

In this paper, we consider biquandle colorings for knotoids in $\mathbb{R}^2$ or $S^2$ and we construct several coloring invariants for knotoids derived as enhancements of the biquandle counting invariant. We first enhance the biquandle counting invariant by using a matrix constructed by utilizing the orientation a knotoid diagram is endowed with. We generalize Niebrzydowski's biquandle longitude invariant for virtual long knots to obtain new invariants for knotoids. We show that biquandle invariants can detect mirror images of knotoids and show that our enhancements are proper in the sense that knotoids which are not distinguished by the counting invariant are distinguished by our enhancements., 13 pages. Version 2 includes typo corrections and changes suggested by referee. To appear in J. Knot Theory Ramifications

For a set theoretical solution of the Yang-Baxter equation $(X,\sigma)$, we define a d.g. bialgebra $B=B(X,\sigma)$, containing the semigroup algebra $A=k\{X\}/\langle xy=zt : \sigma(x,y)=(z,t)\rangle$, such that $k\otimes_A B\otimes_Ak$ and $\mathrm{Hom}_{A-A}(B,k)$ are respectively the homology and cohomology complexes computing biquandle homology and cohomology defined in \cite{CJKS} and other generalizations of cohomology of rack-quanlde case (for example defined in \cite{CES}). This algebraic structure allow us to show the existence of an associative product in the cohomology of biquandles, and a comparison map with Hochschild (co)homology of the algebra $A$., Comment: 23 pages, proof of Theorem 5 written with more details, some references added

In this paper we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to define finite type invariants of virtual knots. The notions of indexed Jones polynomial and indexed quandle are introduced, which generalize the classical Jones polynomial and knot quandle respectively. Some applications of these new invariants are discussed. We also study how to define a generalized chord index via a fixed finite biquandle. Finally the chord index and its applications in twisted knot theory are discussed., 27 pages, 20 figures

En el Capítulo 2 definimos una biálgebra B cuya homología y cohomología coincidencon las de biquandle definidas en [CJKS] y otras generalizaciones de cohomologíadel caso quandle o rack (por ejemplo la definida en [CES2]). La estructura algebraica encontrada permite demostrar con transparencia la existencia de un producto asociativo en la cohomología de biquandles. Este producto era conocido para el caso rack (con una demostración topológica, por lo que nuestra construcción provee de una prueba completamente algebraica e independiente) pero era desconocido en el caso general de biquandles. También esta estructura algebraica descubierta permite mostrar la existencia de morfismos de comparación con cohomología de Hochschild que, eventualmente, podrán proveer de ejemplos de cálculo de cociclos, que (en grado dos para nudos, y en grado tres para superficies) pueden ser utilizados para calcular invariantes. Más aún, explicitamos un morfismo de comparación que se factoriza por un complejo que, como bimódulo, es la extensión de escalares de un álgebra de Nichols. En [AG] se define un 2-cociclo de quandle como una aplicación β : X × X → H donde (X,★) es un quandle y H es un grupo (no necesariamente abeliano) tal queβ(x1,x2)β(x1★x2,x3) = β(x1,x3)β(x1★x3,x2★x3) y β(x,x) = 1. En el Capítulo 3 generalizamos esa definición para biquandles (X,σ) adaptando lasecuaciones existentes y agregando una equación más: Una función f : X × X → H es un 2-cociclo trenzado no conmutativo si • f(x1,x2) f(σ2(x1,x2),x3) = f(x1,σ1(x2,x3)) f(σ2(x1,σ1(x2,x3)),σ2(x2,x3)), y • f(σ1(x1,x2),σ1(σ2(x1,x2),x3))= f(x2,x3) ∀x1,x2,x3 ∈ X. Definimos un grupo, Unc, y un 2-cociclo no conmutativo universal, π, tales que paratodo grupo H y f : X × X → H 2-cociclo no conmutativo, existe un único morfismo degrupos ḟ : Unc → H tal que f = ḟ ◦ π. Mostramos que Unc es funtorial. Definimos unaasignación de pesos a cada cruce en un nudo/link y, probando que cierto producto es invariante por movimientos de Reidemeister obtuvimos un nuevo invariante de nudos/links que generaliza el invariante obtenido en [CEGS]. Para cada grupo Unc definimos cocientes Uᵞnc y mostramos que estos, si bien son engeneral mucho más chicos que Unc, guardan la misma información que el primero conrespecto al cálculo de invariantes. Hemos calculado Unc y Uᵞnc para ciertos ejemplos de biquandles pequeños. Para poder trabajar con ejemplos de cardinal mayor a tres utilizamos GAP (System for Computational Discrete Algebra). Esto último nos permitió colorear links con biquandles (no provenientes de quanldles) de mayor cardinal y así distinguir nudos-links concretos (e.g.: el trebol de su imagen especular, la no trivialidad del link Whitehead, etc). Es decir, encontramos ejemplos que muestran que nuestro invariante generaliza estrictamente el definido en [CEGS]. Estos ejemplos ya se dan con biquandles de tamaño muy chico (cardinal 3) y permiten distinguir sensiblemente nudos distintos (e.g.: link Borromeo de tres "no nudos" separados , link de Whitehead de dos "no nudos", trebol de su imagen especular). Palabras clave: invariante de knot-links, cohomología de Yang-Baxter. Quandle, biquandle, rack, biálgebra, álgebra de Hopf, algebra trenzada. In Chapter 2, we define a bialgebra B such that its homology and cohomology arethe same as the biquandle ones defined in [CJKS] and other genalizations of cohomology of the quandle-rack case (for example defined in [CES2]).This algebraic structure enable us to show an associative product in biquandle cohomology. This product was known for the rack case (with topological proof) but unknown in biquandle case. This algebraic structure also allows to define comparison morphisms with other cohomology theories that could eventually provide cocycle examples (of degree two for knots and degree three for surfaces) for computing invariants. Furthermore, we give an explicit comparison morphism that factorizes by a complex that, as bimodule, is the scalar extension of a Nichols algebra. In [AG] a quandle 2-cocycle is defined as a map β : X × X → H where (X,★) is aquandle and H is a group (not necessarily abelian) such thatβ(x1,x2)β(x1★x2,x3) = β(x1,x3)β(x1★x3,x2★x3) and β(x,x) = 1. In Chapter 3 we generalized this definition to biquandles (X,σ): A function f : X × X → H is a non commutative braided 2-cocycle if verifies both • f(x1,x2) f(σ2(x1,x2),x3) = f(x1,σ1(x2,x3)) f(σ2(x1,σ1(x2,x3)),σ2(x2,x3)), and • f(σ1(x1,x2),σ1(σ2(x1,x2),x3))= f(x2,x3) ∀x1,x2,x3 ∈ X. We define a group, Unc and a universal non commutative 2-cocycle π such that forevery group H and f : X × X → H a non commutative 2-cocycle, exist a unique groupmorphism ḟ : Unc H such that f = ḟ ◦ π. We show that Unc is functorial. Define anassignment of a weight to each crossing in a knot-link. A certain product of these weights is invariant under Reidemester moves, then a new invariant for knot-links is obtained generalising the one obtained in [CEGS]. For each group Unc we defined quotients Uᵞnc which keep the same data when computing the invariant and have smaller cardinal. We calculated Unc and Uᵞnc for certain biquandles of small cardinality. To be able to work with more examples we worked with GAP (System for Computational Discrete Algebra). Creating programs we were able to color links with bigger biquandles (not coming fromquandles) and found examples that show our invariant generalizes the one defined in [CEGS]. This examples are achived using biquandles of cardinality three and distinguishknots-links (e.g.: Borromean link from three separeted unknots, Whitehead link from twounknots, trefoil knot and its mirror). . Fil: García Galofre, Juliana. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings., Comment: 14 pages, 8 figures

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A - 1 is also invertible and that the equation [Formula: see text] called the fundamental equation is satisfied. Then an invariant R-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case have been found by Bartholomew, Budden and Fenn. Solutions in the generalized quaternion case have been found by Fenn in an earlier paper. These latter solutions are only partial in the case of 2 × 2 matrices and the aim of this paper is to provide solutions to the missing cases.

It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper we describe a more subtle invariant: a family of biquandle endomorphisms obtained from the set of colorings and longitudinal information., 9 pages

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

We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle, the knot group or the traditional skein invariants., 19 pages. New examples added,typos corrected. To appear in J. Knot Theory Ramifications

We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. These invariants are determined by pairs consisting of a biquandle 2-cocycle \phi^0 and a map \phi^1 with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide examples to show that the parity cocycle invariants can distinguish virtual knots which are not distinguished by the corresponding non-parity invariants., Comment: 12 pages; v2 includes changes suggested by referee