Marco A. Farinati, Juliana García Galofre
We define a knot/link invariant using set theoretical solutions $(X,σ)$ of the Yang-Baxter equation and non commutative 2-cocycles. We also define, for a given $(X,σ)$, a universal group Unc(X) governing all 2-cocycles in $X$, and we exhibit examples of computations.
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For a given $(X,S,β)$, where $S,β\colon X\times X\to X\times X$ are set theoretical solutions of Yang-Baxter equation with a compatibility condition, we define an invariant for virtual (or classical) knots/links using non commutative 2-cocycles pairs $(f,g)$ that generalizes the one defined in [FG2]. We also define, a group $U_{nc}^{fg}=U_{nc}^{fg}(X,S,β)$ and functions $π_f, π_g\colon X\times X\to U_{nc}^{fg}(X)$ governing all 2-cocycles in $X$. We exhibit examples of computations achieved using GAP.
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.
We explore a knot invariant derived from colorings of corresponding $1$-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle $2$-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding $2$-cocycles. This permits the construction of many $2$-cocycle invariants without exhibiting explicit $2$-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann's knot coloring polynomial. Computatio…
Relations will be described between the quandle cocycle invariant and the minimal number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimal number is given in terms of the quandle cocycle invariant.
In this note we prove that for any finite quandle $X$ and any 2-cocycle $ϕ\in Z^2_{Q\pm}(X; \mathds{Z})$, the cocycle invariant $Φ^{\pm}_ϕ(K)$ is trivial for all knots $K$.
Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial $2$-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also e…
We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak-Viro's formulas for finite-type knot invariants. We conjecture that these formulas always represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to coincide over Z/2 with the Teiblum-Turchin cocycle v_3^1.
Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.
Given a biquandle $(X, S)$, a function $τ$ with certain compatibility and a pair of {\em non commutative cocyles} $f,h:X \times X\to G$ with values in a non necessarily commutative group $G$, we give an invariant for singular knots / links. Given $(X,S,τ)$, we also define a universal group $U_{nc}^{fh}(X)$ and universal functions governing all 2-cocycles in $X$, and exhibit examples of computations. When the target group is abelian, a notion of {\em abelian cocycle pair} is given and the "state sum" is defined for singular knots/links. Computations generalizing linking number for singular knot…
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow world are extended to the positive and twisted cases. Another application is a sharpening of twisted quandle cocycle invariants for multi-component links.
Minimum numbers of Dehn colors of knots and symmetric local biquandle cocycle invariants
本文提出利用对称局部双纽结圈上链不变量评估纽结 Dehn 染色最小数的方法，回答了前文遗留问题，并证明存在能被该最小数区分的纽结。
The theory of Gauss diagrams and Gauss diagram formulas provides convenient ways to compute knot invariants, such as coefficients of the HOMFLYPT polynomial. In \cite{4,5}, the author uses Gauss diagram formulas to find combinatorial 1-cocycles in the moduli space of knots in the solid torus. Evaluated on canonical loops, one can then obtain new, non trivial knot invariants. In those books, the author conjectures that a new formula, based on the Vassiliev invariant $v_3$ also gives a 1-cocycle. We prove that it is in fact true by using the same methods developed by the author in those books.
To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for the tables of knots and tangles, to investigate which tangles may or may not embed in knots in the tables.
State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.
The fundamental problem of knot theory is to know whether two knots are equivalent or not. As a tool to prove that two knots are different, mathematicians have developed various invariants. Knots invariants are just functions that can be computed from the knot and depend only on the topology of the knot. Here we describe quantum invariants, a powerful family of invariants related to the celebrated Yang-Baxter equation.
Let $M^{reg}$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M^{reg}_n$ be the moduli space of all n-cables of framed long knots which are twisted by a string link to a knot in the solid torus $V^3$ . We upgrade the Vassiliev invariant $v_2$ of a knot to an integer valued combinatorial 1-cocycle for $M^{reg}_n$ by a very simple formula. This 1-cocycle depends on a natural number $a \in \mathbb{Z}\cong H_1(V^3;\mathbb{Z})$ with $0<a<n$ as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation pol…
We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow fundamental classes, we show the relation between the shadow quandle cocycle invariants and the based shadow quandle cocycle invariants. Moreover, we will show, for a prime knot, that any third quandle homology classes are considered as images of the shadow fundamental classes of …
We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since dim_{F_p} H_Q^3(R_p; F_p) = 1, our 3-cocycle is a constant multiple of the Mochizuki 3-cocycle up to coboundary. Dually, we construct a group cycle represented by a cyclic branched covering branched along a knot K from the quandle cycle associated with a colored diagram of K.
We give a method to construct non symmetric solutions of a global tetrahedron equation from solutions of the Yang-Baxter equation. The solution in the HOMFLYPT case gives rise to the first combinatorial quantum 1-cocycle which represents a non trivial cohomology class in the topological moduli space of long knots. We conjecture that the quotient of its values on Hatchers loop and on the rotation around the long axis is related to the simplicial volume of the knot complement in the 3-sphere and we prove this for the figure eight knot. Surprisingly, the formula for the solution in the HOMFLYPT c…
Quantum Cocycle Invariants of Knots from Yang-Baxter Cohomology
本文基于杨-巴克斯特上同调的2-上循环，从形变理论视角引入纽结量子上循环不变量，证明其可解释拟群上循环不变量且在部分例子中更强，还推导了琼斯、亚历山大多项式并给出相关实例。
A refined 1-cocycle for regular isotopies and the refined tangle equations
本文改进了正则同胚长结的组合1-余循环，用于定义结方程，其中系数为洛朗多项式而非整数，提供关于结同胚的量化信息。
This paper studies the chirality of knotoids using shadow quandle colorings and the shadow quandle cocycle invariant. The shadow coloring number and the shadow quandle cocycle invariant is shown to distinguish infinitely many knotoids from their mirrors. Specifically, the knot-type knotoid $3_1$ is shown to be chiral. The weight of a quandle 3-cocycle is used to calculate the crossing numbers of infinitely many multi-linkoids.
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinning. We also demonstrate relations between cocycle invariants and Alexander matrices.
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic interpretations of quandle cocycles as deformations of extensions are also given. The proofs rely on colored knot diagrams.
Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.
By 2-twist-spinning the knotted graph that represents the knotted handlebody $5_2$, we obtain a knotted foam in 4-dimensional space with a non-trivial quandle cocycle invariant.
This paper is a survey of several papers in quandle homology theory and cocycle knot invariants that have been published recently. Here we describe cocycle knot invariants that are defined in a state-sum form, quandle homology, and methods of constructing non-trivial cohomology classes.
We give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of some torus knots are not ribbon concordant to their orientation reversed images.
A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.
We construct the first combinatorial 1-cocycle with values in the $ \mathbb{Z} [x,x^{-1}]$-module of isotopy classes of singular long knots in 3-space with a signed planar double point, and which represents a non trivial cohomology class in the topological moduli space of long knots. It can be interpreted as an invariant with values in a $ \mathbb{Z} [x,x^{-1}]$-module generated by 3-manifolds for each element of infinite order of the mapping class group of the complement of a satellite knot in $S^3$. The 1-cocycle seems to be trivial on all loops for long knots which are not satellites but it…
This paper is an extended account of my "Introductory Plenary talk at Knots in Hellas 2016" conference We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram colorings (1956). In the second section we describe how Fox work was generalized to distributive colorings (racks and quandles) and e…
In this note, we consider the problem of constructing knot invariants from Yang-Baxter operators associated to (unitary associative) algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to construct knot invariants, as modified by Murakami, invariably produces from those enhanced Yang-Baxter operators the Alexander polynomial.
Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle co…
Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang-Baxter equation. A homology theory for the set-theoretic Yang-Baxter equation was developed by Carter, Elhamdadi, and Saito in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang-Baxter equation. We obtain some concrete examples of non-trivial $n$-cocycles for Alexander biquandles. For a biquandle $X,$ its geometric realization $BX$ is discussed, which has the potential to build in…
We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Specifically, we consider a Yang-Baxter state model for the sl(n) polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a representation of the singular braid monoid into a matrix algebra, and seek conditions for extending further the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Oh…
Let $M_n$ be the topological moduli space of all parallel n-cables of long framed oriented knots in 3-space. We construct in a combinatorial way for each natural number $n>1$ a 1-cocycle $R_n$ which represents a non trivial class in $H^1(M_n; \mathbb{Z} [x_1,x_2,...,x_1^{-1},x_2^{-1},...])$, where the number of variables $x_m$ depends on $n$. To each generic point in $M_n$ we associate in a canonical way an arc {\em scan} in $M_n$, such that $R_n(scan)$ is already a polynomial knot invariant. We show that $R_3(scan)$ detects the non-invertibility of the knot $8_{17}$ in a very simple way and w…
We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called \emph{oriented singquandles} and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.
This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way. Let $M$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M_n$ be the moduli space of all n-cables $nK$ of framed long knots $K$ which are twisted by a given string link $T$ to close to a knot in the solid torus, with a marked point on the knot at infinity. First we construct integer valued combinat…
Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle invariants of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, a…
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.
A. Ishii and K. Oshiro introduced the notion of an $f$-twisted Alexander matrix. This notion is a quandle version of the twisted Alexander matrix which was introduced by M. Wada. They showed that the twisted Alexander matrix of a pair of a link group and a group representation can be recovered from the $f$-twisted Alexander matrix of a pair of a link quandle and a quandle homomorphism. In this paper, we study a relationship between the $f$-twisted Alexander matrix and the quandle cocycle invariant. As an application, we show that an $f$-twisted Alexander invariant of knot quandles is a really …
Twisted Yang-Baxter sets, cohomology theory, and application to knots
该研究引入扭曲Yang-Baxter集并构建拓展的上同调理论，结合扭曲双广群上循环与Alexander编号构造纽结及曲面的状态和不变量，成功区分2-扭转纺三叶结与其反向构型，验证了方法有效性。
Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any Yang-Baxter solution can be interpreted as a two-qudit quantum gate. Here we show that if this gate is non-entangling, then the resulting invariant of knots is trivial. We thus obtain a general connection between topological entanglement and quantum entanglement, as suggested by Kauffman et al.
The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the…
Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions.
This paper is a sequel to my essay "Distributivity versus associativity in the homology theory of algebraic structures" Demonstratio Math., 44(4), 2011, 821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arc colorings and survey associative and distributive magmas and their homology with relation to knot theory. We outline potential relations to Khovanov homology and categorification, via Yang-Baxter operators. We use here the fact that Yang-Baxter equation can be thought of as a generalization of self-distributivity. We show how to define and visualize Yang-Baxter homology…
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 ϕ^0 and a map ϕ^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.
The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then this invariant can be thought of a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian-longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for gene…
In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not trivial.
This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let π_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in π_K. Given a finite group G and an element x in G, we consider the set of representations ρfrom π_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images ρ(l_K). The resulting invariant maps knots to the gro…
A rack is a set with a binary operation that is right-invertible and self-distributive, properties diagrammatically corresponding to Reidemeister moves II and III, respectively. A rack is said to be an {\it augmented rack} if the operation is written by a group action. Racks and their cohomology theories have been extensively used for knot and knotted surface invariants. Similarly to group cohomology, rack 2-cocycles relate to extensions, and a natural question that arises is to characterize the extensions of augmented racks that are themselves augmented racks. In this paper, we characterize s…
In this article, we adjust the Yang-Baxter operators constructed by Jones for the HOMFLYPT polynomal. Then we compute the second homology for this family of Yang-Baxter operators. It has the potential to yield 2-cocycle invariant for links.
This paper is a brief overview of some of our recent results in collaboration with other authors. The cocycle invariants of classical knots and knotted surfaces are summarized, and some applications are presented.
Our main object of study is a certain degree-one cohomology class of the space K of long knots in R^3. We describe this class in terms of graphs and configuration space integrals, showing the vanishing of some anomalous obstructions. To show that this class is not zero, we integrate it over a cycle studied by Gramain. As a corollary, we establish a relation between this class and (R-valued) Casson's knot invariant. These are R-versions of the results which were previously proved by Teiblyum, Turchin and Vassiliev over Z/2 in a different way from ours.
Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented classical and virtual knots and links via algebraic structures called birack dynamical cocycles. The new invariants can also be understood in terms of partitions of the set of birack labelings of a link diagram determined by a homomorphism $p:X\to Y$ between finite labeling biracks. We provide examples to show that the new invariant is stronger than the unenhanced birack counting invariant and examine connections with other knot and link invariants.
The tangle-valued 1-cocycle for knots
本文构造取值于 tangle 模空间同调的组合 1-上闭链，并通过迭代 scan 得到 Alexander 树不变量，可局部计算且证明 $8_{17}$ 与 Conway 纽结不可逆。
Pairings of combinatorial 1-cocycles with loops in knot spaces
本文用Gauss图构造长纽结空间上的组合1-上循环，证明其有限型不变量性质，并计算与各类环的配对，给出下降判据。
For an oriented knot $K$, we construct a functor from the category of pointed quandles to the category of quandles in three different ways. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariant of knots, and study their properties.
The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.
We consider the classical problem of a position of n-dimensional manifold M in R^{n+2}. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting M to R^{n+2}. In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring, of a diagram of M embedded in R^{n+2} we have (n+1)- and (n+2)-(co)cycle invariants (i.e. invariant under Roseman moves). We speculate on a similar construction for general Yang-Baxter operators.
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