Cocycle Invariants Research Articles

(Co)homology of racks and multiple group racks for compact oriented surfaces in the 3-sphere

We introduce (co)homology theory for multiple group racks and construct cocycle invariants of compact oriented surfaces in the 3-sphere using their 2-cocycles, where a multiple group rack is a rack consisting of a disjoint union of groups. We provide a method to find new rack cocycles and multiple group rack cocycles from existing rack cocycles and quandle cocycles. Evaluating cocycle invariants, we determine several symmetry types, caused by chirality and invertibility, of compact oriented surfaces in the 3-sphere.

Fundamental heaps for surface ribbons and cocycle invariants

Fundamental heaps for surface ribbons and cocycle invariants

Quandle colorings vs. biquandle colorings

Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for virtual links. However, we have not found an essentially refined invariant for classical/surface links so far. In this paper, we give an explicit one-to-one correspondence between biquandle colorings and quandle colorings for classical/surface links. We also show that biquandle homotopy invariants and quandle homotopy invariants are equivalent. As a byproduct, we can interpret biquandle cocycle invariants in terms of shadow quandle cocycle invariants.

Set-theoretic Yang–Baxter (co)homology theory of involutive non-degenerate solutions

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang–Baxter equation and cycle sets. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang–Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology theory of an involutive right non-degenerate solution of the Yang–Baxter equation and compute the normalized set-theoretic Yang–Baxter homology of cyclic racks. Moreover, we explicitly calculate some two-cocycles, which can be used to classify certain families of torus links.

Extensions of augmented racks and surface ribbon cocycle invariants

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 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 such extensions in terms of what we call fibrant and additive cohomology of racks. Simultaneous extensions of racks and groups are considered, where the respective 2-cocycles are related through a certain formula. Furthermore, we construct coloring and cocycle invariants for compact orientable surfaces with boundary in ribbon forms embedded in 3-space.

Fundamental heap for framed links and ribbon cocycle invariants

A heap is a set with a certain ternary operation that is self-distributive and exemplified by a group with the operation [Formula: see text]. We introduce and investigate framed link invariants using heaps. In analogy with the knot group, we define the fundamental heap of framed links using group presentations. The fundamental heap is determined for some classes of links such as certain families of torus and pretzel links. We show that for these families of links there exist epimorphisms from fundamental heaps to Vinberg and Coxeter groups, implying that corresponding groups are infinite. A relation to the Wirtinger presentation is also described. The cocycle invariant is defined using ternary self-distributive (TSD) cohomology, by means of a state sum that uses ternary heap [Formula: see text]-cocycles as weights. This invariant corresponds to a rack cocycle invariant for the rack constructed by doubling of a heap, while colorings can be regarded as heap morphisms from the fundamental heap. For the construction of the invariant, first computational methods for the heap cohomology are developed. It is shown that the cohomology splits into two types, called degenerate and nondegenerate, and that the degenerate part is one-dimensional. Subcomplexes are constructed based on group cosets, that allow computations of the nondegenerate part. Computations of the cocycle invariants are presented using the cocycles constructed, and conversely, it is proved that the invariant values can be used to derive algebraic properties of the cohomology.

Universal cocycle invariants for singular knots and links

Given a biquandle [Formula: see text], a function [Formula: see text] with certain compatibility and a pair of non commutative cocyles [Formula: see text] with values in a non necessarily commutative group [Formula: see text], we give an invariant for singular knots/links. Given [Formula: see text], we also define a universal group [Formula: see text] and universal functions governing all 2-cocycles in [Formula: see text], and exhibit examples of computations. When the target group is abelian, a notion of abelian cocycle pair is given and the “state sum” is defined for singular knots/links. Computations generalizing linking number for singular knots are given. As for virtual knots, a “self-linking number” may be defined for singular knots.

Shifting chain maps in quandle homology and cocycle invariants

Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map σ \sigma on each quandle chain complex that lowers the dimensions by one. By using its pull-back σ ♯ \sigma ^\sharp , each 2 2 -cocycle ϕ \phi gives us the 3 3 -cocycle σ ♯ ϕ \sigma ^\sharp \phi . For oriented classical links in the 3 3 -space, we explore relation between their quandle 2 2 -cocycle invariants associated with ϕ \phi and their shadow 3 3 -cocycle invariants associated with σ ♯ ϕ \sigma ^\sharp \phi . For oriented surface links in the 4 4 -space, we explore how powerful their quandle 3 3 -cocycle invariants associated with σ ♯ ϕ \sigma ^\sharp \phi are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.

Cocycle Invariants and Oriented Singular Knots

We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called 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.

Quantum Enhancements via Tribracket Brackets

We enhance the tribracket counting invariant with 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. Wex provide explicit examples as well as questions for future work.

Shadow biquandles and local biquandles

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 [12–14] 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 (bi)quandle cocycles.

Local biquandles and Niebrzydowski's tribracket theory

We introduce a new algebraic structure called 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 similarly as biquandle (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.

Biquandle cohomology and state-sum invariants of links and surface-links

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.

Quasi-trivial quandles and biquandles, cocycle enhancements and link-homotopy of pretzel links

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 Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications 22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].

Bowling ball representation of virtual string links

In this paper, we investigate the virtual string links via a probabilistic interpretation. This representation can be used to distinguish some virtual string links from classical string links. In order to study the algebraic structure behind this probabilistic interpretation, we introduce the notion of virtual flat biquandle. The cocycle invariants associated with virtual flat biquandle are also discussed.

HOMOLOGY FOR QUANDLES WITH PARTIAL GROUP OPERATIONS.

A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group and quandle homology theories. A quandle that is a union of groups with the operation restricting to conjugation on each group component is called a multiple conjugation quandle (MCQ, defined rigorously within). In this definition, compatibilities between the group and quandle operations are imposed which are motivated by considerations on colorings of handlebody-links. The homology theory defined here for MCQs takes into consideration both group and quandle operations, as well as their compatibility. The first homology group is characterized, and the notion of extensions by 2-cocycles is provided. Degenerate subcomplexes are defined in relation to simplicial decompositions of prismatic (products of simplices) complexes and group inverses. Cocycle invariants are also defined for handlebody-links.

Parity biquandle invariants of virtual knots

We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. The cocycle invariants are determined by pairs consisting of a biquandle 2-cocycle ϕ0 and a map ϕ1 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.

Quandle coloring and cocycle invariants of composite knots and abelian extensions.

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 coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of 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, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.

Integral positive (negative) quandle cocycle invariants are trivial for knots

In this note we prove that for any finite quandle X and any 2-cocycle ϕ∈ZQ±2(X;Z), the cocycle invariant Φϕ±(K) is trivial for all knots K.

The shadow nature of positive and twisted quandle invariants of knots

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.