Skew-rack cocycle invariants of closed 3-manifolds

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.

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.

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.

Extensions of augmented racks and surface ribbon cocycle invariants

Quandle colorings vs. biquandle colorings

We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orientable. The invariants can be used in order to estimate the minimal triple point numbers of non-orientable surface-links.

As one of the problems in his list [20], Ohtsuki proposed to study relations between quandle cocycle invariants and quantum invariants. The aim of this paper is to answer one of those questions. We prove that the coefficient of the finite perturbative expansion of the quandle shadow cocycle invariant defined by -Laurent polynomial quandle is Vassiliev invariant for any braids.

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.

Parity biquandle invariants of virtual knots

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

On homotopy groups of quandle spaces and the quandle homotopy invariant of links

Local biquandles and Niebrzydowski's tribracket theory

Shadow biquandles and local biquandles

We calculate the dihedral quandle cocycle invariants of twist-spins of alternating odd pretzel knots. The calculation leads us to the conclusion that there exist non-ribbon 2-knots which admit a non-trivial coloring by the dihedral quandle Rp and all of whose cocycle invariants derived from ℤp-valued 3-cocycles on Rp take value in ℤ ⊂ ℤ[ℤp] for any odd prime integer p.

In the preceding chapter, we explained \(\mathrm{Col}_X(D)\) as a link invariant. This chapter further equips it with gradings in several ways. The original one is introduced by Fenn-Rourke-Sanderson [FRS1, FRS2], and is graded by a homotopy group. After that, from the homological viewpoints, Carter-Jelsovsky-Kamada-Langford-Saito [CJKLS] used quandle cocycles to introduce computable link-invariants (which are called the cocycle invariants). Furthermore, the invariants are generalized to a shadow version, a non-abelian one, and a bigraded one (see Sects. 4.2–4.4 respectively). In this chapter, we study the invariants with various versions in turn. We assume basic knowledge of CW-complexes (see the textbook [Hat]).