[8] On third homologies of group and of quandle via the Dijkgraaf-Witten invariant and Inoue-Kabaya map, submitted, (arxiv)

We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are shown to be derived from group cocycles of some non-abelian group. As an application, we calculate some $\Z$-equivariant parts of the Dijkgraaf-Witten invariants of some cyclic branched covering spaces, via some cocycle invariant of links.

[7] Homotopical interpretation of link invariants from finite quandles, submitted, (arxiv)

This paper demonstrates a topological meaning of the quandle cocycle invariants of links with respect to finite connected quandles $X$, from a perspective of homotopy theory: Specifically, for any prime $\ell$ which does not divide the type of $X$, the $\ell$-torsion of this invariants is reduced to be a sum of the colouring polynomial and a $\Z$-equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Furthermore, our homotopical approach involves application of computing some third homologies and second homotopy groups of the classifying spaces of quandles, from group cohomology.

[6] Quandle cocycles from invariant theory, (PDF) to appear Advances in Mathematics

Let G be a group. Any G-module M has an algebraic structure called \G-family of Alexander quandles". Given a 2-cocycle of a cohomology of this G-family, topological invariants of (handlebody-)knots in the 3-sphere were de ned. This paper develops a simple algorithm to algebraically construct n-cocycles of this G-family from G-invariant group n-cocycles of the abelian group M. We give many examples of 2-cocycles of these G-families by facts in (modular) invariant theory. .

[5] Quandle homotopy invariants of knotted surfaces. Mathematische Zeitschrift 2013, (Journal page)

Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to "regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of the Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.

[4] Some topological aspects of 4-fold symmetric quandle invariants of 3-manifolds ,(joint work with Eri Hatakenaka), Int. J. Math.23, (Journal page)

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants with topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf-Witten invariants. As an application, for an odd prime $p$, we show that the quandle cocycle invariant of a link in $ S^3$ using the Mochizuki 3-cocycle is equivalent to the Dijkgraaf-Witten invariant with respect to $\Z/p\Z$ of the double covering of $S^3$ branched along the link. We also reconstruct the Chern-Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to \cite{IK}. This is joint work with Eri Hatakenaka.

[3] 4-fold symmetric quandle invariants of 3-manifolds, Algebraic and Geometric Topology 11 (2011) 1601-1648. (Journal page)

The paper introduces 4-fold symmetric quandles, and 4-fold symmetric quandle homotopy invariants of 3-manifolds. We classify 4-fold symmetric quandles and investigate their properties. When the quandle is nite, we explicitly determine a presentation of its inner automorphism group. We calculate the container of the 4-fold symmetric quandle homotopy invariant. We also discuss symmetric quandle cocycle invariants and coloring polynomials of 4-fold symmetric quandles.

[2] On quandle homology groups of Alexander quandles of prime Trans. Amer. Math. Soc. (2013) (Journal page)

In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed Fibonacci conjecture by M. Niebrzydowski and J. H. Przytycki. Further, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Moreover, we construct isomorphic cohomological operations on the quandle cohomology. Furthermore, we prove that the integral quandle homology of a nite connected Alexander quandle is annihilated by the order of the quandle.

[1] On homotopy groups of quandle spaces and the quandle homotopy invariant of links, Topology and its Applications 158 (2011), 996-1011 (Journal page)

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson, and the quandle homotopy invariant of links is defined in \Z[\pi_2(BX)], modifying the rack homotopy invariant. It is known that the cocycle invariants introduced in [3, 5, 6] can be derived from the quandle homotopy invariant.

In this paper, we show that, for a finite quandle X, $\pi_2(BX)$ is finitely generated, and that, for a connected finite quandle X, $\pi_2(BX)$ is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate \pi_2(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links.