Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Authorship：Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.topol.2020.107533

researchmap

Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Hiroshima University - Department of Mathematics  

DOI： 10.32917/hmj/1595901628

researchmap

Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.jalgebra.2019.12.027

researchmap

Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.topol.2019.106991

researchmap

Authorship：Lead author   Publishing type：Research paper (scientific journal)   Publisher：European Mathematical Society Publishing House  

DOI： 10.4171/prims/56-1-7

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles X, from a perspective of homotopy theory: Specifically, for any prime l which does not divide the type of X, the l-torsion of this invariants is equal to a sum of the coloring polynomial and a Z-equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Moreover, our homotopical approach involves applications of computing some third homology groups and second homotopy groups of the classifying spaces of quandles, from results of group cohomology. (C) 2015 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2015.05.087

Web of Science

researchmap

Language：English   Publishing type：Research paper (scientific journal)   Publisher：GEOMETRY & TOPOLOGY PUBLICATIONS  

We propose a simple method for producing quandle cocycles from group cocycles by a modification of the Inoue-Kabaya chain map. Further, we show that, with respect to "universal extension of quandles", the chain map induces an isomorphism between third homologies (modulo some torsion). For example, all Mochizuki's quandle 3-cocycles are shown to be derived from group cocycles. 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.

DOI： 10.2140/agt.2014.14.2655

Web of Science

researchmap

Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Let G be a group. Any G-module M has an algebraic structure called a G-family of Alexander quandles. Given a 2-cocycle of a cohomology associated with this G-family, topological invariants of (handlebody) knots in the 3-sphere are defined. We develop a simple algorithm to algebraically construct n-cocycles of this G-family from G-invariant group n-cocycles of the abelian group M. We present many examples of 2-cocycles of these G-families using facts from (modular) invariant theory. (C) 2013 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2013.05.022

Web of Science

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)  

researchmap

Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to 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 constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf-Witten invariant with respect to Z/pZ 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 [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].

DOI： 10.1142/S0129167X12500644

Web of Science

researchmap

Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995)1131, and the quandle homotopy invariant of links is defined in Z[pi(2)(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13]. It is known that the cocycle invariants introduced in Carter et al. (2005) [3], Carter et al. (2003) [5], Carter et al. (2001) [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. (C) 2011 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.topol.2011.02.006

Web of Science

researchmap

Language：English   Publishing type：Research paper (scientific journal)   Publisher：GEOMETRY & TOPOLOGY PUBLICATIONS  

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 finite, 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.

DOI： 10.2140/agt.2011.11.1601

Web of Science

researchmap