arXiv subjects
Publications and source records attributed to Takefumi Nosaka.
We determine the center of a meta-nilpotent quotient of a mapping-torus group. As a corollary, we introduce two invariants, which are quadratic forms, of knots and of mapping classes.
We define a map from second quandle homology to the Schur multiplier and examine its properties. Furthermore, we express the second homology of Alexander quandles in terms of exterior algebras. Additionally, we present a self-contained proof of its structure and provide some computational examples.
Given a representation of a link group, we introduce a trilinear form, as a topological invariant. We show that, if the link is either hyperbolic or a knot with malnormality, then the trilinear form equals the pairing of the (twisted) triple cup product and the fundamental relative 3-class. Further, we give some examples of the computation.
We focus on the cohomology of the $k$-th nilpotent quotient of the free group, $F/F_k$. This paper describes all the group 2-, 3-cocycles in terms of Massey products, and gives expressions for some of the 3-cocycles. We also give simple proofs of some of the results on Milnor invariants and the Johnson-Morita homomorphisms.
We describe a finite presentation of $ \mathcal{T}_{g,r} $ for $g \geq 3$. % or $(g,r)=(2,0)$. Here $\mathcal{T}_{g,r} $ is the universal central extension of the mapping class group of the surface of genus $g$ with $r$-boundaries. We also investigate the case $g=2$,
We show a de Rham theory for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in \cite{Dup2,DK} produce cocycles of quandles.
As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree $ k $ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k $. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and discuss a method of computing these invariants
We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to every link-group representation. The point is an observation of a bridge between the relative group homology and quandle homology from the viewpoints of Inoue--Kabaya map \cite{IK}. Furthermore, we give an algorithm to algebraically describe the fundamental 3-class of any hyperbolic knot.
This paper develops an approach for describing centrally extended groups, as determining the adjoint groups associated with quandles. Furthermore, we explicitly describe such groups of some quandles. As a corollary, we determine some second quandle homologies.
We introduce invariants of Hurwitz equivalence classes with respect to arbitrary group $G$. The invariants are constructed from any right $G$-modules $M$ and any $G$-invariant bilinear function on $M$, and are of bilinear forms. For instance, when $G$ is the mapping class group of the closed surface, $\mathcal{M}_g$, we get an invariant of 4-dimensional Lefschetz fibrations over the 2-sphere. Moreover, the construction is applicable for the quantum representations of $\mathcal{M}_g $ derived from Chern-Simons field theory. We compute the associated invariants in some cases, and find infinitely many Lefschetz fibrations which have the same Seiberg-Witten invariant and are homeomorphic but not mutually isomorphic as fibrations.
We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every links in the 3-sphere. As a corollary, we construct bilinear forms on the twisted Alexander modules of links.
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 $\ell$ which does not divide the type of $X$, the $\ell$-torsion of this invariants is equal to a sum of the colouring polynomial and a $\Z$-equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Moreover, our homotopical approach involves application of computing some third homology groups and second homotopy groups of the classifying spaces of quandles, from results of group cohomology.
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.
T. Mochizuki determined all 3-cocycles of the third quandle cohomologies of Alexander quandles on finite fields. We show that all the 3-cocycles, except those of 2-cocycle forms, are derived from group 3-cocycles of a meta-abelian group. Further, the quandle cocycle invariant of a link using Mochizuki's 3-cocycle is equivalent to a $\Z$-equivariant part of the Dijkgraaf-Witten invariant of a cyclic covering of $S^3$ branched over the link using the group. We compute some Massey triple products via the former invariant.
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.