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A complete invariant defined for (closed, connected, orientable) 3-manifolds is an invariant defined for the 3-manifolds such that any two 3-manifolds with the same invariant are homeomorphic. Further, if the 3-manifold itself is reconstructed from the data of the complete invariant, then it is called a characteristic invariant defined for the 3-manifolds. In previous papers by the first author, a characteristic lattice point invariant and a characteristic rational invariant defined for the 3-manifolds were constructed which also produced a smooth real function with the definition interval ( − 1 , 1 ) as a characteristic invariant defined for the 3-manifolds. In this paper, a complex number-valued characteristic invariant for the 3-manifolds whose norm is smaller than or equal to one half is introduced by using an embedding of a set of lattice points called the ADelta set, distinct from the PDelta set, into the set of complex numbers. The distributive situation for the invariants of the 3-manifolds of lengths up to 10 is plotted in the complex number plane with radius smaller than or equal to one half. By using this complex number-valued characteristic invariant, a holomorphic function with the unit open disk as the definition domain which is called the characteristic quantity function is constructed as a characteristic invariant defined for the 3-manifolds. [ABSTRACT FROM AUTHOR]

Abstract Let ϕ : M → M be a diffeomorphism of a C ∞ compact connected manifold, and X its mapping torus. There is a natural fibration p : X → S 1 , denote by ξ ∈ H 1 (X , Z) the corresponding cohomology class. Let ρ : π 1 (X , x 0) → GL (n , C) be a representation (here x 0 ∈ M); denote by H ⁎ (X , ρ) the corresponding twisted cohomology of X. Denote by ρ 0 the restriction of ρ to π 1 (M , x 0) , and by ρ 0 ⁎ the antirepresentation conjugate to ρ 0. We construct from these data the twisted monodromy homomorphism ϕ ⁎ of the group H ⁎ (M , ρ ⁎ 0). This homomorphism is a generalization of the homomorphism induced by ϕ in the ordinary homology of M. The aim of the present work is to establish a relation between Massey products in H ⁎ (X , ρ) and Jordan blocks of ϕ ⁎. We have a natural pairing H ⁎ (X , C) ⊗ H ⁎ (X , ρ) → H ⁎ (X , ρ) ; one can define Massey products of the form 〈 ξ , ... , ξ , x 〉 , where x ∈ H ⁎ (X , ρ). The Massey product containing r terms ξ will be denoted by 〈 ξ , x 〉 r ; we say that the length of this product is equal to r. Denote by M k (ρ) the maximal length of a non-zero Massey product 〈 ξ , x 〉 r for x ∈ H k (X , ρ). Given a non-zero complex number λ define a representation ρ λ : π 1 (X , x 0) → GL (n , C) as follows: ρ λ (g) = λ ξ (g) ⋅ ρ (g). Denote by J k (ϕ ⁎ , λ) the maximal size of a Jordan block of eigenvalue λ of the automorphism ϕ ⁎ in the homology of degree k. The main result of the paper says that M k (ρ λ) = J k (ϕ ⁎ , λ). In particular, ϕ ⁎ is diagonalizable, if a suitable formality condition holds for the manifold X. This is the case if X a compact Kähler manifold and ρ is a semisimple representation. The proof of the main theorem is based on the fact that the above Massey products can be identified with differentials in a Massey spectral sequence, which in turn can be explicitly computed in terms of the Jordan normal form of ϕ ⁎. [ABSTRACT FROM AUTHOR]

Abstract: Let be either the real number field or the complex number field and the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given -vector bundle over to be stably extendible to for every . In this paper, we simplify the theorems and apply them to the tangent bundle of , its complexification, the normal bundle associated to an immersion of in , and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5]. [Copyright &y& Elsevier]

Abstract: In this paper, applying Chebyshev polynomials we give a basic proof of the irreducibility over the complex number field of the defining polynomial of -character variety of twist knots in infinitely many cases. The irreducibility, combined with a result in the paper of M. Boileau, S. Boyer, A.W. Reid and S. Wang in 2010, shows the minimality of infinitely many twist knots for a partial order on the set of prime knots defined by using surjective group homomorphisms between knot groups. In Appendix B, we also give a straightforward proof of the result of Boileau et al. [Copyright &y& Elsevier]

Abstract: Let F be the real number field R or the complex number field C, and let denote the real projective n-space. In this paper, we study the conditions for a given F-vector bundle over to be stably extendible to for every , and establish the formulas on the power (r-fold) of an F-vector bundle ζ over . Our results are improvements of the previous papers [T. Kobayashi, H. Yamasaki, T. Yoshida, The power of the tangent bundle of the real projective space, its complexification and extendibility, Proc. Amer. Math. Soc. 134 (2005) 303–310] and [Y. Hemmi, T. Kobayashi, Min Lwin Oo, The power of the normal bundle associated to an immersion of , its complexification and extendibility, Hiroshima Math. J. 37 (2007) 101–109]. Furthermore, we answer the stable splitting problem for F-vector bundles over by means of arithmetic conditions. [Copyright &y& Elsevier]

Abstract: For a surface F, the Kauffman bracket skein module of , denoted , admits a natural multiplication which makes it an algebra. When specialized at a complex number t, nonzero and not a root of unity, we have , a vector space over . In this paper, we will use the product-to-sum formula of Frohman and Gelca to show that the vector space has five distinct traces. One trace, the Yang–Mills measure, is obtained by picking off the coefficient of the empty skein. The other four traces on correspond to the four singular points of the moduli space of flat -connections on the torus. [Copyright &y& Elsevier]

Using the R-matrix formulation of the sl3 invariant of links, we compute the coloured sl3 generalised Jones polynomial for the trefoil. From this, the PSU(3) invariant of the Poincare´ homology sphere is obtained. This takes complex number values at roots of unity. The result obtained is formally an infinite sum, independent of the order of the root of unity, which at roots of unity reduces to a finite sum. This form enables the derivation of the PSU(3) analogue of the Ohtsuki series for the Poincare´ homology sphere, which it was shown by Thang Le could be extracted from the PSU(N) invariants of any rational homology sphere. [Copyright &y& Elsevier]

We study the local structure of the variety of representations of a virtual knot group in SL2(C) near an abelian representation ρ0. To such a representation is attached a complex number ω and there are three cases. If ω and ω−1 are not roots of the Alexander polynomial, there are only abelian representations around ρ0. If ω is a root and ω−1 is not, there are only reducible representations. If both ω and ω−1 are roots and certain homological conditions hold, there are irreducible representations. [Copyright &y& Elsevier]

There is a four-point space S1 weakly homotopy equivalent to the circle. The restriction to S1 of the complex number multiplication is not continuous, nevertheless a continuous model of the multiplication with values in S1 can be defined on an eight-point circle. Applying an analogue of Hopf''s construction we obtain a finite model of Hopf''s famous map S3→S2. [Copyright &y& Elsevier]