arXiv subjects
Publications and source records attributed to Teruaki Kitano.
A basic open question motivated by the study of ribbon knot diagrams asks whether every ribbon knot can be presented as a symmetric union. In this article, we give a negative answer to this question by exhibiting a ribbon Montesinos knot which does not admit a symmetric union presentation.
Thurston constructed codimension one foliations on $S^3$ thereby proved that the homomorphism $gv: \pi_3(B\overline{\Gamma}^\infty_1)\rightarrow \mathbb{R}$ induced by the Godbillon-Vey invariant is surjective. By another real analytic construction, he proved that the homomorphism $gv: H_3(B\overline{\Gamma}^\omega_1)\rightarrow \mathbb{R}$ is also surjective where $B\overline{\Gamma}^\omega_1$ is a $K(\pi,1)$ space by Haefliger. Tsuboi proved that the former surjection splits so that $\pi_3(B\overline{\Gamma}^\infty_1)= \mathbb{R}\oplus \mathrm{Ker}\,gv$. He further showed that the subgroup of $H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})$ generated by all the Thurston's constructions coincides with his direct summand $\mathbb{R}$. In this paper, we prove that Thurston's second surjection splits and also that the subgroup of $H_3(B\overline{\Gamma}^\omega_1;\mathbb{Z})$ generated by all the Thurston's cycles is equal to our direct summand $\mathbb{R}$ which is a lift of Tsuboi's one. To show this, we modify the arguments of Thurston and Tsuboi by replacing Reeb components with a real analytic construction. We prove certain {\it uniqueness} of them by showing acyclicity of the affine group in the Haefliger group $\pi_1(B\overline{\Gamma}^\omega_1)$. We also prove the existence of a new kind of characteristic class of foliations in $H^4(B\overline{\Gamma}^\omega_1;\mathbb{Z})$.
The authors recently introduced a new construction of a knot as an extended symmetric union of a knot with a single tangle region. In this paper, we generalize the construction to include multiple tangle regions. The constructed knot $K$ with a partial knot $\hat{K}$ and multiple tangle regions satisfies the following two properties: its Alexander polynomial is the product of the Alexander polynomials of the numerators of these tangles and the square of the Alexander polynomial of the partial knot $\hat{K}$, and there exists a surjective homomorphism from the knot group of $K$ to that of $\hat{K}$ which maps the longitude of $K$ to the trivial element.
For a link $L$ in the $3$-sphere, the $\pi$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group $G(L)$ of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$ fitting into a certain commutative diagram, we define a relation $L\succeq L'$ and explore the relationships between the two links. Specifically, we prove that if $L\succeq L'$ and $L$ is a Montesinos link with $r$ rational tangles $(r\geq 3)$, then $L'$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L'$ satisfying $L\succeq L'$. In contrast, if $L$ has determinant zero, then $L\succeq L'$ for every $2$-bridge link $L'$. Our main applications concern symmetric unions of knots. In particular, we provide a criterion showing that a given knot does not admit a symmetric union presentation.
For prime knots $K_1$ and $K_2$, we write $K_1 \geq K_2$ if there is an epimorphism from the knot group of $K_1$ to that of $K_2$ which preserves the meridian. We construct a family of pairs of knots with $K_1 \geq K_2$ such that an epimorphism maps the longitude of $K_1$ to the trivial element. This construction is regarded as an extension of a symmetric union with a single full twisted region. In particular, it extends a property of the Alexander polynomial of a symmetric union. We also exhibit that all but two of the knots up to ten crossings in the list of Kitano-Suzuki, which have an epimorphism mapping the longitude to the trivial element, arise from this construction.
In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union.
The Chern-Simons perturbation theory gives an invariant $d(M,ρ)$ for a pair of a closed oriented 3-manifold $M$ and a representation $ρ$ of the fundamental group. We generalize $d(M,ρ)$ for compact oriented 3-manifolds with torus boundaries. Further we give a gluing formula for $d(M,ρ)$.
We describe low dimensional homology groups of $\mathrm{Diff}^δ_+S^1$ in terms of Haefliger's classifying space $B\overlineΓ_1$ by applying a theorem of Thurston. Then we consider the question whether some power of the rational Euler class vanishes for real analytic flat $S^1$-bundles. We show that if it occurs, then the homology group of $\mathrm{Diff}_+^{ω,δ} S^1$ should contain two kinds of many torsion classes which vanish in $\mathrm{Diff}^δ_+S^1$. This is an informal note on our discussions about the above question.
For a 3-manifold $M$ and an acyclic $\mathit{SL}(2,\mathbb{C})$-representation $ρ$ of its fundamental group, the $\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion $τ_ρ(M) \in \mathbb{C}^\times$ is defined. If there are only finitely many conjugacy classes of irreducible representations, then the Reidemeister torsions are known to be algebraic numbers. Furthermore, we prove that the Reidemeister torsions are not only algebraic numbers but also algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3-manifolds. Also, for a knot exterior $E(K)$, we discuss the behavior of $τ_ρ(E(K))$ when the restriction of $ρ$ to the boundary torus is fixed.
The set $\mathit{RT}(M)$ of values of the $\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion of a 3-manifold $M$ can be both finite and infinite. We prove that $\mathit{RT}(M)$ is a finite set if $M$ is the splice of two certain knots in the 3-sphere. The proof is based on an observation on the character varieties and $A$-polynomials of knots.
In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.
We show some computations on representations of the fundamental group in SL(2;C) and Reidemeister torsion for a homology 3-sphere obtained by Dehn surgery along the figure-eight knot. This is the second version. We recorrected several errors in the first version.
In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.
We classify SL(2;C)-representations of a Brieskorn homology 3-sphere. We show any irreducible representation into SL(2;C) is conjugate to that into either SU(2) or SL(2;R). We also give a construction of SL(2;R)-representations for a Brieskorn homology 3-sphere from PSL(2;R)-representations of the base orbifold fundamental group.
This article is based on the lectures in the Winter Braids V (Pau, Feb. 2015). Main puposel of this is to explain how to compute twisted Alexander polynomials for non-experts.
Let $M_n$ be a homology 3-sphere obtained by $\frac1n$-Dehn surgery along a $(p,q)$-torus knot. We consider a polynomial $σ_{(p,q,n)}(t)$ whose zeros are the inverses of the Reideimeister torsion of $M_n$ for $\mathit{SL}(2;\mathbb{C})$-irreducible representations. We give an explicit formula of this polynomial by using Tchebychev polynomials of the first kind. Further we also give a 3-term relations of these polynomials.
Let K be a (2p,q)-torus knot and M_n is a 3-manifold obtained by 1/n-Dehn surgery along K. We consider a polynomial whose zeros are the inverses of the Reideimeister torsion of M_n for SL(2;C)-irreducible representations. Johnson gave a formula for the case of the (2,3)-torus knot under some modification and normalization. We generalize this formula by using Tchebychev polynomials.
Let M be a closed 3-manifold obtained by Dehn surgery along the figure-eight knot. We give a formula of the Reisdemeisiter torsion of M for any irreducible SL(2;C)-representation. It is described as a rational expression of the trace of the image of the meridian.