Tulin Altunoz, Mehmetcik Pamuk, Oguz Yildiz
We showed that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus $g\geq13$ can be generated by two involutions and an element of order $g$ or $g-1$ depending on whether $g$ is odd or even respectively.
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We compute the Reidemeister torsion of the complement of a twist knot in $S^3$ and that of the 3-manifold obtained by a Dehn surgery on a twist knot.
We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expansion of the colored Jones polynomial.
We explain the effect of applying a full Lutz twist along a pre-Lagrangian torus in a contact 3-manifold, on the contact invariant in Heegaard Floer homology with twisted coefficients.
This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.
Let $N_g$ be the non-orientable surface with genus $g$, $\text{MCG}(N_g)$ be the mapping class group of $N_g$, $\mathcal{T}(N_g)$ be the index 2 subgroup generated by all Dehn twists of $\text{MCG}(N_g)$. We prove that for odd genus, $\mathcal{T}(N_g)$ can be generated by three elements of finite orders.
For a compact oriented 3-manifold with torus boundary the adjoint Reidemeister torsion is defined as a function on the $\mathrm{SL}_2(\mathbb{C})$-character variety depending on a choice of a boundary curve. Under reasonable assumptions, it is conjectured that the adjoint torsion satisfies a certain type of vanishing identities. In this paper, we prove that the conjecture holds for all hyperbolic twist knot exteriors by using Jacobi's residue theorem.
We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL(2;C)-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coefficients in the higher dimensional Reidemeister torsion explicitly.
Let $H^n$ be the hyperbolic n-space with $n\geq 2$. Suppose that $Γ<Isom H^n$ is a discrete, torsion free subgroup and $a$ is a point in the domain of discontinuity $Ω(Γ)$. Let $p$ be the projection map from $H^n$ to the quotient manifold $M=H^n/Γ$. In this paper we prove that there exists an open neighborhood $U$ of $a$ in $H^n\cupΩ(Γ)$ such that $p$ is an isometry on $U\cap H^n$.
Generalized torsion elements in the fundamental groups of 3-manifolds obtained by the 0-surgeries along some double twist knots
该研究针对沿双扭结做0-手术得到的3-流形，构造其基本群中广义挠元的候选元并验证存在性，借助JSJ分解完成证明，同时得出该类3-流形基本群可双序等价于无广义挠元的结论，还列举了相关实例。
For a nonorientable surface, the twist subgroup is an index 2 subgroup of the mapping class group. It is generated by Dehn twists about two-sided simple closed curves. In this paper, we study involution generators of the twist subgroup. We give generating sets of involutions with the smallest number of elements our methods allow.
We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.
Fibering of double twist knots via the adjoint hyperbolic torsion polynomial
本文针对S³中双曲纽结的伴随双曲挠率多项式，研究双参数双扭结族的该多项式，利用代数整数证明其可确定该纽结族的亏格与纤维化性质。
We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.
We study knots of order 2 in the grope filtration $\{\G_h\}$ and the solvable filtration $\{\F_h\}$ of the knot concordance group. We show that, for any integer $n\ge4$, there are knots generating a $\Z_2^\infty$ subgroup of $\G_n/\G_{n.5}$. Considering the solvable filtration, our knots generate a $\Z_2^\infty$ subgroup of $\F_n/\F_{n.5}$ $(n\ge2)$ distinct from the subgroup generated by the previously known 2-torsion knots of Cochran, Harvey, and Leidy. We also present a result on the 2-torsion part in the Cochran, Harvey, and Leidy's primary decomposition of the solvable filtration.
It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim for the knot $5_2$, which is the $(-2)$-twist knot, by Naylor and Rolfsen.
Let $N_{g}$ denote the closed non-orientable surface of genus $g$ and let ${\mathcal M} _g$ denote the mapping class group of $N_{g}$. Let ${\mathcal T} _g$ denote the twist subgroup of ${\mathcal M} _g$ which is the subgroup of ${\mathcal M} _g$ is generated by all Dehn twists. In this thesis, we proved that ${\mathcal T} _g$ is generated by six involutions for $g \geq 16$ or $g = 14$.
We show that the normal closure of any periodic element of the mapping class group of a non-orientable surface whose order is greater than 2 contains the commutator subgroup, which for $g\geq 7$ is equal to the twist subgroup, and provide necessary and sufficient conditions for the normal closures of involutions to contain the twist subgroup. Finally, we provide a criterion for a periodic element to normally generate $\mathcal{M}(N_g)$ and give examples.
In this note we give presentations of all finite subgroups of the mapping class group of a closed surface of genus 2 by the Humphries generators up to conjugacy.
Let T(N) be the subgroup of the mapping class group of a nonorientable surface N (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for T(N). As an application we compute the first homology group (abelianization) of T(N).
In this paper we prove that for suitable sequences of congruence subgroups of Bianchi groups, including the standard exhaustive sequences of a congruence subgroup, and even symmetric powers of the standard representation of Sl_2(C) the size of the torsion part in the first homology grows exponentially. This extends results of Bergeron and Venkatesh to a case of non-uniform lattices.
On twisted torsion of compact $3$-manifolds
本文研究具有非空边界的非球形紧致3-流形与另一3-流形去掉3-胞腔后的流形，建立扭曲Reidemeister挠率乘法性质与连通和运算的关系，以理解其拓扑与几何。
We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn twists is one. The lower bounds is obtained from an argument of Hirose [5].
Integrality of genus-$g$ indices with adjoint Reidemeister torsions of twist knots
研究扭结的伴随Reidemeister挠率之和，证明对扭结和子午线该和是整数，并给出生成函数的具体例子。
Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture made in \cite{BV}, lattices in $G$ should have 'little' --- in the very weak sense of 'subexponential in the co-volume' --- torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the \emph{square root} of the volume. This is deduced from a general theorem that compares twisted and untwisted $L^…
On the non-triviality of the torsion subgroup of the abelianized Johnson kernel
本文通过二维方法证明了abelized Johnson核的torsion子群非平凡，并给出了其基数的下界。
We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup generated by three uniform powers of Dehn twists can be either the whole mapping class group, a direct product of a free group of rank two with the cyclic group of order two, or a free group of rank at most three. We characterize subgroups generated by a collection of (respectively squares of) Dehn twists, if there is a pair among so that …
We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove that the level 2 twist subgroup is normally generated in the mapping class group by a crosscap pushing map along a non-separating two-sided simple loop for genus $g\geq 5$ and $g=3$. As an application, we calculate the first homology group of the level 2 twist subgroup for genus $…
Residual torsion-free nilpotence has proven to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of parafree groups. This paper proves Mayland's assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $\mathbb{Z}$. We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link gro…
In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a (2, 2q+1)-torus knot K, we demonstrate that there are infinitely many unknots c such that p-twisting K about c yields a twist family, which consists of hyperbolic knots with generalized torsion whenever |p| > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus…
A finite presentation for the subgroup of the mapping class group of a compact non-orientable surface generated by all Dehn twists was given by Stukow. In this paper, we give an infinite presentation for this group, mainly using the presentation given by Stukow and Birman exact sequences on mapping class groups of non-orientable surfaces.
Generating the twist subgroup of the level $2$ mapping class group of a non-orientable closed surface
本文为不可定向闭曲面的映射类群中由Dehn扭转生成且在Z2同调上平凡作用的子群，构造了两个小生成集。
Let $G$ be a group hyperbolic relative to a finite collection of subgroups $\mathcal P$. Let $\mathcal F$ be the family of subgroups consisting of all the conjugates of subgroups in $\mathcal P$, all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{\mathcal F} G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.
Mod $p$ and torsion homology growth in nonpositive curvature
本文通过计算直角 Artin 群剩余序列的模 p 同调增长并构造闭局部 CAT(0) 流形，证明了同调挠元可呈指数增长，从而揭示 Singer 猜想与 Lück 猜想至少有一个不成立。
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