Stable equivalence of bridge positions of a handlebody-knot

The Gordian distance of handlebody-knots and Alexander biquandle colorings

We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any positive integer $n$.

Knots in handlebodies with nontrivial handlebody surgeries

We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge.

JSJ decomposition for handlebody-knots

The paper applies the JSJ decomposition and Koda-Ozawa's annulus classification to analyze the annulus configuration in a handlebody-knot exterior. We introduce the notion of the annulus diagram, to pack the configuration into a labeled graph, and classify genus two handlebody-knots in terms of their annulus diagrams. Applications to handlebody-knot symmetries are discussed; methods to produce handlebody-knots with various types of annulus diagrams are also presented.

Unknotting annuli and handlebody-knot symmetry

By Thurston's hyperbolization theorem, irreducible handlebody-knots are classified into three classes: hyperbolic, toroidal, and atoroidal cylindrical. It is known that a non-trivial handlebody-knot of genus two has a finite symmetry group if and only if it is atoroidal. The paper investigates the topology of cylindrical handlebody-knots of genus two that admit an unknotting annulus; we show that the symmetry group is trivial if the unknotting annulus is unique and of type $2$.

Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots

We consider the group of isotopy classes of automorphisms of the 3-sphere that preserve a spatial graph or a handlebody-knot embedded in it. We prove that the group is finitely presented for an arbitrary spatial graph or a reducible handlebody-knot of genus two. We also prove that the groups for "most" irreducible genus two handlebody-knots are finite.

Invariants of Handlebody-Knots via Yokota's Invariants

We construct quantum $\mathcal{U}_q(\mathfrak{sl}_{\,2})$ type invariants for handlebody-knots in the 3-sphere $S^3$. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We give a table of calculations of our invariants for genus 2 handlebody-knots up to six crossings. We also show our invariants are identified with special cases of the Witten-Reshetikhin-Turaev invariants.

On inequalities between unknotting numbers and crossing numbers of spatial embeddings of trivializable graphs and handlebody-knots

We study relations between unknotting number and crossing number of a spatial embedding of a handcuff-graph and a theta curve. It is well known that for any non-trivial knot $K$ twice the unknotting number of $K$ is less than or equal to the crossing number of $K$ minus one. We show that this is extended to handlebody-knots. We also characterize the handlebody-knots which satisfy the equality.

亏格为二的柄结的3分解

3-decompositions of genus two handlebody-knots

本文研究了亏格为二的柄结的3分解性质，通过分类τ-和ρ-结节，确定了不同类别下的本质圆环数量，并识别了双曲柄结的双曲性。

On calculations of the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links

There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on…

Rigidity and symmetry of cylindrical handlebody-knots

A recent result of Funayoshi-Koda shows that a handlebody-knot of genus two has a finite symmetry group if and only if it is hyperbolic -- the exterior admits a hyperbolic structure with totally geodesic boundary -- or irreducible, atoroidal, cylindrical -- the exterior contains no essential disks or tori but contains an essential annulus. Based on the Koda-Ozawa classification theorem, essential annuli in an irreducible, atoroidal handlebody-knots of genus two are classified into four classes: type $2$, type $3$-$2$, type $3$-$3$ and type $4$-$1$. We show that under mild condition most genus …

通过缠结替换得到的柄体纽结

Handlebody-knots obtained by tangle replacement

研究通过缠结替换得到的柄体纽结，对在有四个交叉的素空间手铐图上进行缠结替换所得的柄体纽结分类并确定其对称性，以解决小交叉柄体纽结难以区分及手性问题。

一个具有七个交叉的亏格为二的柄体纽结表

A table of genus two handlebody-knots with seven crossings

研究具有七个交叉的亏格为二的柄体纽结，通过枚举扩展石井等人的表格。

关于打结柄体的唯一分解

On unique decomposition of knotted handlebodies

本文研究了3-球中打结柄体沿分解2-球因式分解的唯一性问题，在特定条件下证明了唯一性，并据此确定了$6_{10}$的手性，构造了具有同胚外部的双曲柄体结无穷族。

Numerical irreducibility criteria for handlebody links

In this paper we define a set of numerical criteria for a handlebody link to be irreducible. It provides an effective, easy-to-implement method to determine the irreducibility of handlebody links; particularly, it recognizes the irreducibility of all handlebody knots in the Ishii-Kishimoto-Moriuchi-Suzuki knot table and most handlebody links in the Bellettini-Paolini-Paolini-Wang link table.

Partially Multiplicative Biquandles and Handlebody-Knots

We introduce several algebraic structures related to handlebody-knots, including $G$-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in $Y$-oriented spatial trivalent graph diagrams representing $S^1$-oriented handlebody-knots to obtain computable invariants for handlebody-knots and handlebody-links. In the case of $G$-families of biquandles, we enhance the counting invariant using the group $G$ to obtain a polynomial invariant of handlebody-knots.

Handlebody-knot invariants derived from unimodular Hopf algebras

A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a uniform method to construct invariants for handlebody-links. We introduce the category $\mathcal{T}$ of handlebody-tangles and present it by generators and relations. The result tells us that every functor on $\mathcal{T}$ that gives rise to invariants is derived from what we call a quantum-commutative quantum-symmetric algebra in the target category. The example of such algebras of our main concern is finite-dimensional unimodular Hopf algebras. We investigate how those Hopf algebras give rise to handlebody-knot invari…

单纯复形中的距离估计及其在3-流形和柄体结上的应用

Estimating distances in simplicial complexes with applications to 3-manifolds and handlebody-knots

本文研究了低维流形相关单纯复形中的距离关系，获得了基于分量数和曲线复形距离的距离界，并利用Heegaard分解的分裂距离为闭3-流形和柄体结定义了在稳定化下收敛于非平凡极限的新不变量。

Twisted Virtual Bikeigebras and Twisted Virtual Handlebody-Knots

We generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form $Σ\times [0,1]$ for $Σ$ a compact closed 2-manifold up to stable equivalence. We introduce a related algebraic structure known as twisted virtual bikeigebras whose axioms are motivated by the twisted virtual handlebody-link Reidemeister moves. We use twisted virtual bikeigebras to define $X$-colorability for twisted virtual handlebody-links and define an integer-valued invariant $Φ_{X}^{\mathbb{Z}}$ of twisted virtual handlebody-links. We provid…

Refilling meridians in a genus 2 handlebody complement

Suppose a genus two handlebody is removed from a 3-manifold M and then a single meridian of the handlebody is restored. The result is a knot or link complement in M and it is natural to ask whether geometric properties of the link complement say something about the meridian that was restored. Here we consider what the relation must be between two not necessarily disjoint meridians so that restoring each of them gives a trivial knot or a split link.

虚拟图与柄体链环的桥数

Bridge numbers of virtual graphs and handlebody-links

本文研究图、虚拟图及柄体类的桥指标，构造虚拟ravel等例子，证明多种桥指标差异无界，并给出Suzuki曲线边连通和的精确指标。

On closed oriented surfaces in the 3-sphere

In this paper we study embeddings of oriented connected closed surfaces in $\mathbb S^3$. We define a complete invariant, the fundamental span, for such embeddings, generalizing the notion of the peripheral system of a knot group. From the fundamental span, several computable invariants are derived and employed to study handlebody knots, bi-knotted surfaces, and chirality of knots. These invariants are capable to distinguish inequivalent handlebody knots and bi-knotted surfaces with homeomorphic complements. Particularly, we obtain an alternative proof of the inequivalence of Ishii et al.'s ha…

Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto and Baker.

亏格二柄体外部中的本质环带

Essential annuli in genus two handlebody exteriors

本文综合 JSJ 图与本质环带的既有分类，刻画三维球面中亏格二无环面柄体外部的本质环带构型，并确定无穷多个不同痕环带族中各类环带的数量。

Tunnel-number-one knot exteriors in $S^3$ disjoint from proper power curves

As one of the background papers of the classification project of hyperbolic primitive/Seifert knots in $S^3$ whose complete list is given in [BK20], this paper classifies all possible R-R diagrams of two disjoint simple closed curves $R$ and $β$ lying in the boundary of a genus two handlebody $H$ up to equivalence such that $β$ is a proper power curve and a 2-handle addition $H[R]$ along $R$ embeds in $S^3$ so that $H[R]$ is the exterior of a tunnel-number-one knot. As a consequence, if $R$ is a nonseparating simple closed curve on the boundary of a genus two handlebody such that $H[R]$ embeds…

Knot theory in handlebodies

We consider oriented knots and links in a handlebody of genus $g$ through appropriate braid representatives in $S^3$, which are elements of the braid groups $B_{g,n}$. We prove a geometric version of the Markov theorem for braid equivalence in the handlebody, which is based on the $L$-moves. Using this we then prove two algebraic versions of the Markov theorem. The first one uses the $L$-moves. The second one uses the Markov moves and conjugation in the groups $B_{g,n}$. We show that not all conjugations correspond to isotopies.

Primitive, proper power, and Seifert curves in the boundary of a genus two handlebody

A simple closed curve $α$ in the boundary of a genus two handlebody $H$ is primitive if adding a 2-handle to $H$ along $α$ yields a solid torus. If adding a 2-handle to $H$ along $α$ yields a Seifert-fibered space and not a solid torus, the curve is called Seifert. If $α$ is disjoint from an essential separating disk in $H$, does not bound a disk in $H$, and is not primitive in $H$, then it is said to be proper power. As one of the background papers of the classification project of hyperbolic primitive/Seifert knots in $S^3$ whose complete list is given in [BK20], this paper classifies in term…

Boundary-reducing surgeries and bridge number

Let $M$ be a $3$--dimensional handlebody of genus $g$. This paper gives examples of hyperbolic knots in $M$ with arbitrarily large genus $g$ bridge number which admit Dehn surgeries which are boundary-reducible manifolds.

关于手柄体中交替结的注记

A note on alternating knots in handlebodies

本文研究手柄体中交替结的最小交叉数和旋转数，并推广Jones多项式到该设置中。

混合 Hilden 辫子群与柄体中的 plat 等价性

The mixed Hilden braid group and the plat equivalence in handlebodies

本文证明亏格 g 柄体中的任意纽结或链环均可表示为辫子的 plat 闭包，定义混合 Hilden 辫子群，并给出同一链环同痕类下混合 plat 的代数等价判据。

关于纽结柄体的对称性与外部问题

On symmetry and exterior problems of knotted handlebodies

研究3维球面中亏格2纽结柄体的对称性与外部问题，确定由非整数环面Dehn手术双曲纽结构造的柄体对称群并解决其外部问题，同时推广Lee-Lee族。

Bridge spectra of cables of 2-bridge knots

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Minimal bridge projections for 11-crossing prime knots

We give the bridge indices for 11-crossing prime knots and give a minimal bridge projection for each of these knots. The results on the indices may be easily summarized: all of these knots that are not rational knots or Montesinos knots have bridge index three.

Uniqueness of bridge surfaces for 2-bridge knots

Any 2-bridge knot in the 3-sphere has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.

The 2-bridge knots of up to 16 crossings

For any given number of crossings $c$, there exists a formula to determine the number of 2-bridge knots of $c$ crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of whether a given (prime) knot is a 2-bridge knot remains a nontrivial exercise, and we have no procedure to determine bridge numbers more generally. Herein, we identify the 2-bridge knots within the Hoste--Thistlethwaite--Weeks tables of prime knots of up to 16 crossings by an exhaustive search of a larger set of 2-bridge knots. As the unknot is the only knot wi…

Non-minimal bridge positions of torus knots are stabilized

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.

Genera of two-bridge knots and epimorphisms of their knot groups

Let $K,K'$ be two-bridge knots of genus $n,k$ respectively. We show the necessary and sufficient condition of $n$ in terms of $k$ that there exists an epimorphism from the knot group of $K$ onto that of $K'$.

Generating function on epimorphisms between $2$-bridge knot groups

We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking account into genus as well as crossing number. Next, we determine the number of epimorphisms between fibered $2$-bridge knot groups. Moreover, we discuss degree one maps and $2$-bridge knots uknotting number one.

The Kauffman Polynomials of 2-bridge Knots

The 2-bridge knots are a family of knots with bridge number 2. In this paper, we compute the Kauffman polynomials of 2-bridge knots using the Kauffman skein theory and linear algebra techniques. Our calculation can be easily carried out using Mathematica, Maple, Mathcad, etc.

Epimorphisms between $2$-bridge knot groups and their crossing numbers

Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K'$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K')$. Especially it is shown that $c(K)$ is greater than or equal to $3 c(K')$ and we estimate how many knot groups a $2$-bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot.

Bridge numbers and meridional ranks of knotted surfaces and welded knots

The Meridional Rank Conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the analogous conjecture for knotted surfaces in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order and which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, …

Twisted Alexander polynomials of 2-bridge knots

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

On 3-super bridge knots

It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure eight knot. In this paper, we prove that the $5_2$ knot and the $6_2$ knot are also 3-super bridge knots by providing a polynomial representation of these knots in degree $6.$ This also answers a question asked by Durfee and O'Shea in their paper on polynomial knots: is there any 5-crossing knot in degree 6?

关于双桥纽结的辫指数

On the braid index of a two-bridge knot

本文研究双桥纽结的辫指数，证明了纽结群间存在满同态时双桥纽结辫指数的不等式，并给出给定交叉数的所有双桥纽结的平均辫指数。

On minimality of two-bridge knots

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $\mathfrak{b}(p,q)$ is minimal where $q \leq 6$ or $p \leq 100$.

A note on Riley polynomials of $2$-bridge knots

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.

Closed incompressible surfaces of genus two in 3-bridge knot complements

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.

Crosscap numbers of 2-bridge knots

We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of them).

The disk complex and 2-bridge knots

We give an alternative proof of a result of Kobayashi and Saeki that every genus one $1$-bridge position of a non-trivial $2$-bridge knot is a stabilization.

Uniqueness of higher genus bridge surfaces for torus knots

We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.

Adjoint twisted Alexander polynomials of genus one two-bridge knots

We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.

Riley's Conjecture on SL(2,R) Representations of 2-Bridge Knots

We prove Riley's conjecture on the number of parabolic SL(2,R) representations of 2-bridge knot groups.

A knot with destabilized bridge spheres of arbitrarily high bridge number

We show that there exists an infinite family of knots, each of which has, for each integer k>=0, a destabilized (2k+5)-bridge sphere. We also show that, for each integer n>=4, there exists a knot with a destabilized 3-bridge sphere and a destabilized n-bridge sphere.

Bridge spectra of twisted torus knots

We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result, the bridge spectra of these knots have two gaps which can be chosen to be arbitrarily large, providing the first known examples of hyperbolic knots exhibiting this property. In addition, we show that there are Berge and Dean knots with arbitrarily large genus one bridge numbers, and as a result, we give solutions to problems of Eudave-Muñoz concerning tunnel number one knots.

Commensurability classes of 2-bridge knot complements

We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.