Jesse Johnson
We show that there are hyperbolic tunnel-number one knots with arbitrarily high bridge number and that "most" tunnel-number one knots are not one-bridge with respect to an unknotted torus. The proof relies on a connection between bridge number and a certain distance in the curve complex of a genus-two surface.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study smooth isotopy classes of complex curves in complex surfaces from the perspective of the theory of bridge trisections, with a special focus on curves in $\mathbb{CP}^2$ and $\mathbb{CP}^1\times\mathbb{CP}^1$. We are especially interested in bridge trisections and trisections that are as simple as possible, which we call "efficient". We show that any curve in $\mathbb{CP}^2$ or $\mathbb{CP}^1\times\mathbb{CP}^1$ admits an efficient bridge trisection. Because bridge trisections and trisections are nicely related via branched covering operations, we are able to give many examples of comp…
We study the twisted Alexander polynomial from the viewpoint of the SL(2,C)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2,C)-representations are all monic. In this paper, we show that the converse holds for 2-bridge knots. Furthermore we show that for a 2-bridge knot there exists a curve component in the SL(2,C)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexand…
We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [KP10] and [BKP14]. We also give a sharp lower bound for the lexicographic degree of any knot, using real polynomial curves properties.
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.
We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yielding an invariant of the manifold-knot pair. We also give evidence toward the relationship between the pants distance of a bridge splitting and the hyperbolic volume of the exterior of a knot.
The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal …
Connected sum and trivalent vertex sum are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the tunnel number of a composite genus 2 spatial graph. For the tunnel number of a composite Brunnian $θ$-curve, our result implies that the tunnel number is at least the number of summands, as in the knot case. We also prove a version of a theorem of Morimoto for knots: the tunnel number of a composite m-small genus 2 spatial graph is at least the sum of the tunnel numb…
Given any closed, connected, orientable $3$--manifold and integers $g\geq g(M), D > 0$, we show the existence of knots in $M$ whose genus $g$ bridge number is greater than $D$. These knots lie in a page of an open book decomposition of $M$, and the proof proceeds by examining the action of the map induced by the monodromy on the arc and curve complex of a page. A corollary is that there are Berge knots of arbitrarily large genus one bridge number.
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。
If a knot K in a closed, orientable 3-manifold M has a bridge surface T with distance at least 3 in the curve complex of T - K, then the genus of any essential surface in its exterior with non-empty, non-meridional boundary gives rise to an upper bound for the bridge number of K with respect to T. In particular, a nontrivial, aspherical, and atoroidal knot K with such a bridge surface has its bridge number bounded by 5 if K has a non-trivial reducing surgery; 6 if K has a non-trivial toroidal surgery; and 4g + 2 if K is null-homologous and has Seifert genus g.
We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in $\mathbb{CP}^2$ are closely related: We prove that a positive genus surface $\mathcal{K}$ in $\mathbb{CP}^2$ that minimizes genus in its homology class is isotopic to a complex curve $\mathcal{C}_d$ if and only if $\mathcal{K}$ admits a hexagonal lattice diagram, a special type of shadow diagram in which arcs meet only at bridge points and tile the central surface of the standard trisection of $\mathbb{CP}^2$ by hexagons. There are eight families of these diagrams, two of which represent surfa…
Let $K= K(w,b,t)$ be a 1-bridge braid in a solid torus $V$, and let $γ$ be a $(p,q)$ curve on the torus $T = \partial V$ of the exterior $M_K$ of $K$. It will be shown that Dehn filling on $T$ along $γ$ produces a solid torus if and only if $p$ and $q$ satisfy one of four conditions determined by the parameters $(w,b,t)$ of the knot $K$. This solves the classification problem raised by Menasco and Zhang for such Dehn fillings.
We show that essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bridge number of a tangle product is at least the sum of the bridge numbers of the two factor links up to a constant error.
We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the covering for the Matveev complexity of the covering manifold. The lower estimate uses the hyperbolic volume and results of Cao-Meyerhoff and Gueritaud-Futer (who recently improved previous work of Lackenby), while the upper estimate is based on an explicit triangulation, which also allows us to give a bound on the Delzant T-invariant of the fundamental group …
We show that the complex of weak reducing disks for the unknot in $3$-bridge position is contractible.
We study the degree of polynomial representations of knots. We give the lexicographic degree of all two-bridge knots with 11 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane curves, and in particular to use the braid theoretical method developed by Orevkov.
We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.
J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudo-Anosov maps have minimal bridge presentations which are thin.
In a 3-manifold M, let K be a knot and R be an annulus which meets K transversely. We define the notion of the pair (R,K) being caught by a surface Q in the exterior of the link given by K and the boundary curves of R. For a caught pair (R,K), we consider the knot K^n gotten by twisting K n times along R and give a lower bound on the bridge number of K^n with respect to Heegaard splittings of M -- as a function of n, the genus of the splitting, and the catching surface Q. As a result, the bridge number of K^n tends to infinity with n. In application, we look at a family of knots K^n found by T…
Extending Azumaya algebras associated to arithmetic 2-bridge links
本文将Chinburg-Reid-Stover关于双曲3-流形基本群特征标簇典范曲线的结果部分推广到某些双曲链环补空间，证明相关典范四元数代数无法拓展为典范曲面上的阿祖梅亚代数。
We study multisections of embedded surfaces in 4-manifolds admitting effective torus actions. We show that a simply-connected 4-manifold admits a genus one multisection if and only if it admits an effective torus action. Orlik and Raymond showed that these 4-manifolds are precisely the connected sums of copies of $\mathbb{CP}^2$, $\overline{\mathbb{CP}^2}$, and $S^2\times S^2$. Therefore, embedded surfaces in these 4-manifolds can be encoded diagrammatically on a genus one surface. Our main result is that every smooth, complex curve in $\mathbb{CP}^1\times\mathbb{CP}^1$ can be put in efficient…
Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the knot. We provide an efficient formula for the colored Jones function of the simplest hyperbolic non-2-bridge knot, and using this formula, we provide numerical evidence for the Hyperbolic Volume Conjecture for the simplest hyperbolic non-2-bridge knot.
We give a variation of McShane's identity, which describes the cusp shape of a hyperbolic 2-bridge link in terms of the complex translation lengths of simple loops on the bridge sphere. We also explicitly determine the set of end invariants of $SL(2,\mathbb{C})$-characters of the once-punctured torus corresponding to the holonomy representations of the complete hyperbolic structures of 2-bridge link complements.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。