On topologically minimal surfaces of high genus

关于四维流形中的最小亏格问题

On the minimal genus problem in four-manifolds

本文研究某些光滑四维流形中的最小亏格问题，旨在刻画其中嵌入曲面的最小亏格。

A note on the concordance $\mathbb{Z}$-genus

We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden-Livingston-Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.

Knots with many minimal genus Seifert surfaces

Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.

极小亏格Seifert面上的素theta曲线

Prime theta-curves on minimal genus Seifert surfaces

研究极小亏格Seifert面上素结与本质弧的结合性质，证明其为素theta曲线。

Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings

We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.

Normalizing Topologically Minimal Surfaces III: Bounded Combinatorics

We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a particular normal form with respect to any triangulation.

柄数并非总能由最小亏格的Seifert曲面实现

Handle number is not always realized by a minimal genus Seifert surface

构造亏格一纽结，其柄数仅由非最小亏格Seifert曲面实现，反驳Seifert亏格等于Morse-Novikov亏格的猜想，并证明差异可任意大。

On the existence of high index topologically minimal surfaces

The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.

最小亏格问题

The Minimal Genus Problem

本文综述了自Lawson调查以来最小亏格问题的研究进展，涵盖关键成果与方向。

Lower bounds on virtual crossing number and minimal surface genus

We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.

Minimal genus in circle bundles over 3-manifolds

An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the self-intersection of the class and the Thurston norm of the underlying 3-manifold.

Minimal genus problem for $T^{2}$-bundles over surfaces

For any positive integer $g$, we completely determine the minimal genus function for $Σ_{g}\times T^{2}$. We show that the lower bound given by the adjunction inequality is not sharp for some class in $H_{2}(Σ_{g}\times T^{2})$. However, we construct a suitable embedded surface for each class and we have exact values of minimal genus functions.

Topological minimal genus and $L^2$-signatures

We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov $ρ$-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples with arbitrarily large slice genus for which our lower bound is optimal but all previously known invariants vanish.

Minimally intersecting filling pairs on the punctured surface of genus two

In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for all surfaces completing the work of Aougab-Huang and Aougab-Taylor.

Knots with unique minimal genus Seifert surface and depth of knots

We describe a procedure for creating infinite families of hyperbolic knots having unique minimal genus Seifert surface. A large subset of these knots have the further property that the surface cannot be the sole compact leaf of a depth one foliation of the knot exterior.

Minimal genus four manifolds

In 2018, M. Chu and S. Tillmann gave a lower bound for the trisection genus of a closed 4-manifold in terms of the Euler characteristic of $M$ and the rank of its fundamental group. We show that given a group $G$, there exist a 4-manifold $M$ with fundamental group $G$ with trisection genus achieving Chu-Tillmann's lower bound.

小亏格曲面的反向定向最小拉伸因子问题

The orientation-reversing minimal stretch factor problem in small genus

该研究针对亏格2、3、4的闭可定向曲面，确定了反向定向伪Anosov映射类的最小拉伸因子，为相关几何问题提供了关键结果。

同一4-流形的非微分同胚极小亏格相对三分解

Non-diffeomorphic minimal genus relative trisections of the same 4-manifold

本文证明存在一个带边4-流形，它允许两个非微分同胚且具有相同类型的极小亏格相对三分解，并提出了一种从相对三分解图生成闭4-流形三分解图的简单操作。

Minimal crossing number implies minimal supporting genus

A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and ascending numbers of a classical link do not decrease when it is regarded as a virtual link. This extends corresponding results in the case of virtual knots due to Manturov and Chernov.

软木塞的最小亏格相对三截面

Minimal genus relative trisections of corks

本文证明了Akbulut软木塞的三截面亏格为3，构造了无穷多个此类软木塞，并给出了带边界4-流形三截面亏格的下界，以及异种对的低亏格相对三截面图。

Minimal genus in 4-manifolds with a free circle action

Let N be a closed irreducible 3-manifold and assume N is not a graph manifold. We improve for all but finitely many S^1-bundles M over N the adjunction inequality for the minimal complexity of embedded surfaces. This allows us to completely determine the minimal complexity of embedded surfaces in all but finitely many S^1-bundles over a large class of 3-manifolds.

拓扑无结环面的例子

Examples of topologically unknotted tori

本文证明了 $S^4$ 中具有群 $\mathbb{Z}$ 的特定光滑环面在拓扑上是无结的，为相关猜想提供了证据，并探讨了其光滑无结性及分支双重覆盖的奇异性质。

交替链接的最小亏格Seifert面

Minimal genus Seifert surfaces for alternating links

本文通过完整证明，明确描述了非分裂、素、特殊、交替链接的Kakimizu复数。

Bridge surfaces with the topological minimality preserved by perturbation

We show that except for $n = 2$ if a bridge surface for a knot is an index $n$ topologically minimal surface, then after a perturbation it is still topologically minimal with index at most $n+1$.

Plane algebraic curves of arbitrary genus via Heegaard Floer homology

Suppose C is a singular curve in CP^2 and it is topologically an embedded surface of genus g; such curves are called cuspidal. The singularities of C are cones on knots K_i. We apply Heegaard Floer theory to find new constraints on the sets of knots {K_i} that can arise as the links of singularities of cuspidal curves. We combine algebro-geometric constraints with ours to solve the existence problem for curves with genus one, d>33, that possess exactly one singularity which has exactly one Puiseux pair (p;q). The realized triples (p,d,q) are expressed as successive even terms in the Fibonacci …

Genus-minimal crystallizations of PL 4-manifolds

For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every 3-manifold equals its Heegaard genus. For every closed connected PL $4$-manifold $M$, it is known that its regular genus $\mathcal G(M)$ is at least $2 χ(M) + 5m -4$, where $m$ is the rank of the fundamental group of $M$. In this article, we introduce the concept of "weak semi-s…

椭圆曲面 $E(n)$ 通过柄图的最小亏格三分割图

Minimal genus trisection diagrams of the elliptic surfaces $E(n)$ via handle diagrams

本文通过柄图明确构造了椭圆曲面 $E(n)$ 的 $(12n-2,0)$-三分割图，该三分割图具有最小亏格。

Minimal genus for 4-manifolds with $b^+=1$

We derive an adjunction inequality for any smooth, closed, connected, oriented 4-manifold $X$ with $b^+=1$. This inequality depends only on the cohomology algebra and generalizes the inequality of Strle in the case of $b_1=0$. We demonstrate that the inequality is especially powerful when $2\tilde χ+3σ\geq 0$, where $\tilde χ$ is the modified Euler number taking account of the cup product on $H^1$.

Normalizing Topologically Minimal Surfaces II: Disks

We show that a topologically minimal disk in a tetrahedron with index $n$ is either a normal triangle, a normal quadrilateral, or a normal helicoid with boundary length 4(n+1). This mirrors geometric results of Colding and Minicozzi.

Genus bounds for min-max minimal surfaces

We prove optimal genus bounds for minimal surfaces arising from the min-max construction of Simon-Smith. This confirms a conjecture made by Pitts-Rubinstein in 1986.

Normalizing Topologically Minimal Surfaces I: Global to Local Index

We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of Kneser and Haken, and more recent theorems of Rubinstein and Stocking, and is the first step in a program to show that every topologically minimal surface has a normal form with respect to any triangulation.

The existence of an indecomposable minimal genus two Lefschetz fibration

It was shown by Usher that any fiber sum of Lefschetz fibrations over $S^2$ is minimal, which was conjectured by Stipsicz. We prove that the converse does not hold by showing that there exists an indecomposable minimal genus-2 Lefschetz fibration.

The minimal genus problem for elliptic surfaces

We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second homology. In the case we consider we get the minimal possible genus allowed by the adjunction inequality.

解开亏格为4和5的不可定向曲面

Unknotting Nonorientable Surfaces of Genus 4 and 5

本文利用Sage计算修正手术障碍，证明D^4中纽结群为Z_2、亏格4或5且边界行列式绝对值为1的不可定向局部平坦曲面相对边界环境同痕，从而对应S^4闭曲面拓扑不打结，并指出方法在亏格6、7时失效。

Symplectic genus, minimal genus and diffeomorphisms

In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least -1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine…

Minimal number of singular fibers in a nonorientable Lefschetz fibration

We show that there exists an admissible nonorientable genus $g$ Lefschetz fibration with only one singular fiber over a closed orientable surface of genus $h$ if and only if $g \geq 4$ and $h \geq 1$.

The Heegaard genus of bundles over S^1

This paper explores connections between Heegaard genus, minimal surfaces, and pseudo-Anosov monodromies. Fixing a pseudo-Anosov map phi and an integer n, let M_n be the 3-manifold fibered over S^1 with monodromy phi^n. JH Rubinstein showed that for a large enough n every minimal surface of genus at most h in M_n is homotopic into a fiber; as a consequence Rubinstein concludes that every Heegaard surface of genus at most h for M_n is standard, that is, obtained by tubing together two fibers. We prove this result and also discuss related results of Lackenby and Souto.

大映射类群的小拓扑生成集

Small Sets of Topological Generators for Big Mapping Class Groups

本文研究无限型曲面映射类群的拓扑生成元最小个数，证明对于具有无限亏格和n个端点的曲面S(n)，当n≥8时可由3个元素拓扑生成，n≥3时可由4个元素拓扑生成，并给出Loch Ness Monster曲面S(1)和Jacob's Ladder曲面S(2)的生成集。

The four genus of a link, Levine-Tristram signatures and satellites

We give a new proof that the Levine-Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Friedl and Teichner to show that the 4-genus of a link does not increase under infection by a string link, which is a generalised satellite construction, provided that certain homotopy triviality conditions hold on the axis curves, and that enough Milnor's invariants of the infection…

Small genus knots in lens spaces have small bridge number

In a lens space X of order r a knot K representing an element of the fundamental group pi_1 X = Z/rZ of order s <= r contains a connected orientable surface S properly embedded in its exterior X-N(K) such that the boundary of S intersects the meridian of K minimally s times. Assume S has just one boundary component. Let g be the minimal genus of such surfaces for K, and assume s >= 4g-1. Then with respect to the genus one Heegaard splitting of X, K has bridge number at most 1.

结在最小亏格Seifert面上的托管与友谊

Amicable Knots on Minimal Genus Seifert Surfaces

研究结在最小亏格Seifert面上的托管关系及其友谊，揭示了结的托管现象和友谊现象的框架，指出不存在普遍托管的单个结。

A genus 4 origami with minimal hitting time and an intersection property

In a minimal flow, the hitting time is the exponent of the power law, as r goes to zero, for the time needed by orbits to become r-dense. We show that on the so-called Ornithorynque origami the hitting time of the flow in an irrational slope equals the diophantine type of the slope. We give a general criterion for such equality. In general, for genus at least two, hitting time is strictly bigger than diophantine type.

The Minimal Genus of Homology Classes in a Finite 2-Complex

We study surface representatives of homology classes of finite complexes which minimize certain complexity measures, including its genus and Euler characteristic. Our main result is that up to surgery at nullhomotopic curves minimizers are homotopic to cellwise coverings to the 2-skeleton. From this we conclude that the minimizing problem is in general algorithmically undecidable, but can be solved for 2-dimensional CAT(-1)-complexes.

On minimal higher genus fillings

In this article, we prove that if $(M,g)$ is a genus $G$ orientable surface with a single boundary component $S^1$, and if $(D,g_0)$ is a disc such that interior points are connected by unique geodesics and $$d_{(D,g_0)}(x,y) \geq d_{(M,g)}(x,y)$$ for all $x,y \in \partial M = \partial D$, then $$(1 + \frac{2 G}π) \textrm{Area}(M,g) \geq \textrm{Area}(D,g_0).$$

A note on minimal finite quotients of mapping class groups

We prove that the minimal nontrivial finite quotient group of the mapping class group M_g of a closed orientable surface of genus g is the symplectic group PSp(2g,Z_2), for g = 3 and 4 (this might remain true, however, for arbitrary genus g > 2). We discuss also some results for arbitrary genus g.

正Ricci曲率三维球面中任意亏格的极小曲面

Minimal surfaces with arbitrary genus in 3-spheres of positive Ricci curvature

本文研究正Ricci曲率三维球面中的极小曲面，证明对任意亏格g，存在面积不超过第一Simon-Smith宽度两倍的嵌入极小曲面。

直角 Artin 群的最小亏格问题

The minimal genus problem for right angled Artin groups

本文研究 RAAG 二阶同调类的最小亏格，给出基于卡积矩阵秩的下界，证明其在多类图中由环面不交并实现，并刻画单环面可表示类及反例。

无限亏格的把手体与肖特基群

Handlebodies of Infinite Genus and Schottky Groups

本文通过定义展开对无限亏格把手体、无限生成肖特基群及单值化问题的研究，证明相关曲面和把手体的拓扑单值化、特定黎曼曲面的拟共形单值化及唯一性，并给出反例。

Genus bounds for minimal surfaces arising from min-max constructions

In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a famous result of Meeks, Simon and Yau on the convergence of minimizing sequences of isotopic surfaces. This result is proved in the second part of the paper.

极小曲面与给定亏格的极小理论

Min-max theory and minimal surfaces with prescribed genus

本文通过极小理论构造具有给定亏格的极小曲面，利用正各向同性曲率3流形中的拓扑最优族方法，为构造多个具有给定亏格的极小曲面提供关键工具。

Minimal number of singular fibers in a Lefschetz fibration

There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g>2 and h>1. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist on any closed surface is equal to a single commutator.

Algebraic and topological properties of big mapping class groups

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$, denoted $\mathrm{PMap}(S)$, detects the homeomorphism type of $S$. As a corollary, every automorphism of $\mathrm{PMap}(S)$ is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that $\mathrm{PMap}(S)$ is residually finite if and only if $S$ has finite genus, demonstra…

The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups

We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of…

亏格2的Teichmüller曲线的极小双曲面积

Minimal Hyperbolic Area of Teichmuller Curves in Genus Two

该研究确定亏格2的Teichmüller曲线的极小双曲面积为3π/5，明确其取得条件，通过结合轨形分类与仿射下降构造，排除非平方零点模式完成证明。