Handle number one links and generalized property R

On invariants of surfaces in the $3$-sphere

In this paper we study isotopy classes of closed connected orientable surfaces in the standard $3$-sphere. Such a surface splits the $3$-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a $2$-component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial $1$-handles can recover the original surface. Therefore, we can reduce the study of surfaces in the $3$-sphere to that of $2$-component handlebody-links up to stabilizations. Then, by using $G$-families of quandles…

Traces of links and simply connected 4-manifolds

We study the set $\widehat{\mathcal S}_M$ of framed smoothly slice links which lie on the boundary of the complement of a 1-handlebody in a closed, simply connected, smooth 4-manifold $M$. We show that $\widehat{\mathcal S}_M$ is well-defined and describe how it relates to exotic phenomena in dimension four. In particular, in the case when $X$ is smooth, with a handle decompositions with no 1-handles and homeomorphic to but not smoothly embeddable in $D^4$, our results tell us that $X$ is exotic if and only if there is a link $L\hookrightarrow S^3$ which is smoothly slice in $X$, but not in $D…

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。

通过A型Soergel双模构造带柄体中链环的HOMFLYPT同调

HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

本文利用经典辫群子群与奇异Soergel双模的范畴作用，将3-球中着色HOMFLYPT同调推广为亏格g带柄体中链环的三重分次不变量。

关于GST链接稳定等价的一些实验结果：针对广义性质R猜想

Some experimental results on stable equivalence of GST Links for the Generalized Property R Conjecture

研究通过算法验证部分GST链接的稳定handleslide平凡性，并展示许多链接的稳定handleslide等价性，为广义性质R猜想提供新见解。

同调广义Property R猜想是错误的

A homological generalized Property R conjecture is false

该研究驳斥了广义Property R猜想的进一步推广，证明存在2组成部分的链接通过手术得到连通和的流形，但不与split链接handleslide等价。

On converting a side-pairing to a handle decomposition

We give a method for obtaining a handle decomposition of an $n$-manifold if the manifold is given by isometric side-pairings of a polyhedron in $\en$, $\sn$ or $\hn$. Every cycle of $k$-faces on the polyhedron corresponds to an $(n-k)$-handle of the manifold. Two applications of the method are given. One helps recognize when a noncompact hyperbolic 3-manifold is a complement of a link in $S^3$ (and automatically produces the link diagram), the other shows that a topological $S^4$ described by the author in \cite{Ivansic3} is diffeomorphic to the standard differentiable $S^4$.

Fibered knots and Property 2R, II

A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that converts L into a 0-framed unlink. The Generalized Property R Conjecture is that all knots have Property nR for all n. The simplest plausible counterexample could be the square knot. Exploiting the remarkable symmetry of the square knot, we characterize all two-component links that contain it and which surger to S^1 x S^2 # S^1 x S^2. We argue that at least o…

A New property of quasi-alternating links

We show that the crossing number of any link that is known to be quasi-alternating is less than or equal to its determinant. Based on this, we conjecture that the crossing number of any quasi-alternating link is less than or equal to its determinant. Thus, if this conjecture is proved then it would give an easier obstruction for quasi-alternateness than the ones already known.

Intrinsically linked graphs and even linking number

We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with lk(A,L) = k2^r, k not 0, a non-split n-component link where all linking numbers are even, or an n-component link with components L, A_i where lk(L,A_i) = 3k, k not 0. Links with other properties are considered as well. For a given property, we prove that every embedding of a certain complete graph contains a link with that property. The size o…

Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link

There is a question asking whether a handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link. This question for the case of a trivial surface-link is affirmatively answered. That is, a handle-irreducible summand of every stably trivial surface-link is only a trivial 2-link. By combining this result with an old result of F. Hosowaka and the author that every surface-knot with infinite cyclic fundamental group is a stably trivial surface-knot, it is concluded that every surface-knot with infinite cyclic fundamental group is a trivial (i.e., an unknotted) su…

广义方结与同伦4-球面

Generalized square knots and homotopy 4-spheres

本文通过分析nR-链环研究同伦4-球面，证明在特殊2-柄附着于广义方结时，无1-柄且仅有两个2-柄的同伦4-球面微分同胚于标准4-球面，并构造了广义性质R猜想的潜在反例族。

Dual decompositions of 4-manifolds II: linear invariants

This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq2$. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction th…

Symplectic 2-handles and transverse links

A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching 2-handles to convex boundaries of symplectic 4-manifolds along links transverse to the induced contact structures. One method results in concave boundaries and depends on a fibration of the link complement over S^1; in this case the handles can be attached with any framing larger than a lower bound determined by…

Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to (S^1 x S^2) # (S^1 x S^2). We exhibit a family of such links that are probably counterexamples to General…

Contact handles, duality, and sutured Floer homology

We give an explicit construction of the Honda--Kazez--Matić gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.

Cutting arcs for torus links and trees

Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., $A_n$, $D_n$, $E_6$, $E_7$ and $E_8$) among all positive tree-like Hopf plumbings.

GridPyM：一个用于处理网格图的Python模块

GridPyM: a Python module to handle grid diagrams

本文提出了GridPyM，一个兼容Sage的Python模块，用于处理纽结理论中的网格图，支持网格的生成、简化及局部变换建模。

Refined Kirby calculus for integral homology spheres

A theorem of Kirby states that two framed links in the 3-sphere produce orientation-preserving homeomorphic results of surgery if they are related by a sequence of stabilization and handle-slide moves. The purpose of the present paper is twofold: First, we give a sufficient condition for a sequence of handle-slides on framed links to be able to be replaced with a sequences of algebraically canceling pairs of handle-slides. Then, using the first result, we obtain a refinement of Kirby's calculus for integral homology spheres which involves only +/-1-framed links with zero linking numbers.

Equivalent characterizations of handle-ribbon knots

The stable Kauffman conjecture posits that a knot in $S^3$ is slice if and only if it admits a slice derivative. We prove a related statement: A knot is handle-ribbon (also called strongly homotopy-ribbon) in a homotopy 4-ball $B$ if and only if it admits an R-link derivative; i.e. an $n$-component derivative $L$ with the property that zero-framed surgery on $L$ yields $\#^n(S^1\times S^2)$. We also show that $K$ bounds a handle-ribbon disk $D \subset B$ if and only if the 3-manifold obtained by zero-surgery on $K$ admits a singular fibration that extends over handlebodies in $B \setminus D$, …

Morse-Novikov numbers, tunnel numbers, and handle numbers of sutured manifolds

Developed from geometric arguments for bounding the Morse-Novikov number of a link in terms of its tunnel number, we obtain upper and lower bounds on the handle number of a Heegaard splitting of a sutured manifold $(M,γ)$ in terms of the handle number of its decompositions along a surface representing a given 2nd homology class. Fixing the sutured structure $(M,γ)$, this leads us to develop the handle number function $h \colon H_2(M,\partial M;\mathbb{R}) \to \mathbb{N}$ which is bounded, constant on rays from the origin, and locally maximal. Furthermore, for an integral class $ξ$, $h(ξ)=0$ if…

边界表面链的Ribbonness，修订版

Fusion of boundary surface-link and ribbonness

本文修订了作者先前的结果，证明了在4球体中，若通过非平凡融合1柄系统手术得到的表面链是Ribbon表面链，则原边界表面链也是Ribbon表面链。此结果回答了Cochran关于非Ribbon球体结的猜想。

Comparing 2-handle additions to a genus 2 boundary component

We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary component of a compact, orientable 3-manifold. These results give a near complete solution to a conjecture of Scharlemann and provide evidence for a conjecture of Scharlemann and Wu. The proofs make use of a new theorem concerning the effects of attaching a 2-handle to a suture in th…

Proposed Property 2R counterexamples classified

Earlier work with Robert Gompf and Abigail Thompson classified, via a natural slope indexed by the rationals, all two-component links which contain the square knot and from which $(S^1 \times S^2) \# (S^1 \times S^2)$ can be obtained by surgery. It was argued that a certain family $L_n$ of such links probably contradict the Generalized Property R Conjecture. Left unresolved was how the family $L_n$ fits into the classification scheme. This question is resolved here, in part by giving varied perspectives and more detail on the construction of the $L_n$.

A classification of spanning surfaces for alternating links

A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus, orientability, and aggregate slope that a surface spanning that link can have. To this end, we describe a straightforward algorithm, much like Seifert's Algorithm, through which to construct certain spanning surfaces called layered surfaces. A particularly important subset of these will be what we call basic layered surfaces. We can alter these surface by perform…

A complete invariant for closed surfaces in the three-sphere

Associated to an embedded surface in the $3$-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, wherefrom we deduce complete invariants of handlebody links, tunnels of handlebody links, and spatial graphs.The main ingredients in the proof of the completeness are a generalization of the Kneser conjecture for $3$-manifolds with boundary proved also here, and extensions of Waldhausen's theorem by Evans, Tucker and Swarup. Computable invariants of handlebody links derived therefrom are calculated.

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。

Triple linking numbers and Heegaard Floer homology

We establish some new relationships between Milnor invariants and Heegaard Floer homology. This includes a formula for the Milnor triple linking number from the link Floer complex, detection results for the Whitehead link and Borromean rings, and a structural property of the $d$-invariants of surgeries on certain algebraically split links.

The disjoint curve property

A Heegaard splitting of a closed, orientable three-manifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve property. This paper shows that in a closed, orientable three-manifold all splittings of sufficiently large genus have the disjoint curve property. From this and a solution to the generalized Waldhausen conjecture it would follow that any closed, orientable three manifold contains only…

Stein fillable 3-manifolds admit positive open book decompositions along arbitrary links

It is known by A. Loi and R. Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L' to L such that the union of the links L and L' is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian cu…

Band taut sutured manifolds

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surface and one of the sutures, and the cocore of the 2-handle in the manifold after attaching a 2-handle along the suture. We use this result to show that tunnels for tunnel number one knots or links in any 3-manifold can be isotoped to lie on a branched surface corresponding to a ce…

具有多条枝的树状链环的桥数

The bridge number of arborescent links with many twigs

本文证明了满足所有分支点均带有通向至少三片叶子的直枝这一条件的平面树相关联树状链环的子午秩猜想，并给出了适用于所有树状链环的桥数上界。

Invariants of links of Conway type

The purpose of this paper is to present a certain combinatorial method of constructing invariants of isotopy classes of oriented tame links. This arises as a generalization of the known polynomial invariants of Conway and Jones. These invariants have one striking common feature. If L+, L- and L0 are diagrams of oriented links which are identical, except near one crossing point (as in Conway or Jones polynomials), then an invariant w(L) has the property: w(L+) is uniquely determined by w(L-) and w(L0), and also w(L-) is uniquely determined by w(L+) and w(L0). To formalize this property we intro…

三维球面中双曲链补集内的不可压缩平面曲面

Incompressible planar surfaces in hyperbolic link exteriors in the 3-sphere

对于每个整数N≥3，构造N分支双曲链的例子，其外部包含一个不可压缩的跨越平面曲面，该曲面在每个边界环面上具有非经向且非整数斜率，从而反驳了Eudave-Muñoz和Ozawa的猜想。

曲面链的自由带状引理

Free ribbon lemma for surface-link

本文证明4维球面中自由球面链必为带状球面链，给出四种证明，并推出曲面链可嵌入自由曲面链当且仅当它是带状球面链的稳定化。

Contractible 3-manifolds and the double 3-space property

Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to $\mathbb R^3$ and whose intersection is again homeomorphic to $\mathbb R^3$. Using a family of generalizations of the Whitehead Link, we show that there are uncountably many contractible 3-manifolds with this double 3-space property. Using a separate family of generalizations of the Whitehead Link and using an extension of interlacing theory, we also show that there are uncountably many contractible 3-manifolds that fail to have this property.

Wheeling: A diagrammatic analogue of the Duflo isomorphism

We construct and prove a diagrammatic version of the Duflo isomorphism between the invariant subalgebra of the symmetric algebra of a Lie algebra and the center of the universal enveloping algebra. This version implies the original for metrized Lie algebras (Lie algebras with an invariant non-degenerate bilinear form). As an application of this isomorphism, we will compute the Kontsevich integral of the unknot and the Hopf link to all orders. At the core of the proof, we use an elementary property of the Hopf link which can be summarized by the equation ``1+1=2'' in abacus arithmetic: doubling…

广义 Alexander 多项式的梯形性质

Trapezodial property of the generalized Alexander polynomial

本文研究特殊交错链环 Alexander 多项式的广义性质，证明其对特定向量配置统计量的生成函数关系，并据此推导出对数凹性与梯形性质，从而为 Fox 猜想在该特例下提供了新证明。

A combinatorial property of generic immersions of curves

A divide is a relative generic immersion of a finite union of copies of the unit interval in the unit disk. A divide defines a classical link in the 3- sphere, which is a fibered link if the image of the immersion is connected. We prove in this paper, that the Lefschetz number of the monodromy is 0. This result was known for divides, which correspond to real morsifications of complexe plane curve singularities. The proof uses the space of gradient lines of a morse function.

Explicit concave fillings of contact three-manifolds

In this paper we give explicit, handle-by-handle constructions of concave symplectic fillings of all closed, oriented contact 3-manifolds. These constructions combine recent results of Giroux relating contact structures and open book decompositions of 3-manifolds, earlier results of the author on attaching 4-dimensional symplectic 2-handles along transverse links, and some tricks with mapping class groups of compact surfaces with non-empty boundary.

Cube diagrams and 3-dimensional Reidemeister-like moves for knots

In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.