Jonathan A. Hillman
We show that if $B$ is an aspherical 2-orbifold in one of the families known to have orbifold fundamental groups of weight 1 then $B$ is the base of a Seifert fibration of a 2-knot manifold $M(K)$.
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 prove that there are exactly $6$ Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in $S^3$, with $\{60, 144, 156, 288, 300\}$ as the exact set of all such surgery slopes up to taking the mirror images of the knots. We conjecture that there are exactly $4$ specific hyperbolic knots in $S^3$ which admit Nil Seifert fibred surgery. We also give some more general results and a more general conjecture concerning Seifert fibred surgeries on hyperbolic knots in $S^3$.
It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3-manifold into any other triangulation of the same 3-manifold exists. In this paper we describe a procedure yielding an explicit formula for such a function if the 3-manifold in question is a Seifert fibred space.
For every genus $g\geq 2$, we construct an infinite family of strongly quasipositive fibred knots having the same Seifert form as the torus knot $T(2,2g+1)$. In particular, their signatures and four-genera are maximal and their homological monodromies (hence their Alexander module structures) agree. On the other hand, the geometric stretching factors are pairwise distinct and the knots are pairwise not ribbon concordant.
We consider the question of when is the closed manifold obtained by elementary surgery on an $n$-knot Seifert fibred over a 2-orbifold. After some observations on the classical case, we concentrate on the cases n=2 and 3. We have found a new family of 2-knots with torsion-free, solvable group, overlooked in earlier work. We know of no higher dimensional examples.
The present paper gives an example of a rigid spherical cone-manifold and that of a flexible one which are both Seifert fibred.
Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery
本文利用 Dijkgraaf-Witten 不变量研究沿纽结的 k/ℓ 手术产生的小 Seifert 3-流形，证明其重数与手术系数受 Alexander 多项式约束。
We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander polynomial, there exist infinitely many non-fibered knots with the given Seifert matrix. We illustrate examples of knots that have trivial Alexander polynomials but do not have twisted Alexander invariants of fibred knots.
We consider Chern-Simons theory with complex gauge group and present a complete non-perturbative evaluation of the path integral (the partition function and certain expectation values of Wilson loops) on Seifert fibred 3-Manifolds. We use the method of Abelianisation. In certain cases the path integral can be seen to factorize neatly into holomorphic and anti-holomorphic parts. We obtain closed formulae of this factorization for the expectation values of torus knots.
Fillable structures on negative-definite Seifert fibred spaces
本文研究负定星形连接空间上的可填充接触结构，证明此类Seifert纤维空间具有唯一的负最大扭数，并通过晶格上同调的Alexander滤子计算该扭数，进一步展示这些流形上的负扭紧结构由复表面奇点的最小解析中的Stein结构诱导。
We prove that basket links, whose symmetrized Seifert form is congruent to the Cartan matrix of the simply laced Dynkin diagram $A_{n}$, are isotopic to the torus link $T(2,n+1)$. In addition, we provide examples of links, constructed by plumbing $n$ positive Hopf bands the core curves of which intersect at most once, with symmetrized Seifert form congruent to the Cartan matrix $A_{n}$, that are not isotopic to $T(2,n+1)$.
We explore algebraic characterizations of 2-knots whose associated knot manifolds fibre over lower-dimensional orbifolds, and consider also some issues related to the groups of higher-dimensional fibred knots.
Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in $S^3$. We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if $K$ is a knot in a closed 3--manifold $Y$, $Y-K$ is irreducible, and $\hat{HFK}(Y,K)$ is monic, then $K$ is fibred. The proof relies on previous works due to Gabai, Ozsváth--Szabó, Ghiggini and the author. A corollary is that if a knot in $S^3$ admits a lens space surgery, then the knot is fibred.
We correct a mistake on the citation of JSJ theory in \cite{Ni}. Some arguments in \cite{Ni} are also slightly modified accordingly.
In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangul…
A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question of when K(2a,b|r) is fibred. In most cases, the Alexander polynomials determine the genus and fibredness of these knots. We develop both algebraic and geometric techniques to describe the genus and fibredness by means of continued fraction expansions of b/2a. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same …
Large volume fibred knots of fixed genus
该研究构造固定亏格却具有任意大体积的三维球面双曲纤维纽结，证明体积与亏格、强拟正性及 Seifert 形式均无关。
We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.
This paper gives a new obstruction for ribbon-move equivalence of 2-knots. Let $K$ and $K'$ be 2-knots. Let $K$ and $K'$ are ribbon-move equivalent. One corollary to our main theorem is as follows. A 2-dimensional fibered knot whose fiber is the punctured 3-dimensional torus is not ribbon-move equivalent to any 2-dimensional knot whose Seifert hypersurface is a punctured homology sphere.
A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those on a trefoil knot by twisting along their seiferters. Although Seifert surgeries on a trefoil knot are the most basic ones, this family is rich in variety. For any m which is not -2 it contains a successive triple of Seifert surgeries (K, m), (K, m +1), (K, m +2) on a hyperbolic k…
Tight fibred knots without L-space surgeries
本文证明对每个固定亏格$g\geq 2$，存在无穷多个纤维且强拟正的结，它们与同亏格环面结代数协边，但不承认L-空间手术。
How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a "hyperbolic seiferter" or a "hyperbolic annular pair" produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate …
A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pairs of seiferters". One problem of the network is whether there is a path from each vertex to a vertex on a torus knot, the most basic Seifert surgery. We give a method to find seiferters and annular pairs of seiferters for Seifert surgeries obtained by taking two--fold branched co…
The twisted torus knots lie on the standard genus 2 Heegaard surface for $S^3$, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted torus knots, we consider the twisted torus knots to partially answer the question of which primitive/Seifert knots are fibered. A braid computation shows that a particular family of twisted torus knots is fibered, and that computation is then used to generalize the results of a prev…
We prove that every knot in the 3-space bounds an embedded punctured Moebius band whose other boundary component is a quasipositive fibred knot.
We show that if a Montesinos knot admits a Dehn surgery yielding a toroidal Seifert fibered 3-manifold, then the knot is the trefoil knot and the surgery slope is 0.
Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.
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.
We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger-Gromov-von Neumann rho-invariants for amenable groups developed by Cha and Orr and polynomial splittings of metabelian rho-invariants.
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus knot and the surgery slope is 2(p+q) with |p|, |q| at least three.
We show that a knot in $S^3$ with an infinite number of distinct incompressible Seifert surfaces contains a closed incompressible surface in its complement.
Recently Kearton showed that any Seifert matrix of a knot is S--equivalent to the Seifert matrix of a prime knot. We show in this note that such a matrix is in fact S--equivalent to the Seifert matrix of a hyperbolic knot. This result follows from reinterpreting this problem in terms of Blanchfield pairings and by applying results of Kawauchi.
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.
Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, $F$, in $S^3$; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to $F$; surgery on these knots at the surface slope yields a Seifert fibered space. Here we construct a two-parameter family of knots that have distinct primitive/Seifert embeddings in $F$ with the same surface slope, as well as a family of torus knots that have a primitive/primitive representative and a primitive/Seifert representativ…
In this paper, we define the primitive/Seifert-fibered property for a knot in S^3. If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base S^2 and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of examples of primitive/Seifert-fibered knots. By analyzing the twisted torus knots, we prove that nearly all possible triples of multiplicities of the critical fibers arise via Dehn surgery on primitive/Seifert-fibered knots.
Exceptional Dehn surgeries on arborescent knots have been classified except for Seifert fibered surgeries on Montesinos knots of length 3. There are infinitely many of them as it is known that 4n+6 and 4n+7 surgeries on a (-2, 3, 2n+1) pretzel knot are Seifert fibered. It will be shown that there are only finitely many others. A list of 20 surgeries will be given and proved to be Seifert fibered. We conjecture that this is a complete list.
We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev's filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial.
We will use immersed surfaces to study Seifert fibered surgery on Montesinos knots, and show that if $\frac 1{q_1-1} + \frac 1{q_2-1} + \frac 1{q_3-1} \leq 1$ then a Montesinos knot $K(\frac{p_1}{q_1}, \frac{p_2}{q_2}, \frac{p_3}{q_3})$ admits no atoroidal Seifert fibered surgery.
We study invariant Seifert surfaces for strongly invertible knots, and prove that the gap between the equivariant genus (the minimum of the genera of invariant Seifert surfaces) of a strongly invertible knot and the (usual) genus of the underlying knot can be arbitrary large. This forms a sharp contrast with Edmonds' theorem that every periodic knot admits an invariant minimal genus Seifert surface. We also prove variants of Edmonds' theorem, which are useful in studying invariant Seifert surfaces for strongly invertible knots.
We consider the question of how many essential Seifert Klein bottles with common boundary slope a knot in S^3 can bound, up to ambient isotopy. We prove that any hyperbolic knot in S^3 bounds at most six Seifert Klein bottles with a given boundary slope. The Seifert Klein bottles in a minimal projection of hyperbolic pretzel knots of length 3 are shown to be unique and pi_1-injective, with surgery along their boundary slope producing irreducible toroidal manifolds. The cable knots which bound essential Seifert Klein bottles are classified; their Seifert Klein bottles are shown to be non-pi_1-i…
Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot $K$ gives the Poincare homology sphere $Σ(2,3,5)$, then $K$ is the left-handed trefoil knot.
Suppose $α$ and $R$ are disjoint simple closed curves in the boundary of a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $α$ is Seifert in $H$(i.e., a 2-handle addition $H[α]$ is a Seifert-fibered space) and not the meridian of $H[R]$. Then for a slope $γ$ of $k$ represented by $α$, $γ$-Dehn surgery $k(γ)$ is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was i…
Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the 3-sphere S^3? It is conjectured that if r-surgery on a hyperbolic knot in S^3 yields a Seifert fiber space, then r is an integer. We show that for each integer n, there exists a tunnel number one, hyperbolic knot K_n in S^3 such that n-surgery on K_n produces a small Seifert fiber space.
By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this paper, we extend this result to the full extent. We show that if the knot has nontrivial Alexander polynomial, then there exists an infinite family of non-concordant knots having the same Seifert form as the knot. As a corollary, no nontrivial Alexander polynomial determines a unique knot concordance class. We use Cochran-Orr-Teichner's recent result on the k…
Let K be a knot in S^3 of genus g and let n>0. We show that if rk HFK(K,g) < 2^{n+1} (where HFK denotes knot Floer homology), in particular if K is an alternating knot such that the leading coefficient a_g of its Alexander polynomial satisfies |a_g| <2^{n+1}, then K has at most n pairwise disjoint non-isotopic genus g Seifert surfaces. For n=1 this implies that K has a unique minimal genus Seifert surface up to isotopy.
We complete the classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. This is the final step in understanding all exceptional surgeries on hyperbolic pretzel knots. We also present results toward similar classifications for non-pretzel Montesinos knots of length three.
We give a new criterion for a given knot to be a Montesinos knot by using the Rasmussen invariant and the signature. We apply the criterion to study Seifert fibered surgery on a strongly invertible knot, and show that a $(p,q,q)$-pretzel knot with integers $p,q \ge 2$ admits no Seifert fibered surgery.
We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic Seifert surfaces by giving bounds on the width invariant in the presence of such a surface. Finally, we utilize these examples to demonstrate that the Six Theorem is sharp for knot complements in the 3-sphere.
The aim of this paper is to investigate the relations between Seifert manifolds and (1,1)-knots. In particular, we prove that every orientable Seifert manifold with invariants {Oo,0|-1;(p,q),...,(p,q),(l, l-1)} has a cyclically presented fundamental group and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq-p|,q), branched over the (1,1)-knot K(q,q(nl-2),p-2q,p-q) if p>=2q, and over the (1,1)-knot K(p-q,2q-p,q(nl-2),p-q) if p<2q.
A virtual knot that has a homologically trivial representative $\mathscr{K}$ in a thickened surface $Σ\times [0,1]$ is said to be an almost classical (AC) knot. $\mathscr{K}$ then bounds a Seifert surface $F\subset Σ\times [0,1]$. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in $Σ\times [0,1]$ are difficult to construct. Here we introduce virtual Seifert surfaces of AC knots. These are planar figures representing $F \subset Σ\times [0,1]$. An algorithm for constructing a virtual Seifert surface from a Gauss diagr…
This article was originally published in Topology 31 (1992). The present hyperTeXed redaction corrects a few typographical errors and updates the references.
For a knot $K$ with $Δ_K(t)\doteq t^2-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that an appropriate assumption on the Reidemeister torsion of the universal abelian covering of $M$ implies $q=\pm 1$, if $M$ is a Seifert fibered space.
For a given smooth $2$-knot in $S^4$, we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible $ SU(2)$-representations of its knot group. For example, we see that any smooth $2$-knot having the Poincaré homology $3$-sphere as a Seifert hypersurface has at least four irreducible $SU(2)$-representations of its knot group. This result is false in the topological category. The proof uses a quantitative formulation of instanton Floer homology. Using similar techniques, we also obtain similar results about codimension-$1$ embeddings of homology $3$…
For a knot $K$ with $Δ_K(t)\doteq t^2-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that appropriate assumptions on the Reidemeister torsion and the Casson-Walker-Lescop invariant of the universal abelian covering of $M$ imply $q=\pm 1$, if $M$ is a Seifert fibered space.
We examine surgery on a knot in $S^3$ to determine surgery obstructions to Seifert fibered integral homology spheres. We find such surgery obstructions using Heegaard Floer, Knot Floer homology and the mapping cone formula for computing Heegaard Floer homology of surgery on a knot. Here however, we take a different approach and use the number of singular fibers of a Seifert fibered integral homology sphere, which is the toroidal structure, to find obstructions. This approach allows us to show that genus one knots cannot yield Seifert fibered integral homology spheres with six or more singular …
We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over $S^2$. As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information.
We show that the Vassiliev invariants of orders $\leq n$ of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. As a consequence of this, we obtain that the Vassiliev invariants of the knot $K=\partial S$ are null-concordance obstructions of certain links that can be obtained from regular spines of S. We also discuss various generalizations of these results, and we conjecture a geometric characterization of knots whose invariants of all orders vanish.
We consider vector fields on knot/link complements in $S^3$ which are transverse to the fibres of a fibration of the complement over a circle. We prove that a large class of fibred knots/links, including all non-torus fibred 2-bridge knots, has the following property: any vector field transverse to the fibres of the fibration of the complement must possess periodic orbits representing all possible knot and link types.
Fibred realizations of the prism quandle $P_7$ in the standard four-sphere
本文证明由四元数棱柱流形构造的两个同伦四维球面均为标准，并给出棱柱quandle阶五十六的两个不等价纤维实现，解决了阶七情形。
A persistent lamination for a knot K is an essential lamination in the complement of K, which remains essential after every non-trivial Dehn surgery along K. Having a persistent lamination implies, for example, that every manifold obtained by non-trivial Dehn surgery along K has universal cover R^3. In this paper we present a method for building persistent laminations for knots from an incompressible Seifert surface for some `parent' knot. Using this construction, we can, for example, currently build persistent laminations for approximately 45 percent of the knots in the standard knot tables; …
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。