Stefan Friedl
We explain an algorithm for finding a boundary link Seifert matrix for a given Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander polynomial.
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We show some properties of a Seifert matrix of an $n$-component Brunnian link. In particular, we give a necessary and sufficient condition for a matrix to be a Seifert matrix of a 2-component Brunnian link up to S-equivalence.
We develop an algorithm for computing generalized Seifert matrices for colored links given as closures of colored braids. The algorithm has been implemented by the second author as a computer program called Clasper. Clasper also outputs the Conway potential function, the multivariable Alexander polynomial and the Cimasoni-Florens signatures of a link, and displays a visualization of the C-complex used for producing the generalized Seifert matrices.
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.
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.
The Goeritz matrix of a link is obtained from the Jacobian matrix of a modified Dehn presentation associated to a diagram using Fox's free differential calculus. When the diagram is special the Seifert matrix can also be determined from the presentation.
We generalize H. Seifert's algorithm for finding a Seifert surface for a knot or link. The generalization applies to "framed oriented measured lamination links." For knots, a Seifert surface determines a unique framing. In our setting, we analyze the set of framed lamination links which bound Seifert laminations and are carried by an $S^1$-fibered tube neighborhood of an oriented train track embedded in a 3-manifold.
We study compatible contact structures of fibered Seifert multilinks in homology 3-spheres and especially give a necessary and sufficient condition for the contact structure to be tight in the case where the Seifert fibration is positively twisted. As a corollary we determine the strongly quasipositivity of fibered Seifert links in the 3-sphere. We also study the compatible contact structures of cablings along links in any 3-manifolds.
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 introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of the link.
We introduce and define "oriented framed measured lamination links" in a 3-manifold $M$. These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as some framed links bound Seifert surfaces, so also some framed lamination links bound 2-dimensional measured and oriented "Seifert laminations." We show that any lamination link which bounds a 2-dimensional Seifert lamination, bounds a "taut" Seifert lamination, i.e. one of maximum Euler characteristic, subject to the condition that the Seifert lamination is car…
We introduce a new standard form of a Seifert surface $F$. In that standard form, $F$ is obtained by successively plumbing flat annuli to a disk $D$, where the gluing regions are all in $D$. We show that any link has a Seifert surface in the standard form, and thereby present a new way of coding a link. We present an algorithm to read the code directly from a braid presentation.
Homogeneous links were introduced by Peter Cromwell, who proved that the projection surface of these links, that given by the Seifert algorithm, has minimal genus. Here we provide a different proof, with a geometric rather than combinatorial flavor. To do this, we first show a direct relation between the Seifert matrix and the decomposition into blocks of the Seifert graph. Precisely, we prove that the Seifert matrix can be arranged in a block triangular form, with small boxes in the diagonal corresponding to the blocks of the Seifert graph. Then we prove that the boxes in the diagonal has non…
This article was originally published in Topology 31 (1992). The present hyperTeXed redaction corrects a few typographical errors and updates the references.
Cyclic branched covers of Seifert links and properties related to the $ADE$ link conjecture
本文证明了Seifert链环的循环分支覆盖具有左序基本群当且仅当该链环在定向意义下不是ADE链环，从而证明了ADE链环猜想，并进一步分类了相关覆盖、强拟正链环及亏格性质。
Non-split, alternating links bound unique Seifert surfaces in the 4-ball
该研究证明非分裂交替链接的Seifert表面在推入4球体后成为光滑同胚，且粘合后的表面总是光滑无结。
An algorithm for Seifert surfaces in 3-manifolds via surgery presentations
本文提出了一种通过手术表示法在任意3流形中构造无界链接Seifert表面的算法，扩展了经典的Seifert算法。
Let $\barγ$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $γ$ in $\mathcal O.$ We prove that the complement $M_{\barγ}$ of $\barγ$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.
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.
We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for twisted Whitehead doubles of non-cable knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 12 crossings.
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。
Seifert forms and slice Euler characteristic of links
本文定义了链环的Witt余指标作为协变不变量，给出切片欧拉示性数的上界，并刻画余指标为1的形式，用于证明两分支链环不界定局部平坦圆柱。
We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so, we are able to generalise the result, replacing `minimal genus' with `incompressible' and `alternating' with `homogeneous'. We also examine the implications of our proof for alternating links in general.
Alternating links have at most polynomially many Seifert surfaces of fixed genus
本文证明非分裂素交错链环的固定亏格Seifert曲面数量受显式多项式界定，改进了先前的指数级界限。
In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S^3. This yields an integral valued function on the m-dimensional torus, where m is the number of colors of the link. The case m=1 corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this m-variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4-dimensional interpretation and th…
The flat plumbing basket presentation of a link is introduced by Furihata, Hirasawa and Kobayashi. In this paper, we show that the pass-equivalence and an equivalence introduced by using the flat plumbing basket presentation are the same relation. Furthermore, we obtain an evaluation of the minimal number of bands used for the flat pluming basket presentation for a knot coming from the degree of the Alexander polynomial and the three genus of the knot.
Two Seifert surfaces of links in $S^3$ are said to be twist equivalent if one can be obtained from the other, up to isotopy, by repeatedly performing operations consisting of cutting along an embedded arc, applying a full twist near one copy of the arc, and re-gluing. By using bridge spheres for their boundary links, we provide a new method of distinguishing twist equivalence classes of Seifert surfaces of any given genus. Given a Seifert surface $F$ of a link $L$, we show that the bridge numbers of the boundary links of Seifert surfaces twist equivalent to $F$ are uniformly bounded above by a…
It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Murasugi had conjectured that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link. This conjecture turned out to be false. In this paper we prove the next best thing that one could hope for: we characterize exactly those alternating links for which their braid ind…
We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either "trivial", in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high D…
Mock Seifert matrices and unoriented algebraic concordance
本文通过Mock Seifert矩阵定义无向代数协边群，证明其阿贝尔且无限生成，并用于研究虚拟纽结协边等不变量。
Seifert circles, crossing number and the braid index of generalized knots and links
本文引入广义交叉和广义Reidemeister型移动，研究与交叉类型无关的链环性质，证明了广义纽结和链环总交叉数与辫指数间的不等式，并适用于虚拟奇链环。
Rectangular diagrams of links are link diagrams in the plane ${\mathbb R}^2$ such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. P. R. Cromwell and I. A. Dynnikov showed that rectangular diagrams of links are useful for deciding whether a given link is split or not, and whether a given knot is trivial or not. We show in this paper that an oriented link diagram $D$ with $c(D)$ crossings and $s(D)$ Seifert circles can be deformed by an ambient isotopy of ${\mathbb R}^2$ into a rectangular diagram w…
Let $L$ be an oriented link with an alternating diagram $D$. It is known that $L$ is a fibered link if and only if the surface $R$ obtained by applying Seifert's algorithm to $D$ is a Hopf plumbing. Here, we call $R$ a Hopf plumbing if $R$ is obtained by successively plumbing finite number of Hopf bands to a disk. In this paper, we discuss its extension so that we show the following theorem. Let $R$ be a Seifert surface obtained by applying Seifert's algorithm to an almost alternating diagrams. Then $R$ is a fiber surface if and only if $R$ is a Hopf plumbing. We also show that the above theor…
Recently, Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or equivalently, embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a characterization of Seifert graphs in terms of Eulerian subgraphs. This characterization can be viewed as a refinement of the fact that Seifert graphs are bipartite. We go on to examine the family of ribbon graphs that arises by forming the parallels of a link diagram and determine…
Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we generalize Seifert's algorithm to explicitly construct a rational Seifert surface for any rationally null-homologous knot. As an application of the techniques developed in the paper, we derive closed formulae for the rational Thurston-Bennequin and rotation numbers of a Legendri…
This paper presents a new algorithm "A" for constructing Seifert surfaces from n-bridge projections of links. The algorithm produces minimal complexity surfaces for large classes of braids and alternating links. In addition, we consider a family of knots for which the canonical genus is strictly greater than the genus, (g_c(K) > g(K)), and show that A builds surfaces realizing the knot genus g(K). We also present a generalization of Seifert's algorithm which may be used to construct surfaces representing arbitrary relative second homology classes in a link complement.
We study the effect of Nielsen moves and their geometric counterparts, handle slides, on good boundary links. A collection of links, universal for 4-dimensional surgery, is shown to admit Seifert surfaces with trivial Lagrangian. They are good boundary links, with Seifert matrices of a more general form than in known constructions of slice links. We show that a certain more restrictive condition on Seifert matrices is sufficient for proving the links are slice. We also give a correction of a Kirby calculus identity in \cite{FK2}, useful for constructing surgery kernels associated to link-slice…
For a compact connected 3-submanifold with connected boundary in the 3-sphere, we relate the existence of a Seifert surface system for a surface with a Dehn surgery along a null-homologous link. As its corollary, we obtain a refinement of the Fox's re-embedding theorem.
We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert form is represented by 3 curves on the Seifert surfaces that form a link with nonvanishing Milnor triple linking number. We also give a generalisation to m-component links and higher Milnor invariants. We prove that our examples are ribbon and that all ribbon links are boundary slice.
Infinitely many pairs of spatial surfaces
本文提出为每个链接构造无限多对Seifert曲面，通过环境同胚和单模合同不变量区分曲面对。
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 $T(2,n+1)$.
It is well known that the Blanchfield pairing of a knot can be expressed using Seifert matrices. In this paper, we compute the Blanchfield pairing of a colored link with non-zero Alexander polynomial. More precisely, we show that the Blanchfield pairing of such a link can be written in terms of generalized Seifert matrices which arise from the use of C-complexes.
In this survey paper we present results about link diagrams in Seifert manifolds using arrow diagrams, starting with link diagrams in $F\times S^1$ and $N\hat{\times}S^1$, where $F$ is an orientable and $N$ an unorientable surface. Reidemeister moves for such arrow diagrams make the study of link invariants possible. Transitions between arrow diagrams and alternative diagrams are presented. We recall results about %the knot group presentation for lens spaces and the Kauffman bracket and HOMFLYPT skein modules of some Seifert manifolds using arrow diagrams, namely lens spaces, a product of a di…
The classification of high-dimensional mu-component boundary links motivates decomposition theorems for the algebraic K-groups of the group ring A[F_mu] and the noncommutative Cohn localization Sigma^{-1}A[F_mu], for any mu>0 and an arbitrary ring A, with F_mu the free group on mu generators and Sigma the set of matrices over A[F_mu] which become invertible over A under the augmentation A[F_mu] to A. Blanchfield A[F_mu]-modules and Seifert A-modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for A[F_mu]-module chain comple…
The existence of basket, flat plumbing and flat plumbing basket surfaces of a link was first proven from a braid representative of the link. In the present article, we show the existence of such surfaces from an induced graph of the link. Consequently, we define the basket number, flat plumbing number and flat plumbing basket number of a link. Then we provide several upper bounds for these plumbing numbers and study the relation between these plumbing numbers and the genera of links.
We characterise positive braid links with positive Seifert form via a finite number of forbidden minors. From this we deduce a one-to-one correspondence between prime positive braid links with positive Seifert form and simply laced Dynkin diagrams, as well as a simple classification of alternating positive braid knots.
We show that $|MS(L_1 # L_2)|=|MS(L_1)|\times|MS(L_2)|\times\mathbb{R}$ when $L_1$ and $L_2$ are any non-split and non-fibred links. Here $MS(L)$ denotes the Kakimizu complex of a link $L$, which records the taut Seifert surfaces for $L$. We also show that the analogous result holds if we study incompressible Seifert surfaces instead of taut ones.
Non-isotopic Seifert surfaces in the 4-ball
本文构造了多族在4-球中具有拓扑非同位Seifert曲面配对的纽结与链环，推广了Hayden等人的例子，并基于二重分支覆盖与Seifert形式给出证明，同时利用Conway拓扑图理论实现了二元整二次型同构判定算法。
A Seifert algorithm for integral homology spheres
本文将经典纽结理论中的 Seifert 算法推广到任意闭积分同调 3-球面，利用 Heegaard 分裂的平面图表示和有向缠结的正则投影，构造出 Seifert 圆及半扭转带，从而证明每个有向链环都是有向嵌入曲面的边界。
Spaces of Legendrian cables and Seifert fibered links
本文确定了标准接触3-球面中具最大瑟斯顿-本内昆不变量的若干勒让德纽结和链环嵌入空间的同伦型，给出充分正缆线的递归公式，证明相关接触结构空间的同伦等价性与可缩性，还发现满足单射h-原理的无穷多新分支。
Algebraic concordance of links
从同调手术和布兰奇菲尔德形式视角系统研究链环的代数协调，得出两个协调障碍，即同调手术不变量和布兰奇菲尔德不变量，还对\(\mu\leq2\)用广义赛弗特矩阵描述了这些不变量。
On elementary invariants of genus one knots and Seifert surfaces
本文提出了一种用于研究一阶结和塞弗特面的亚历山大形式不变量，通过分析三维同调柄体的性质来证明其不变性。
Signatures of iterated torus links
本文计算了塞弗特链环（含带核环面链环）的多元符号，将其用于链环缆线多元符号计算的拼接公式，并将相关公式用平行四边形整点表示，类似Hirzebruch公式。
We show that the twisted signature invariants of boundary link concordance derived from unitary representations of the free group are actually ordinary link concordance invariants. We also show how the discontinuity locus of this signature function is determined by Seifert matrices of the link.
We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured F…
Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the `local Thom Conjecture', has various interesting consequences; for instance, it yields an easily-computed estimate for the slice euler characteristic of the link L(D) (where D is arbitrary) that extends and often improves the `slice-Bennequin inequality' for closed-braid diagram…
We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of Levine on cobordism of sphere knots, we define disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cob…
Minimal genus Seifert surface for 11 crossing alternating knots
本文计算并明确描述了所有11个交叉点的素交错纽结的Kakimizu复形，结合已知算法与Murasugi和及缝合流形理论，完整刻画了这些纽结的最小亏格Seifert曲面结构。
We extend several classical invariants of links in the 3-sphere to links in so-called quasi-cylinders. These invariants include the linking number, the Seifert form, the Alexander module, the Alexander-Conway polynomial and the Murasugi-Tristram-Levine signatures.
We prove the following: Let $2p + 1$ be no less than 5 and $p$ be a natural number. Let $K$ and $J$ be closed, oriented, $(2p+1)$-dimensional connected, $(p-1)$-connected, simple submanifolds of the standard $(2p+3)$-sphere. Then $K$ is equivalent to $J$ if and only if a Seifert matrix associated with a simple Seifert hypersurface for $K$ is $(-1)^p$-$S$-equivalent to that for $J$. We also discuss the $2p+1=3$ case. This result implies one of our main results: Let $μ$ be a natural number. A 1-link $A$ is pass-move equivalent to a 1-link $B$ if and only if the knot product of $A$ and $μ$ copies…
It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals the braid index $\textbf{b}(D)$ of $D$ if $D$ contains no {\em lone crossings} (a crossing in $D$ is called a {\em lone crossing} if it is the only crossing between two Seifert circles in $D$). If $D$ contains lone crossings, then $\textbf{b}(D)$ is strictly less than $s(D)$. However in general it is not known how $s(D)$ is related to $\textbf{b}(D)$. In this …
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