Seifert Fibered Research Articles

On continuous families of geometric Seifert conemanifold structures

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space [Formula: see text] and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot [Formula: see text]. As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on [Formula: see text], the result of [Formula: see text] surgery in the left-handed trefoil knot [Formula: see text], J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

Volume formulae for fibred cone-manifolds with spherical geometry

We classify cone-manifold structures on a three-dimensional sphere whose singular set consists of fibres of a Seifert fibration. We describe domains of existence of a spherical structure on these cone-manifolds in terms of cone angles, and obtain explicit analytic formulae for their volumes. Bibliography: 18 titles.

Profinite rigidity for Seifert fibre spaces

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.

Integral foliated simplicial volume of hyperbolic 3-manifolds

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical manifolds and give refined upper bounds of integral foliated simplicial volume in terms of stable integral simplicial volume. This allows us to compute the integral foliated simplicial volume of hyperbolic 3-manifolds. This is complemented by the calculation of the integral foliated simplicial volume of Seifert 3-manifolds.

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

We study in detail the large N expansion of {mathrm {SU}}(N) and {mathrm {SO}}(N)/{mathrm {Sp}}(2N) Chern–Simons partition function Z_N(M) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. This partition function admits a matrix model-like representation, whose spectral curve can be characterized in terms of a certain scalar, linear, non-local Riemann-Hilbert problem (RHP). We develop tools necessary to address a class of such RHPs involving finite subgroups of mathrm{PSL}_{2}({mathbb {C}}). We associate with such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. These techniques are applied to the RHP relevant for Chern–Simons theory on Seifert spaces. When pi _1(M) is finite—i.e., for manifolds M that are quotients of {mathbb {S}}_{3} by a finite isometry group of type ADE—we find that the Weyl group associated with the RHP is finite and the spectral curve is algebraic and can be in principle computed. We then show that the large N expansion of Z_N(M) is computed by the topological recursion. This has consequences for the analyticity properties of {mathrm {SU}}/{mathrm {SO}}/{mathrm {Sp}} perturbative invariants of knots along fibers in M.

The Dijkgraaf–Witten invariants of Seifert 3-manifolds with orientable bases

We derive a formula for the Dijkgraaf–Witten invariants of Seifert 3-manifolds with orientable bases.

Alternating Heegaard diagrams and Williams solenoid attractors in $3$-manifolds

We find all Heegaard diagrams with the property ``alternating'' or ``weakly alternating'' on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two $3$-manifolds, each admits an automorphism whose non-wandering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincaré's homology $3$-sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums. The result shows that many kinds of $3$-manifolds admit a kind of ``translation'' with certain stability.

Bridge number and integral Dehn surgery

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 Teragaito that live in a small Seifert fiber space M and where each K^n admits a Dehn surgery giving the 3-sphere. We show that the bridge number of K^n with respect to any genus 2 Heegaard splitting of M tends to infinity with n. This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving the 3-sphere.

TWISTED TORUS KNOTS WITH GRAPH MANIFOLD DEHN SURGERIES

In this paper, we classify all twisted torus knots which are doubly middle Seifert-fibered. Also we show that all of these knots possibly except a few admit Dehn surgery producing a non-Seifert-fibered graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. This provides another infinite family of knots in admitting Dehn surgery yielding such manifolds as done in [5].

Braid equivalences in 3-manifolds with rational surgery description

In this paper we provide algebraic mixed braid classification of links in any c.c.o. 3-manifold M obtained by rational surgery along a framed link in S3. We do this by representing M by a closed framed braid in S3 and links in M by closed mixed braids in S3. We first prove an analogue of the Reidemeister theorem for links in M. We then give geometric formulations of the mixed braid equivalence using the L-moves and the braid band moves. Finally we formulate the algebraic braid equivalence in terms of the mixed braid groups Bm,n, using cabling and the parting and combing techniques for mixed braids. Our results set a homogeneous algebraic ground for studying links in 3-manifolds and in families of 3-manifolds using computational tools. We provide concrete formuli of the braid equivalences in lens spaces, in Seifert manifolds, in homology spheres obtained from the trefoil and in manifolds obtained from torus knots.Our setting is appropriate for constructing Jones-type invariants for links in families of 3-manifolds via quotient algebras of the mixed braid groups Bm,n, as well as for studying skein modules of 3-manifolds, since they provide a controlled algebraic framework and much of the diagrammatic complexity has been absorbed into the proofs. Further, our moves can be used in a braid analogue of Rolfsen's rational calculus and potentially in computing Witten invariants.

The fixed subgroups of homeomorphisms of Seifert manifolds

Let M be a compact connected orientable Seifert manifold with hyperbolic orbifold B M , and f π : π 1(M) → π 1(M) be an automorphism induced by an orientation-reversing homeomorphism f of M. We give a bound on the rank of the fixed subgroup of f π , namely, rankFix(f π ) < 2rankπ 1(M), which is an analogue of inequalities on surface groups and hyperbolic 3-manifold groups.

Asymptotic properties of the quantum representations of the mapping class group

For any surface with genus $\geqslant 2$, the monodromy of Hitchin’s connection is a projective representation of the mapping class group of the surface. We establish two results on the large level limit of these representations. First we prove that these projective representations lift to asymptotic representations. Second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion. This proves the Witten’s asymptotic conjecture for mapping tori of surface diffeomorphisms. Our result is not limited to Seifert manifolds and applies to hyperbolic manifolds.

KNOTS ADMITTING SEIFERT-FIBERED SURGERIES OVER S2WITH FOUR EXCEPTIONAL FIBERS

In this paper, we construct infinite families of knots in <TEX>$S^3$</TEX> which admit Dehn surgery producing a Seifert-fibered space over <TEX>$S^2$</TEX> with four exceptional fibers. Also we show that these knots are turned out to be satellite knots, which supports the conjecture that no hyperbolic knot in <TEX>$S^3$</TEX> admits a Seifert-fibered space over <TEX>$S^2$</TEX> with four exceptional fibers as Dehn surgery.

On a higher dimensional generalization of Seifert fibrations

The notion of generalized Seifert fibration is introduced; it is shown that the projections of certain Eschenburg 7-manifolds \(W_{\bar n}^7 \) onto ℂP2 define such fibrations; and their characteristic classes corresponding to the generators of H2(B(U(2)/ℤ2n);ℤ) are defined.

KNOTS IN S3ADMITTING GRAPH MANIFOLD DEHN SURGERIES

In this paper, we construct infinite families of knots in <TEX>$S^3$</TEX> which admit Dehn surgery producing a graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. In particular, we show that for any natural numbers a, b, c, and d with <TEX>$a{\geq}3$</TEX> and <TEX>$b,c,d{\geq}2$</TEX>, there are knots in <TEX>$S^3$</TEX> admitting a graph manifold Dehn surgery consisting of two Seifert-fibered spaces over the disk with two exceptional fibers of indexes a, b, and c, d, respectively.

On the holomorphic torus-Bott tower of aspherical manifolds

We introduce a notion of holomorphic torus-Bott tower which is an iterated holomorphic Seifert fiber space with fiber a complex torus. This is thought of as a holomorphic version of a real Bott tower. The top space of the holomorphic torus-Bott tower is called a holomorphic torus-Bott manifold. We discuss the structure of holomorphic torus-Bott manifolds and particularly the holomorphic rigidity of holomorphic torus-Bott manifolds.

Embedding Seifert manifolds in 𝑆⁴

Using an obstruction based on Donaldson’s theorem on the intersection forms of definite 4-manifolds, we determine which connected sums of lens spaces smoothly embed in S 4 S^4 . We also find constraints on the Seifert invariants of Seifert 3-manifolds which embed in S 4 S^4 when either the base orbifold is non-orientable or the first Betti number is odd. In addition, we construct some new embeddings and use these, along with the d d and μ ¯ \overline {\mu } invariants, to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds.

Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

A Seifert surgery is a pair $$(K, m)$$ of a knot $$K$$ in $$S^3$$ 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 $$m \ne -2$$ it contains a successive triple of Seifert surgeries $$(K, m)$$ , $$(K, m +1)$$ , $$(K, m +2)$$ on a hyperbolic knot $$K$$ , e.g. $$17$$ -, $$18$$ -, $$19$$ -surgeries on the $$(-2, 3, 7)$$ pretzel knot. It contains infinitely many Seifert surgeries on strongly invertible hyperbolic knots none of which arises from the primitive/Seifert-fibered construction, e.g. $$(-1)$$ -surgery on the $$(3, -3, -3)$$ pretzel knot.

Open-book decompositions of 𝕊5 and real singularities

In this article, we study the topology of the family of real analytic germs F : (ℂ3, 0) → (ℂ, 0) given by [Formula: see text] with p, q, r ∈ ℕ, p, q, r ≥ 2 and (p, q) = 1. Such a germ has an isolated singularity at 0 and gives rise to a Milnor fibration [Formula: see text]. Moreover, it is known that the link LF is a Seifert manifold and that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of 𝕊5 given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ℂ3.

Rectangular Seifert circles and arcs system

Rectangular diagrams of links are link diagrams in the plane ℝ2 such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. Cromwell and 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 ℝ2 into a rectangular diagram with at most c(D) + 2s(D) vertical segments, and that, if D is connected, at most 2c(D) + 2 - w(D) vertical segments, where w(D) is a certain non-negative integer. In order to obtain these results, we show that the system of Seifert circles and arcs substituting for crossings can be deformed by an ambient isotopy of ℝ2 so that Seifert circles are rectangles composed of two vertical line segments and two horizontal line segments and arcs are vertical line segments, and that we can obtain a single circle from a connected link diagram by smoothing operations at the crossings regardless of orientation.