Seifert Fibered Research Articles

An Elliptic Generalization of A1 Spherical DAHA at K = 2

Abstract We construct an algebra that is an elliptic generalization of $A_1$ spherical DAHA acting on its finite-dimensional module at $t=-q^{-K/2}$ with $K=2$. We prove that $PSL(2,\mathbb{Z})$ acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by $3 \times 3$ matrices with matrix elements given by products of elliptic functions. A relation of this construction to the $K$-theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to $SL(3,\mathbb{Z})$ symmetry of Felder–Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.

Every noncompact surface is a leaf of a minimal foliation

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3 -manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of noncompact surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely C^{2} -smoothable.

Tight contact structures on some families of small Seifert fiber spaces

Suppose K is a knot in a 3-manifold Y, and that Y admits a pair of distinct contact structures. Assume that K has Legendrian representatives in each of these contact structures, such that the corresponding Thurston-Bennequin framings are equivalent. This paper provides a method to prove that the contact structures resulting from Legendrian surgery along these two representatives remain distinct. Applying this method to the situation where the starting manifold is -Σ(2,3,6m+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\Sigma(2,3,6m+1)$$\\end{document} and the knot is a singular fiber, together with convex surface theory we can classify the tight contact structures on certain families of Seifert fiber spaces.

Torus knot filtered embedded contact homology of the tight contact 3‐sphere

AbstractKnot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3‐sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the knot filtered embedded contact homology, for odd and positive.

Quantum Modular $\widehat Z{}^G$-Invariants

We study the quantum modular properties of $\widehat Z{}^G$-invariants of closed three-manifolds. Higher depth quantum modular forms are expected to play a central role for general three-manifolds and gauge groups $G$. In particular, we conjecture that for plumbed three-manifolds whose plumbing graphs have $n$ junction nodes with definite signature and for rank $r$ gauge group $G$, that $\widehat Z{}^G$ is related to a quantum modular form of depth $nr$. We prove this for $G={\rm SU}(3)$ and for an infinite class of three-manifolds (weakly negative Seifert with three exceptional fibers). We also investigate the relation between the quantum modularity of $\widehat Z{}^G$-invariants of the same three-manifold with different gauge group $G$. We conjecture a recursive relation among the iterated Eichler integrals relevant for $\widehat Z{}^G$ with $G={\rm SU}(2)$ and ${\rm SU}(3)$, for negative Seifert manifolds with three exceptional fibers. This is reminiscent of the recursive structure among mock modular forms playing the role of Vafa-Witten invariants for ${\rm SU}(N)$. We prove the conjecture when the three-manifold is moreover an integral homological sphere.

Volume of Seifert representations for graph manifolds and their finite covers

AbstractFor any closed orientable 3‐manifold, there is a volume function defined on the space of all Seifert representations of the fundamental group. The maximum absolute value of this function agrees with the Seifert volume of the manifold due to Brooks and Goldman. For any Seifert representation of a graph manifold, the authors establish an effective formula for computing its volume, and obtain restrictions to the representation as analogous to the Milnor–Wood inequality (about transversely projective foliations on Seifert fiber spaces). It is shown that the Seifert volume of any graph manifold is a rational multiple of . Among all finite covers of a given nongeometric graph manifold, the supremum ratio of the Seifert volume over the covering degree can be a positive number, and can be infinite. Examples of both possibilities are discovered, and confirmed with the explicit values determined for the finite ones.

On the geometry of rod packings in the 3‐torus

AbstractRod packings in the 3‐torus encode information of some crystal structures in crystallography. They can be viewed as links in the 3‐torus, and tools from 3‐manifold geometry and topology can be used to study their complements. In this paper, we initiate the use of geometrisation to study such packings. We analyse the geometric structures of the complements of simple rod packings, and find families that are hyperbolic and Seifert fibred.

Modular Transformations of Homological Blocks for Seifert Fibered Homology 3-Spheres

Modular Transformations of Homological Blocks for Seifert Fibered Homology 3-Spheres

Bott‐integrable Reeb flows on 3‐manifolds

AbstractThis paper is devoted to studying a notion of Bott integrability for Reeb flows on contact 3‐manifolds. We show, in analogy with work of Fomenko–Zieschang on Hamiltonian flows in dimension 4, that Bott‐integrable Reeb flows exist precisely on graph manifolds. We also show that all ‐invariant contact structures on Seifert manifolds, as well as all contact structures on the 3‐sphere, on the 3‐torus and on , admit Bott‐integrable Reeb flows. Along the way, we establish some general Liouville‐type theorems for Bott‐integrable Reeb flows, and a number of topological constructions (connected sum, open books, Dehn surgery) that may be expected to have wider applications.

A generalized 4d Chern-Simons theory

A generalization of the 4d Chern-Simons theory action introduced by Costello and Yamazaki is presented. We apply general arguments from symplectic geometry concerning the Hamiltonian action of a symmetry group on the space of gauge connections defined on a 4d manifold and construct an action functional that is quadratic in the moment map associated to the group action. The generalization relies on the use of contact 1-forms defined on non-trivial circle bundles over Riemann surfaces and mimics closely the approach used by Beasley and Witten to reformulate conventional 3d Chern-Simons theories on Seifert manifolds. We also show that the path integral of the generalized theory associated to integrable field theories of the PCM type, takes the canonical form of a symplectic integral over a subspace of the space of gauge connections, turning it a potential candidate for using the method of non-Abelian localization. Alternatively, this new quadratic completion of the 4d Chern-Simons theory can also be deduced in an intuitive way from manipulations similar to those used in T-duality. Further details on how to recover the original 4d Chern-Simons theory data, from the point of view of the Hamiltonian formalism applied to the generalized theory, are included as well.

A lower bound on the average genus of a 2-bridge knot

Experimental data from Dunfield et al. using random grid diagrams suggest that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the average Seifert genus of a 2-bridge knot of a given crossing number.

Essential surfaces in Seifert fiber spaces with singular surfaces

Two-sided incompressible surfaces in Seifert fiber spaces with isolated singular fibers are well-understood. Frohman [3] and Rannard [6] have shown that one-sided incompressible surfaces in Seifert fiber spaces which have isolated singular fibers are either pseudo-horizontal or psuedo-vertical. We extend their result to characterise essential surfaces in Seifert fiber spaces which may have singular surfaces, i.e., in S1-foliated 3-manifolds which have fibered model neighbourhoods that are isomorphic to either a fibered solid torus or a fibered solid Klein bottle.

H′-splittings of Seifert Manifolds with Boundary

H′-splittings of Seifert Manifolds with Boundary

On mathcal{N} = 4 supersymmetry enhancements in three dimensions

We introduce a class of 3d theories consisting of strongly-coupled mathcal{N} = 4 systems coupled to mathcal{N} = 3 Chern-Simons gauge multiplets, which exhibit mathcal{N} = 4 enhancements when a peculiar condition on the Chern-Simons levels is met. An example is the SU(N)3 Chern-Simons theory coupled to the 3d TN theory, which enhances to mathcal{N} = 4 when 1/k1 + 1/k2 + 1/k3 = 0. We also show that some but not all of these mathcal{N} = 4 enhancements can be understood by considering M5-branes on a special class of Seifert manifolds. Our construction provides a large class of mathcal{N} = 4 theories which have not been studied previously.

Topology of Ambient 3-Manifolds of Non-Singular Flows with Twisted Saddle Orbit

In the present paper, nonsingular Morse – Smale flows on closed orientable 3-manifolds are considered under the assumption that among the periodic orbits of the flow there is only one saddle and that it is twisted. An exhaustive description of the topology of such manifolds is obtained. Namely, it is established that any manifold admitting such flows is either a lens space or a connected sum of a lens space with a projective space, or Seifert manifolds with a base homeomorphic to a sphere and three singular fibers. Since the latter are prime manifolds, the result obtained refutes the claim that, among prime manifolds, the flows considered admit only lens spaces.

Cosmological BF Theory on Topological Graph Manifold with Seifert Fibered Homology Spheres

Cosmological BF Theory on Topological Graph Manifold with Seifert Fibered Homology Spheres

BPS invariants for a Knot in Seifert manifolds

We calculate homological blocks for a knot in Seifert manifolds when the gauge group is SU(N). We obtain the homological blocks with a given representation of the gauge group from the expectation value of the Wilson loop operator by analytically continuing the Chern-Simons level. We also obtain homological blocks with the analytically continued level and representation for a knot in the Seifert integer homology spheres.

The quest of null electromagnetics knots from Seifert fibration

In this work we find new null electromagnetic fields that are exact solutions of Maxwell equations in vacuum and generalize the hopfion. The hopfion is an exact solution of Maxwell equations in vacuum in which all the field lines (both electric and magnetic) are topologically equivalent to closed and linked circles, forming a mathematical structure called Hopf fibration. Here we present a generalization to include other field lines topology, such as the Seifert fibration in which the field lines form linked torus knots. Included in this generalization are fields that ergodically fill torus surfaces.

On homogeneous quasipositive links

We discuss when homogeneous quasipositive links are positive. We show that a homogeneous diagram of a quasipositive link whose number of Seifert circles is equal to the braid index is a positive diagram, and show that the truth of Murasugi–Przytycki’s conjecture on braid index of alternating links implies that alternating quasipositive links are positive.

On the quotient of the homology cobordism group by Seifert spaces

We prove that the quotient of the integer homology cobordism group by the subgroup generated by the Seifert fibered spaces is infinitely generated.