arXiv research

We define the surface complex for $3$-manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex is…

Classifies generalized Seifert fiber spaces and their branched covers.

Study flippable Heegaard splittings in Seifert fibered spaces.

Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.

We give an algorithm to find vertical essential tori in small Seifert fiber spaces with infinite fundamental groups. This implies that there are algorithms to decide whether a 3-manifold is a Seifert fiber space.

Notes for a one semester course. The notes contain a description of compact three dimensional Seifert fibered spaces and a classification up to homeomorphism of compact three dimensional Seifert fibered spaces with non-empty boundary.

The study shows a way to represent Seifert fiber space groups with a limited field size.

Prism complexes help classify 3-manifolds, especially Seifert fiber spaces.

Study Kauffman bracket skein modules of Seifert fibered spaces.

Researchers found a way to measure the complexity of Seifert fibered spaces with boundaries.

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

Recognition of Seifert fibered spaces with boundary is computationally tractable.

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 tha…

We calculate the $L^2$-Alexander torsion for Seifert fiber spaces and graph manifolds in terms of the Thurston norm.

Study shows Seifert fibered spaces don't bound rational homology balls.

The paper classifies reversible elements in Seifert-fibered spaces and braid groups.

We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold…

We give a surgery formula for the asymptotic behavior of the sequence given by the logarithm of the higher dimensional Reidemeister torsion. Applying the resulting formula to Seifert fibered spaces, we show that the growth of the sequences has the same order as the indices and we give the explicit values for the limits…

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 exa…

The Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.

This paper constructs Seifert-fibered Dehn surgeries for hyperbolic tunnel-number-one knots.

We establish an inequality which gives strong restrictions on when the standard definite plumbing intersection lattice of a Seifert fibered space over $S^2$ can embed into a standard diagonal lattice, and give two applications. First, we answer a question of Neumann-Zagier on the relationship between Donaldson's theore…

The paper classifies tight contact structures on Seifert fiber spaces.

Study shows quotient group is infinitely generated.

We characterize L-spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e_0=-1 to have nonzero contact Ozsvath--Szabo invariants. This yields an algorith…

On small Seifert fibered spaces $M(e_0;r_1,r_2,r_3)$ with $e_0\neq-1,-2,$ all tight contact structures are Stein fillable. This is not the case for $e_0=-1$ or $-2$. However, for negative twisting structures it is expected that they are all symplectically fillable. Here, we characterize fillable structures among zero-t…

We show that if an orientable Seifert fibered space $M$ with an orientable genus $g$ base space admits a strongly irreducible horizontal Heegaard splitting then there is a one-to-one correspondence between isotopy classes of strongly irreducible horizontal Heegaard splittings and elements of $\mathbf{Z}^{2g}$. The corr…

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a na…

We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3-manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space M has infinitely many isotopy classes of Heegaard spli…

We explore certain restrictions on knots in the three-sphere which admit non-trivial Seifert fibered surgeries. These restrictions stem from the Heegaard Floer homology for Seifert fibered spaces, and hence they have consequences for both the Alexander polynomial of such knots, and also their knot Floer homology. In pa…

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…

Study $SL(2,\mathbb{C})$ connections on Seifert-fibered spaces using gauge theory.

A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifer…

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link in S^3. Our main tool is a careful use of the Aarhus integral and the (now proven) "Wheels" and "Wheeling" conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give …

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 th…

In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient flow lines between monopoles are identified with holomorphic data on an associated ruled surface, and…

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (-3,3,5)-pretzel…

It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds …

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 bo…

Using an obstruction based on Donaldson's theorem, we derive strong restrictions on when a Seifert fibered space $Y = F(e; \frac{p_1}{q_1}, \ldots, \frac{p_k}{q_k})$ over an orientable base surface $F$ can smoothly embed in $S^4$. This allows us to classify precisely when $Y$ smoothly embeds provided $e > k/2$, where $…

Conjecturally, the only knots in $S^3$ with non-integer surgeries producing Seifert fibered spaces are torus knots and cables of torus knots. In this paper, we make progress on the associated realization problem. Let $Y$ be a small Seifert fibered space arising by $p/q$-surgery on a knot in $S^3$, where $p/q$ is positi…

Study monodromy relations in Seifert fibered spaces for 3-manifold fillings.

Study finds all Brieskorn spheres with at most two fillable contact structures.

A special knot in the Poincaré sphere leads to a unique connected sum of lens spaces.

Study the spaces of Legendrian knots and links with maximal Thurston-Bennequin invariant.

Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.

Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.

We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbi…