arXiv research

New method constructs Seifert solids from bridge trisections.

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

Identifies Heegaard Floer homology solid tori via Dehn fillings.

Formula connects knot invariant to Lefschetz number, proving special case for Seifert solids.

Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $Φ\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of $Φ$ is an explicit Seifert hypersurface for $M$.

In a paper published in 2002, the author gave a criterion to determine whether there is a fiber-preserving branched covering between two given orientable Seifert manifolds with orientable bases. Here we supply some details of the proof of two claims in that paper. We give an explicit construction of fiber-preserving br…

This paper classifies curves in genus two handlebodies.

This paper gives a proof that the universal cover of a closed 3-manifold built from three $π_1$-injective handlebodies is homeomorphic to $\mathbb R^3$. This construction is an extension to handlebodies of the conditions for gluing of three solid tori to produce non-Haken Seifert fibered manifolds with infinite fundame…

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

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. T…

Study shows Heegaard genus relation in 3-manifold amalgamation.

Proves left-orderability of certain Dehn fillings on 3-manifolds.

Suppose $K$ is a hyperbolic knot in a solid torus $V$ intersecting a meridian disk $D$ twice. We will show that if $K$ is not the Whitehead knot and the frontier of a regular neighborhood of $K \cup D$ is incompressible in the knot exterior, then $K$ admits at most one exceptional surgery, which must be toroidal. Embed…

Under the relation of $0$-concordance, the set of knotted 2-spheres in $S^4$ forms a commutative monoid $\mathcal{M}_0$ with the operation of connected sum. Sunukjian has recently shown that $\mathcal{M}_0$ contains a submonoid isomorphic to $\mathbb{Z}^{\ge 0}$. In this note, we show that $\mathcal{M}_0$ contains a su…

This paper looks at a class of closed orientable 3-manifolds constructed from a gluing of three handlebodies, such that the inclusion of each handlebody is $π_1$-injective. This construction is the generalisation to handlebodies of the condition for gluing three solid tori to produce non-Haken Seifert fibered 3-manifol…

It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in the complementary solid tori of it, and second smooth the intersections of the boundary of meridian disks uniformly. Then we obtain a torus knot or link on the unknotted torus and its Seifer…

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 pair…

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 ot…

Proves NP and co-NP status for knot core recognition in solid torus.

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.

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…

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 o…

New theorem for 4D links simplifies characterisation problem.

We consider the solid angle that a planar compact subset subtends at a point in a level set of height h and study two extremal problems for the solid angle. One of the variables is a point in such a plane, that is, we study the properties of the solid angle maximizer. The other is the pair of a planar compact subset an…

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…

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

We classify the Seifert fibrations of any given lens space L(p,q). We give an algorithmic construction of a Seifert fibration of L(p,q) over the base orbifold S^2(m,n) with the coprime parts of m and n arbitrarily prescribed. This algorithm produces all possible Seifert fibrations, and the equivalences between the resu…

Classifies small links in an unmarked solid torus.

Extends Seifert algorithm to 3-manifolds via surgery.

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…

Dominant knots have isomorphic Seifert and Tait graphs.

The paper explores representations of graph manifolds to Seifert motion groups.

Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.

Study on Seifert fibered spherical 3-orbifolds, determining their unique fibrations.

Classifies generalized Seifert fiber spaces and their branched covers.

We show that in any triangulation of a solid torus, there is a pre-core curve that lies in the 2-skeleton and that intersects the interior of each face in at most 10 straight arcs. By definition, a pre-core curve is a simple closed curve that becomes a core curve when a collar is attached to the boundary of the solid t…

The paper classifies all tight contact structures on a solid torus.

Study Murasugi sum in 4D for knotted surfaces, defining arborescent surfaces.

New knots found with Seifert genus not matching minimal genus Seifert surfaces.

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…

In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $K\subset Y$ and a base point $p\in K$, we can associate the minus versions, $\underline{\rm KHM}^-(Y,K,p)$ and $\underline{\rm KHI}^-(Y,K,p)$, to the triple $(Y,K,p)$. We pr…

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 ar…

Paper constructs infinitely many pairs of Seifert surfaces for each link.

Geometrically reformulates Cosserat solid mechanics using differential geometry.

Researchers calculate the volume of Seifert representations for graph manifolds and their covers.

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 boun…

Method to create rational Seifert surfaces for knots in Lens space.

Corrects classification of Seifert fibrations for lens spaces with non-orientable bases.