arXiv research

Study finds infinite non-fibered twisted torus knots.

New knot homologies detect non-fibered knots, expanding on previous results.

We construct an infinite family of knots in rational homology spheres with irreducible, non-fibered complements, for which every non-longitudinal filling is an L-space.

New Legendrian bounds for non-fibered knots in 3-manifolds.

We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, no…

We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic…

We study the twisted Alexander polynomial $Δ_{K,ρ}$ of a knot $K$ associated to a non-abelian representation $ρ$ of the knot group into $SL_2(\BC)$. It is known for every knot $K$ that if $K$ is fibered, then for every non-abelian representation, $Δ_{K,ρ}$ is monic and has degree $4g(K)-2$ where $g(K)$ is the genus of …

We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander polynomial, there exist infinitely many non-fibered knots with the given Seifert matrix. …

A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the we…

Defines a new knot invariant and studies its properties.

We study knots in $\mathbb{S}^3$ obtained by the intersection of a minimal surface in $\mathbb{R}^4$ with a small 3-sphere centered at a branch point. We construct examples of new minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or …

In this paper we prove that if $M_K$ is the complement of a non-fibered twist knot $K$ in $\mathbb S^3$, then $M_K$ is not commensurable to a fibered knot complement in a $\mathbb Z/ 2 \mathbb Z$-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then pro…

The paper studies twisted Alexander polynomials for knot groups in various extensions.

It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of "knot adjacency", studied in the paper "Knot adjacency, genus and essential tori" by the authors, can be used to obtain…

Study non-fibered links' relation to tight contact structures.

Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. In our previous paper, we observed that for such a knot, higher-order Alexander invariants defined by Cochran, Harvey and Friedl are generally factorized into the part of the Magnus matrix and that of a certa…

Study knot invariants to deduce Hopf invariant and propose a slope conjecture.

New theorem connects handle-ribbon knots to slice derivatives.

The study proves knots and certain links support taut foliations.

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We conver…

We show that if M is a surface bundle over S^1 with fiber of genus 2, then for any integer n, M has a finite cover tilde(M) with b_1(tilde(M)) > n. A corollary is that M can be geometrized using only the `non-fiber' case of Thurston's Geometrization Theorem for Haken manifolds.

We classify the $5$-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 2 of 3) classifies those in which the linear isotropy representation is either irreducible or trivial. The $5$-dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while …

New flows represent Thurston norm ball faces, differing by veering mutations.

In this paper we show that given any 3-manifold N and any non-fibered class in H^1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. This is obtained by extending earlier work of the authors, together with results of Agol and Wise on separability of 3-manifold groups. …

New triangulations encode flows with vanishing polynomial.

In early 1930s Seifert and Threlfall classified up to conjugacy the finite subgroups of $\mathrm{SO}(4)$, this gives an algebraic classification of orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are Seifert fibered. The underlying topological space and singular set of non-fibered spherical 3…

Closed self-covering manifolds with abelian fundamental groups fiber over tori.

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects…

A polynomial invariant for veering triangulations helps in understanding 3-manifold fibers.

New symmetries found for scalar and vector ODEs of arbitrary dimensions.

Polynomially parametrize interesting knotted surfaces.

New 2-knots found with same knot group but different quandles.

New knot quandles distinguish ribbon knots with isomorphic groups.

Proved colored HOMFLY-PT polynomials for specific knots.

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …

The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transver…

Study concordance of alternating torus knots to L-space knots.

This paper studies how knots combine using Alexander Polynomials.

The paper classifies a special family of knots in lens spaces using knot Floer homology.

The study confirms conjectures about slopes of knots using knot Floer homology.

Defines slice depth for 2-knots and sets upper bounds for specific knots.

New diagonal knots found with non-torus structure.

New hyperbolic knots not concordant to algebraic ones found.

Formula for Alexander polynomial of twisted torus knots derived.

A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecan…

Expanded Legendrian knot atlas for 10-arc index knots.

The paper conjectures Khovanov homology can distinguish torus and twist knots.