arXiv research

Classifies changes between genus one fibered knots.

The study establishes a link between the complexity of fibered knots and the genus of their Heegaard splittings.

Every lens space has a simple knot with specific properties.

Study shows surgeries on certain knots yield left-orderable 3-manifolds.

We prove new results about unknotting fibered positive knots and braids.

The braid axis of a closed 3-braid lifts to a genus one fibered knot in the double cover of S^3 branched over the closed braid. Every (null homologous) genus one fibered knot in a 3-manifold may be obtained in this way. Using this perspective we answer a question of Morimoto about the number of genus one fibered knots …

Study on fibered knots in 3-manifolds, proving unrelated volume and genus.

Study surgery obstructions for Seifert fibered homology spheres using knot and Heegaard Floer homology.

Knots can only be fibered if each component is fibered and their genus is at least the connected sum.

We create Lefschetz fibrations for knot traces of specific types of knots.

It is shown, using sutured manifold theory, that if there are any 2-component counterexamples to the Generalized Property R Conjecture, then any knot of least genus among components of such counterexamples is not a fibered knot. The general question of what fibered knots might appear as a component of such a counterexa…

Classifies certain 3D knots with specific properties.

The study computes invariants of satellite knots using bordered Floer homology.

Classifies knots in the Poincaré sphere, using fixed points and folding automata.

Sharp knots and iterated cables lead to ribbon knots or failure of slice-ribbon conjecture.

New research finds infinitely many hyperbolic knots not almost-fibered.

The study identifies GOF-knots in 3-manifolds with reducible genus two Heegaard splittings.

The study explores how surface diffeomorphisms of knots relate to their topological properties.

We determine the genus one fibered knots in lens spaces that have tunnel number one. We also show that every tunnel number one, once-punctured torus bundle is the result of Dehn filling a component of the Whitehead link in the 3-sphere.

We prove the nugatory crossing conjecture for fibered knots. We also show that if a knot $K$ is $n$-adjacent to a fibered knot $K'$, for some $n>1$, then either the genus of $K$ is larger than that of $K'$ or $K$ is isotopic to $K'$.

Study on reducing surgeries on knots, developing thickness and genus bounds.

We determine the structure of the circular handle decompositions of the family of free genus one knots. Namely, if k is a free genus one knot, then the handle number h(k)= 0, 1 or 2, and, if k is not fibered (that is, if h(k)>0), then k is almost fibered. For this, we develop practical techniques to construct circular …

Supose that $Y$ is a lens space with $|H_1(Y; \mathbb{Z})|$ prime, and $Y$ does not contain a genus one fibered knot. We show that $Y$ contains a knot whose exterior is a once-punctured torus bundle if and only if $Y$ is the result of $p/q$-surgery on the trefoil. This partially answers a question posed by Ken Baker in…

Bounds on knot polynomials for Lie superalgebras of type I.

The twisted torus knots lie on the standard genus 2 Heegaard surface for $S^3$, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted tor…

Paper refines generating function for 2-bridge knot groups.

We prove relative versions of the symplectic capping theorem and sufficiency of Giroux's criterion for Stein fillability and use these to study the 4-genus of knots.

We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the manifolds so obtained, we then examine whether they are of Heegaard genus two or not. As…

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 on knot behavior under twisting, linking winding numbers, and braid axes.

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

Using the mapping cone of a rational surgery, we give several obstructions for Seifert fibered surgeries, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.

Paper analyzes double twist knots using adjoint hyperbolic torsion polynomial.

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

Study of knots with generalized Mazur patterns and their invariants.

Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.

This paper shows that only finitely many knots can be ribbon concordant to any given knot.

In this article we construct a family of knot surgery $4$-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface $E(2)$ using connected sums of fibered knots obtained by Stallings twist from a…

A simple method constructs Lefschetz fibrations on compact Stein surfaces.

A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexa…

Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then pi(K) splits over every free group of rank at …

We find explicit models for the PSL(2,C)- and SL(2,C)-character varieties of the fundamental groups of complements in S^3 of an infinite family of two-bridge knots that contains the twist knots. We compute the genus of the components of these character varieties, and deduce upper bounds on the degree of the associated …

We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable knots which bound a properly embedded complex curve in the four-ball satisfying a ge…

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as …

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Gene…

By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two families of Berge knots. By way of tangle descriptions we also obtain surgery de…

Paper conjectures Links-Gould invariant generalizes Alexander polynomial.

New method constructs Lefschetz fibrations with different regular fibers.