arXiv research

Positive braid knots have simple knot Floer homology.

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

Paper constructs infinitely many non-braid positive hyperbolic L-space knots.

The study connects twist positivity to L-space knots and concordance.

Positive knots are minimal in a specific knot ordering.

Study proves bridge and braid indices match for twist positive knots.

We find braid positive presentations for most L-space knots, except one, and explore related knot properties.

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splitability and the primeness of positive knots and links can be seen from their positive diagrams.

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we p…

Bridge positions of handlebody-knots are equivalent when stable.

Study identifies prime strongly positive amphicheiral knots with double symmetry.

New satellite knots found that can't be represented by positive braids with full twists.

Positive braids with at least two twists form hyperbolic knots.

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

The paper finds infinite knots with surfaces of any positive genus.

We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.

A knot $K_1$ is said to be Gordian adjacent to a knot $K_2$ if $K_1$ is an intermediate knot on an unknotting sequence of $K_2$. We extend previous results on Gordian adjacency by showing sufficient conditions for Gordian adjacency between classes of positive braid knots through manipulations of braid words. In additio…

Positive braids minimize knot untangling steps.

We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus for all positive braid knots with up to 12 crossings.

New upper bound on Jones polynomial for fibered positive links.

Study positive braid knots and their taut foliations, proving some L-space conjecture evidence.

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every…

Positive surgery on knots in weakly fillable 3-manifolds yields weakly fillable contact structures.

A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot…

Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.

Using the Fiedler-Polyak-Viro Gauss diagram formulas we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to t…

New examples of knots with special bridge positions found.

We prove that positive knots and knots with braid index three in the 3-sphere satisfy the Property P conjecture.

Shows large unknotting number for simple knots.

Study Khovanov homology of positive links and L-space knots, finding vanishing conditions.

The paper calculates the Δ-unknotting number for positive pretzel knots.

Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of …

In answer to a question of Long, Flapan constructed an example of a prime strongly positive amphicheiral knot that is not slice. Long had proved that all such knots are algebraically slice. Here we show that the concordance group of algebraically slice knots contains an infinitely generated free subgroup that is genera…

We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.

Perturbing non-minimal bridge positions of a knot ensures similar behavior for its cable links.

We show that the difference between the Seifert genus and the topological 4-genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot. We deduce that among prime positive braid knots, the property of having such a g…

In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invar…

Let $\mathcal{L}$ be a knot with a fixed positive crossing and $\mathcal{L}_n$ the link obtained by replacing this crossing with $n$ positive twists. We prove that the knot Floer homology $\widehat{\text{HFK}}(\mathcal{L}_n)$ `stabilizes' as $n$ goes to infinity. This categorifies a similar stabilization phenomenon of …

The Links-Gould polynomial of alternating knots is shown to be log-concave and positive.

It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves…

The paper defines and computes a knot complement invariant for simple links.

It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if the…

The study confirms that most positive 2-bridge knots up to 31 crossings do not have chirally cosmetic surgeries.

Positive braids linked to knot invariants and geometric monodromy groups.

New diagonal knots found with non-torus structure.

New proof for a knot type not admitting certain surgeries.

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphe…

In this paper, we give the trivializing number of all minimal diagrams of positive 2-bridge knots, and study the relation between the trivializing number and the unknotting number for a part of these knots.