arXiv research

Ozsváth and Szabó conjectured that knot Floer homology detects fibred links. We will verify this conjecture for closed 3-braids, by classifying fibred closed 3-braids. In particular, given a nontrivial closed 3-braid, either it is fibred, or it differs from a fibred link by a half twist. The proof uses Gabai's method o…

The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of knots that are closed braids is shown to lead to a classification theorem for tran…

We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid wo…

Formula found for knot determinant in 3-braid weaving.

Paper computes a specific term of knot homology for 3-braids.

We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation…

A crossing in a knot is nugatory if changing the crossing does not change the knot type. Using an invariant of certain types of closed 3-braid diagrams, we show that if a closed 3-braid contains a nugatory crossing then its braid index is one or two. This proves a special case of a conjecture on nugatory crossings due …

We classify closed 3-braids which are L-space knots.

The paper finds Artin presentations for the trivial group and identifies hyperbolic 3-braids.

In this paper, we show that the volumes for a family of A-adequate closed braids can be bounded above and below in terms of the twist number, the number of braid strings, and a quantity that can be read from the combinatorics of a given closed braid diagram. We also show that the volumes for many of these closed braids…

A knot (or link) diagram is said to be everywhere equivalent if all the diagrams obtained by switching one crossing represent the same knot (or link). We classify such diagrams of a closed 3-braid.

Using the band representation of the 3-strand braid group, it is shown that the genus of 3-braid links can be read off their skein polynomial. Some applications are given, in particular a simple proof of Morton's conjectured inequality and a condition to decide that some polynomials, like the one of 9_{49}, are not adm…

Study positive 3-braids to compute Khovanov homology.

Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate in…

We prove that if an alternating 3-braid knot has unknotting number one, then there must exist an unknotting crossing in any alternating diagram of it, and we enumerate such knots. The argument combines the obstruction to unknotting number one developed by Ozsváth and Szabó using Heegaard Floer homology, together with o…

We are interested in knowing what type of manifolds are obtained by doing Dehn surgery on closed pure 3-braids in the 3-sphere. In particular, we want to determine when we get the 3-sphere by surgery on such a link. We consider links which are small closed pure 3-braids; these are the closure of 3-braids of the form $(…

We classify 3-braids up to (2,2)-move equivalence and in particular we show how to adjust the Harikae-Nakanishi-Uchida conjecture so it holds for closed 3-braids. As important steps to classify 3-braids up to (2,2)-move equivalence we prove the conjecture for 2-algebraic links and classify (2,2)-equivalence classes for…

Classifies 3-braid knots with maximal 4-genus using McCoy's method.

Study on knot classification using 3-braid closures and ribbon surfaces.

The paper calculates a knot invariant for 3-braid knots.

Study Agol cycles for pseudo-Anosov 3-braids.

We study 3-braid knots of finite smooth concordance order. A corollary of our main result is that a chiral 3-braid knot of finite concordance order is ribbon.

Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. The goal of this…

Classifies 3-braids from choreographic motions on Lissajous curves, linking them to mapping classes and geodesics.

Study Agol cycles in pseudo-Anosov 3-braids.

We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obt…

New link detection results using closures of 3-braids.

Study cobordism distances between 3-braid links and trefoil knots.

3-braid knots can't have purely cosmetic surgeries.

Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.

Proves Khovanov homology conjecture for 3-stranded braids.

We show that if a knot has a minimal spanning surface that admits certain Gabai disks, then this knot has Property P. As one of the applications we extend and simplify a recent result of Menasco and Zhang that closed 3-braid knots have Property P. Other applications are given.

The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not transversally simple. The method of proof is topological and indirect.

The study shows knots from 3-braids cannot be concordant to a specific Legendrian unknot.

The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which can be represented as closed 3-braids. Partial results are obtained about spann…

We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Applying the same perspective to the knot Floer invariant $Υ_K(t)$, we show that for a fixed concordance genus of $K$ there are only finitely many possibilities for $Υ_K(t)$. Fo…

New knot invariant from 3-braids and 6-valent graphs.

We show that the correction terms in Heegaard Floer homology give a lower bound to the the genus of one-sided Heegaard splittings and the $\mathbb Z_2$--Thurston norm. Using a result of Jaco--Rubinstein--Tillmann, this gives a lower bound to the complexity of certain closed $3$--manifolds. As an application, we compute…

Random walks on braid groups are transient, with specific closure properties for certain braids.

Let $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror ima…

We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how the Jones slopes and the linear term behave under cabling. As an application we ve…

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 …

In a recent paper, McMullen showed an inequality between the Thurston norm and the Alexander norm of a 3-manifold. This generalizes the well-known fact that twice the genus of a knot is bounded from below by the degree of the Alexander polynomial. We extend the Bennequin inequality for links to an inequality for all po…

Study links' arc index and Turaev genus, proving conjectures.

We examine the conjecture, due to Champanerkar, Kofman, and Purcell that $\text{vol}(K) < 2 π\log \det (K)$ for alternating hyperbolic links, where $\text{vol}(K) = \text{vol}(S^3\backslash K)$ is the hyperbolic volume and $\det(K)$ is the determinant of $K$. We prove that the conjecture holds for 2-bridge links, alter…

The paper classifies links based on signature and crossing number properties.

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…

Investigates polynomial time algorithms for computing Khovanov homology of braids.