arXiv research

Study positive 3-braids to compute Khovanov homology.

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

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…

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

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…

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

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

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…

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…

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

Study Agol cycles for pseudo-Anosov 3-braids.

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…

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.

Formula found for knot determinant in 3-braid weaving.

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

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

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

Study Agol cycles in pseudo-Anosov 3-braids.

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…

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

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

New link detection results using closures of 3-braids.

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 …

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

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

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…

Proves Khovanov homology conjecture for 3-stranded 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.

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

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 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…

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…

Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prov…

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 $(…

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 consider a conformal invariant of braids, the extremal length with totally real horizontal boundary values $λ_{tr}$. The invariant descends to an invariant of elements of $\mathcal{B}_n\diagup\mathcal{Z}_n$, the braid group modulo its center. We prove that the number of elements of $\mathcal{B}_3\diagup\mathcal{Z}_3…

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 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 study of random positive 3-strand braids reveals patterns in the roots of their Alexander polynomials.

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…

Investigates polynomial time algorithms for computing Khovanov homology of braids.

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots, algebraic alter…

We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from $3$-braid invariants and their application.

Discrete Morse theory simplifies Khovanov homology calculations.

In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a…

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.