arXiv research

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

New link detection results using closures of 3-braids.

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

Study positive 3-braids to compute Khovanov homology.

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…

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

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

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

Proves Khovanov homology conjecture for 3-stranded braids.

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

Investigates polynomial time algorithms for computing Khovanov homology of braids.

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if $δ_n = σ_1 σ_2 \ldots σ_{n-1}$ is the dual …

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 $(…

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

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

Classifies torus bundles bounding 4-manifolds with rational homology.

Discrete Morse theory simplifies Khovanov homology calculations.

Let $β:=σ_1σ_2^{-1}$ be a braid in $B_3$, where $B_3$ is the braid group on 3 strings and $σ_1, σ_2$ are the standard Artin generators. We use Gauss diagram formulas to show that for each natural number $n$ not divisible by $3$ the knot which is represented by the closure of the braid $β^n$ is algebraically slice if an…

Study on Fox's trapezoidal conjecture for specific alternating links.

We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by Plamenevskaya for their closures, which are naturally transverse links in the standard contact $3$-sphere. For any $3$-braid…

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

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

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.

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…

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…

Compactifies stability space for $A_2$ category, introducing $q$-deformed rational numbers.

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

Algorithm confirms non-order-preserving braids, proving infinite family not order-preserving.

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.

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 …

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…

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 involutory quandles of certain links are not left-orderable.

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…

The study proves knots and certain links support taut foliations.

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…

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…

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…

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