These notes were prepared to supplement the talk that I gave on Feb 19, 2004, at the First East Asian School of Knots and Related Topics, Seoul, South Korea. In this article I review aspects of the interconnections between braids, knots and contact structures on Euclidean 3-space. I discuss my recent work with William Menasco (arXiv math.GT/0310279)} and (arXiv math.GT/0310280). In the latter we prove that there are distinct transversal knot types in contact 3-space having the same topological knot type and the same Bennequin invariant.
Withdrawn and replaced by two related manuscripts: (1) "Stabilization in the braid groups I:MTWS", published in Geometry and Topology Volume 10 (2006), 413-540, arXiv:math.GT/0310279, and (2) "Stabilization in the braid groups II: Transversal simplicity of knots", Geometry and Topology Volume 10 (2006), to appear, arXiv:math,GT/0310280.
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 transversal knots that are closed 3-braids. A list is given of all low crossing examples of transversally non-simple knots that are closed 3-braids.
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 notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are proved. Similar results for welded knots and braids are also proved.
We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Dehornoy ordering of the braid groups.
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 image.
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.
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.
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 addition, we explore unknotting sequences of positive braid knots and give a proof that there are only finitely many positive braid knots for a given unknotting number.
The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.
We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots. We give a counterexample to, and a corrected version of a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot…
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 genus difference less than any fixed constant is characterised by finitely many forbidden surface minors.
We prove the existence of a knot whose braid index the Morton-Franks-Williams inequality fails to detect but a related inequality (KR-MFW inequality), which uses new information of Khovanov-Rozansky homology, detects. We also prove, by examples, that there exists infinitely many knots for which the KR-MFW inequality fails to detect the braid indices.
This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be effective both when the evolution is carried out for individual knots and when a generic sequence of moves is evolved for a set of knots.
We show that a transverse link in a contact structure supported by an open book decomposition can be transversely braided. We also generalize Markov's theorem on when the closures of two braids represent (transversely) isotopic links.
Virtual knot theory has experienced a lot of nice features that did not appear in classical knot theory, e.g., parity and picture-valued invariants. In the present paper we use virtual knot theory effects to construct new representations of classical (pure) braids.
We give some conditions on positive braids with at least two full twists that ensure their closure is a hyperbolic knot, with applications to the geometric classification of T-links, arising from dynamics, and twisted torus knots.
In this paper, we construct invariants of braids, knots and links by studying dynamics of points in $\R^{2}$ and applying the Ptolemy relation $ac+bd=xy$.
We use Ozsváth, Stipsicz, and Szabó's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a positive full-twist realize the braid index of their closure. We also establish that quasi-positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure. Finally, we provide inductive formulas for the Upsilon invariant of torus knots and compare it to t…
A pair of surgeries on a knot is called purely cosmetic if the pair of resulting 3-manifolds are homeomorphic as oriented manifolds. Using recent work of Hanselman, we show that (nontrivial) knots which arise as the closure of a 3-stranded braid do not admit any purely cosmetic surgeries.
Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is a representation of knot (link) as closure of braid with n threads and length of this braid does not exceed n(4n-5)+2.
We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in terms of the structure of the braid groups. We also prove some results about knots modulo the nth derived subgroups of the pure braid groups, and about knots modulo braid subgroups in general.
We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsváth, Stipsicz, and Szabó. For the upper bound, we use a known bound for braid index $3$ and a new bound for braid index $4$. Both bounds coincide, so that we obtain a sharp result.
For a knot K, let b_n(K) be the minimum length of an n-stranded braid representative of K. Examples of knots exist for which b_n(K) is a non-increasing function. We investigate the behavior of b_n(K). We develop bounds on the function in terms of the genus of K, with stronger results for homogeneous knots and braid positive knots. For knots of nine or fewer crossings, we show that b_n(K) is an increasing function and determine it completely.
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum braids to study graph trees, amphicheirality, unknotting numbers, and periodic tables.
In this survey paper we present the $L$--moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3--manifolds and in handlebodies, as well as for virtual knots, for flat virtuals, for welded knots and for singular knots. The $L$--moves are local and they provide a uniform ground for formulating and proving braid equivalence theorems for any diagrammatic setting where the notion of braid and diagrammatic isotopy is defined, the statements…
We define the property (D) for nontrivial knots. We show that the fundamental group of the manifold obtained by Dehn surgery on a knot $K$ with property (D) with slope $\frac{p}{q}\ge 2g(K)-1$ is not left orderable. By making full use of the fixed point method, we prove that (1) nontrivial knots which are closures of positive $1$-bridge braids have property (D); (2) L-space satellite knots, with positive $1$-bridge braid patterns, and companion with property (D), have property (D); (3) the fundamental group of the manifold obtained by Dehn filling on $v2503$ is not left orderable. Additionally…
We study a certain type of braid closure which resembles the plat closure but has certain advantages; for example, it maps pure braids to knots. The main results of this note are a Markov-type theorem and a description of how Vassiliev invariants behave under this braid closure.
We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However, that approach was unable to explicitly identify the invariant of transverse links that naturally appears in braid diagrams. In this paper, we improve the previous approach in order to compute the transverse invariant. We define a new combinatorial complex that computes knot Floer homology and identify the BRAID invariant of transverse knots and links in the ho…
In this report, I will start by first giving a brief introduction on knots to build some intuition before beginning the more rigorous review in the Literature Review section. There, I will define knot equivalence, the Jones polynomial invariant, braid groups, Hecke algebras, and the HOMFLY polynomial invariant. Next, I will cover some of my independent research regarding the computation of a general formula for the HOMFLY polynomial of the closure of looped coxeter braids. Lastly, I will loosely discuss how this research connects to future research regarding the universal trace in the Plan for…
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the…
We prove that the so-called t algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial and the invariant of singular knots is an extension of an invariant of singular knots previously defined by S. Lambropoulou and the second author.
We estimate from above the set of knots, $Ω(n,μ)$, generated by closure of n-string 1+1- and 2+1-dimensional braids of irreducible length $μ$ ($μ>>1$) in the limit n>>1.
It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type. We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and L, then it is also true for the (p,q)-cable…
Given a knot in $S^3$, one can associate to it a surface diffeomorphism in two different ways. First, an arbitrary knot in $S^{3}$ can be represented by braids, which can be thought of as diffeomorphisms of punctured disks. Second, if the knot is fibered -- that is, if its complement fibers over $S^1$ -- one can consider the monodromy of the fibration. One can ask to what extent properties of these surface diffeomorphisms dictate topological properties of the corresponding knot. In this article we collect observations, conjectures, and questions addressing this, from both the braid perspective…
Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refine this result for (2,2k+1) cable knots and, as a simple corollary, compute the sl(2) Rasmussen invariants of (2,2k+1) cables of slice and amphicheiral knots.
We present a braid-theoretic approach to combinatorially computing knot Floer homology. To a knot or link K, which is braided about the standard disk open book decomposition for (S^3,ξ_std), we associate a corresponding multi-pointed nice Heegaard diagram. We then describe an explicit algorithm for computing the associated knot Floer homology groups. We compute explicit bounds for the computational complexity of our algorithm and demonstrate that, in many cases, it is significantly faster than the previous approach using grid diagrams.
