Rectangular diagrams of surfaces: representability

Bypasses for rectangular diagrams. Proof of Jones' conjecture and related questions

In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a minimal rectangular diagram maximizes the Thurston--Bennequin number for the corresponding Legendrian links. Jones' conjecture about the invariance of the algebraic number of intersections of a minimal braid representing a fixed link type is proved.

弱矩形图、多交叉数和弧指数

Weak rectangular diagrams, multi-crossing number, and arc index

本文通过引入弱矩形图的概念，推导出一个关于弧指数和多交叉数的不等式，并展示了如何将松散矩形图转换为常规矩形图以保持弧指数。

Rectangular knot diagrams classification with deep learning

In this article we discuss applications of neural networks to recognising knots and, in particular, to the unknotting problem. One of motivations for this study is to understand how neural networks work on the example of a problem for which rigorous mathematical algorithms for its solution are known. We represent knots by rectangular Dynnikov diagrams and apply neural networks to distinguish a given diagram class from the given finite families of topological types. The data presented to the program is generated by applying Dynnikov moves to initial samples. The significance of using these diag…

Rectangular diagrams of surfaces: distinguishing Legendrian knots

In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce moves that allow transition between diagrams representing isotopic surfaces. We also introduce more general combinatorial objects called mirror diagrams and various moves for them that can be used to transform presentations of isotopic surfaces to each other. The moves can be divided into two types so that, vaguely speaking, type~I moves commute with type~II …

On rectangular diagrams, Legendrian knots and transverse knots

A correspondence is studied by H. Matsuda between front projections of Legendrian links in the standard contact structure for 3-space and rectangular diagrams. In this paper, we introduce braided rectangular diagrams, and study a relationship with Legendrian links in the standard contact structure for 3-space. We show Alexander and Markov Theorems for Legendrian links in 3-space.

Rectangular Seifert circles and arcs system

Rectangular diagrams of links are link diagrams in the plane ${\mathbb R}^2$ such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. P. R. Cromwell and I. A. Dynnikov showed that rectangular diagrams of links are useful for deciding whether a given link is split or not, and whether a given knot is trivial or not. We show in this paper that an oriented link diagram $D$ with $c(D)$ crossings and $s(D)$ Seifert circles can be deformed by an ambient isotopy of ${\mathbb R}^2$ into a rectangular diagram w…

Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams

We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any knot.

Essential tori in link complements: impossibility of detecting the satellite structure by monotonic simplification

In a recent work "Arc-presentation of links: Monotonic simplification" Ivan Dynnikov showed that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible to simplify monotonically a rectangular diagram of a satellite knot or link into one where the satellite structure is seen? Here we give a negative answer to that question.

Rectangular superpolynomials for the figure-eight knot

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $λ$ of $R$ with extraordinary simple coefficients $D_{λ^{tr}}(r)\cdot D_λ(s)$ in front of the $Z$-factors. Somewhat miraculously, these coefficients are made from quantum dimensions of symmetric representations of the groups $SL(r)$ and $SL(s)$ and restrict summation to diagrams with no more than $s$ rows and $r$ columns. They possess a natural $β$-deformation to Macdonald dimensions and produces positive…

Colored HOMFLY polynomials of knots presented as double fat diagrams

Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first …

Climbing a Legendrian mountain range without Stabilization

We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specific Legendrian and transversal MTWS by enhancing the Legendrian mountain range for the (2,3)-cable of a (2,3)-torus knot provided by Etnyre and Honda, and showing that elementary negative flypes allow us to move toward maximal tb value without having to use Legendrian stabilizatio…