Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distinguishing Legendrian knots of topological type $7_4$, $9_{48}$ and $10_{136}$

Maxim Prasolov, Vladimir Shastin

Abstract

In a recent work of I. Dynnikov and M. Prasolov a new method of comparing Legendrian knots with nontrivial symmetry group is proposed. Using this method we confirm conjectures of Ng and Chongchitmate about Legendrian knots in topological types $7_4$, $9_{48}$ and $10_{136}$. This completes the classification of Legendrian types of rectangular diagrams of knots of complexity up to 9.

Distinguishing Legendrian knots of topological type $7_4$, $9_{48}$ and $10_{136}$

Abstract

In a recent work of I. Dynnikov and M. Prasolov a new method of comparing Legendrian knots with nontrivial symmetry group is proposed. Using this method we confirm conjectures of Ng and Chongchitmate about Legendrian knots in topological types , and . This completes the classification of Legendrian types of rectangular diagrams of knots of complexity up to 9.
Paper Structure (5 sections, 16 theorems, 25 equations, 15 figures)

This paper contains 5 sections, 16 theorems, 25 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Some notations, definitions and conventions
  3. Realizing the symmetry group by type II elementary moves
  4. Distinguishing
  5. Lemmas on morphisms

Key Result

Theorem 1

For any rectangular diagram $R$ the number of exchange classes $[R']$ such that $\mathscr L_{\pm} (R) = \mathscr L_{\pm} (R')$ is equal to the cardinality of the double coset $\mathrm{Sym}_- (R) \backslash \mathrm{Sym}(R) / \mathrm{Sym}_+(R)$.

Figures (15)

  • Figure 1: Exchange moves and (de)stabilizations of type I and II
  • Figure 2: A $\mathbb D^2$-move
  • Figure 3: A representation of a generator of the symmetry group in the topological type $7_4$ by a composition of two stabilizations and two destabilizations of type II
  • Figure 4: The action on the fundamental group of a homeomorphism representing $\eta^2$
  • Figure 5: A representation of a generator of the symmetry group in the topological type $9_{48}$ by a composition of a stabilization and a destabilization of type II
  • ...and 10 more figures

Theorems & Definitions (32)

  • Theorem 1: Corollary 4.9 of dypra202?
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Definition 5
  • Lemma 6
  • proof
  • Remark 1
  • Lemma 7
  • proof
  • ...and 22 more