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Generalised Legendrian racks of Legendrian links

Biswadeep Karmakar, Deepanshi Saraf, Mahender Singh

TL;DR

The paper introduces generalised Legendrian racks (GL-racks) as rack-based algebraic objects equipped with a Legendrian structure to encode front-diagram cusps and Legendrian Reidemeister moves. It builds a purely algebraic invariant, the generalized Legendrian rack $GLR(K)$, from front diagrams of oriented Legendrian links and proves invariance under Legendrian isotopy, with applications distinguishing infinitely many Legendrian unknots and trefoils. Beyond invariants, it develops the algebraic theory: every GL-rack admits a homogeneous representation, defines modules over GL-racks via trunks, and proves an equivalence between the category of GL-rack modules and Beck modules over a fixed GL-rack, thereby laying groundwork for (co)homology theories in this setting. Together, these results provide a robust, purely rack-theoretic framework for studying Legendrian links, connecting to quandle theory and offering new tools for distinguishing Legendrian links while suggesting a categorical approach to potential homology theories.

Abstract

A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of maps encoding the information of Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian link. Employing a purely rack theoretic approach, we associate a generalised Legendrian rack (or a GL-rack) to an oriented Legendrian link, and prove that it is an invariant under Legendrian isotopy. As immediate applications, we prove that this invariant distinguishes infinitely many oriented Legendrian unknots and oriented Legendrian trefoils. To comprehend their algebraic structure, we prove that every GL-rack admits a homogeneous representation. Further, using the idea of trunks, we define modules over GL-racks, and prove the equivalence of the category of GL-rack modules and the category of Beck modules over a fixed GL-rack.

Generalised Legendrian racks of Legendrian links

TL;DR

The paper introduces generalised Legendrian racks (GL-racks) as rack-based algebraic objects equipped with a Legendrian structure to encode front-diagram cusps and Legendrian Reidemeister moves. It builds a purely algebraic invariant, the generalized Legendrian rack , from front diagrams of oriented Legendrian links and proves invariance under Legendrian isotopy, with applications distinguishing infinitely many Legendrian unknots and trefoils. Beyond invariants, it develops the algebraic theory: every GL-rack admits a homogeneous representation, defines modules over GL-racks via trunks, and proves an equivalence between the category of GL-rack modules and Beck modules over a fixed GL-rack, thereby laying groundwork for (co)homology theories in this setting. Together, these results provide a robust, purely rack-theoretic framework for studying Legendrian links, connecting to quandle theory and offering new tools for distinguishing Legendrian links while suggesting a categorical approach to potential homology theories.

Abstract

A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of maps encoding the information of Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian link. Employing a purely rack theoretic approach, we associate a generalised Legendrian rack (or a GL-rack) to an oriented Legendrian link, and prove that it is an invariant under Legendrian isotopy. As immediate applications, we prove that this invariant distinguishes infinitely many oriented Legendrian unknots and oriented Legendrian trefoils. To comprehend their algebraic structure, we prove that every GL-rack admits a homogeneous representation. Further, using the idea of trunks, we define modules over GL-racks, and prove the equivalence of the category of GL-rack modules and the category of Beck modules over a fixed GL-rack.
Paper Structure (7 sections, 18 theorems, 61 equations, 12 figures)

This paper contains 7 sections, 18 theorems, 61 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Contact structure and Legendrian links
  3. generalised Legendrian racks
  4. generalised Legendrian rack of a Legendrian link
  5. Homogeneous representation of a generalised Legendrian rack
  6. Modules over $\operatorname{GL}$-racks
  7. Category of modules over $\operatorname{GL}$-racks

Key Result

Theorem 2.6

Two Legendrian links $K$ and $K'$ are Legendrian equivalent if and only if any front diagram $D(K)$ of $K$ can be obtained from any front diagram $D(K')$ of $K'$ by a finite sequence of Legendrian Reidemeister moves and planar Legendrian isotopy.

Figures (12)

  • Figure 1: Standard contact structure on $\mathbb{R}^3$.
  • Figure 2: A Legendrian trefoil.
  • Figure 3: Legendrian Reidemeister moves.
  • Figure 4: Relations corresponding to crossings and cusps.
  • Figure 5: First LR1 move.
  • ...and 7 more figures

Theorems & Definitions (54)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 44 more