Distinguishing Power of 4-Legendrian Permutation Racks

TL;DR

This work analyzes 4-Legendrian racks as invariants for Legendrian knots, proving that permutation-based 4-Legendrian racks cannot distinguish Legendrian knots sharing the same topological type and classical invariants $tb$ and $rot$. The authors develop a purely algebraic, combinatorial approach, deriving a canonical form for relations in colorings by permutation racks and showing that the counting of colorings cannot differ for knots with identical classical data. Building on prior results for 4-Legendrian quandles and GL-racks, the paper situates the negative result within a broader landscape of Legendrian rack invariants and provides computational classifications for small rack orders. The findings answer open questions about the distinguishing power of 4-Legendrian racks in the negative for a wide class of examples, and they highlight intrinsic obstructions arising from the permutation structure $X_{\sigma}$ and its automorphism group. These insights contribute to understanding the limits of rack-based invariants in Legendrian knot theory and guide future exploration of more powerful (non-permutation) 4-Legendrian structures.

Abstract

We explore 4-Legendrian rack structures and the effectiveness of 4-Legendrian racks to distinguish Legendrian knots. We prove that permutation racks with a 4-Legendrain rack structure cannot distinguish sets of Legendrian knots with the same knot type, Thurston-Bennequin number, and rotation number.

Figures (3)

• Figure 1: The standard contact structure on $\mathbb{R}^3$, depicted as an assignment of a plane to each point.
• Figure 2: Relations imposed on $F(L)$ between arcs at negative and positive crossings.
• Figure 3: Legendrian left handed trefoil with arcs labeled.

Theorems & Definitions (47)

• Theorem 1.1: Theorem \ref{['perm4LRcannotdist']}
• Definition 2.1: Rack, Elhamdadi_Nelson_2015
• Example 2.2: Trivial quandle
• Example 2.3: Dihedral quandle
• Example 2.4: Alexander quandle
• Example 2.5: Conjugation quandle
• Example 2.6: Core quandle
• Example 2.7: Takasaki quandle
• Remark 2.8
• Example 2.9: Permutation rack