On The Cost Function Associated With Legendrian Knots

TL;DR

The paper introduces the $Cost$ function on Legendrian knots to quantify the obstruction to turning a topological isotopy into a Legendrian isotopy, proving it defines a metric $d_C$ on the space of Legendrian representatives within a fixed topological knot type and yielding a graph structure $G_{\mathcal{K}}$. It provides lower bounds in terms of Thurston-Bennequin number $tb$ and rotation number $rot$, and fully characterizes Legendrian simple knots via $Cost$, giving a quantitative version of Fuchs-Tabachnikov's stabilization theorem. The authors compute $Cost$ for several knot families (torus, twist, 2-bridge) and analyze its behavior under connected sums, including exact formulas in certain destabilization contexts and implications for prime decompositions. They conclude with open questions about the interplay between $Cost$, the standard contact structure, and DGAs, and suggest computational avenues to build the Cost-graph for a given knot type, highlighting a new combinatorial framework for the Legendrian knot landscape.

Abstract

In this article, we introduce a non-negative integer-valued function that measures the obstruction for converting topological isotopy between two Legendrian knots into a Legendrian isotopy. We refer to this function as the Cost function. We show that the Cost function induces a metric on the set of topologically isotopic Legendrian knots. Hence, the set of topologically isotopic Legendrian knots can be seen as a graph with path-metric given by the Cost function. Legendrian simple knot types are shown to be characterized using the Cost function. We also get a quantitative version of Fuchs-Tabachnikov's Theorem that says any two Legendrian knots in $(\mathbb{S}^3,ξ_{std})$ in the same topological knot type become Legendrian isotopic after sufficiently many stabilizations. We compute the Cost function for Legendrian simple knots (for example torus knots) and we note the behavior of Cost function for twist knots and cables of torus knots (some of which are Legendrian non-simple). We also construct examples of Legendrian representatives of 2-bridge knots and compute the Cost between them. Further, we investigate the behavior of the Cost function under the connect sum operation. We conclude with some questions about the Cost function, its relation with the standard contact structure, and the topological knot type.

Figures (22)

• Figure 1: Converting a knot diagram into a front projection.
• Figure 2: The moves shown in this figure are called Legendrian Reidemeister (LR) moves. We will denote each move in the first row by LR1, each move in the second and third rows by LR2, and the move in the last row by LR3.
• Figure 3: Positive and a negative stabilization
• Figure 4: Connected sum of $F_1$ and $F_2$.
• Figure 5: Basic cases of front projections $F_0$ and $F_1$ connected by an R1 move.

Theorems & Definitions (70)

• Theorem 2.1: SJ
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Definition 3.1
• Theorem 3.2
• proof
• Remark 3.3
• Remark 3.4