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A three-variable transcendental invariant of planar knotoids via Gauss diagrams

Wandi Feng, Fengling Li, Andrei Vesnin

TL;DR

The paper introduces $H_D(t,y,z)$, a three-variable transcendental invariant for planar knotoids defined from a Gauss-diagram index function $\operatorname{Ind}_c^n(z)$, and proves its invariance under oriented Reidemeister moves. It shows $H_D$ is a Vassiliev invariant of order one and uses it to derive lower bounds on the Gordian distance between planar knotoids via a crossing-change analysis. The authors develop the behavior of $H_D$ under inversion and mirror, relate it to zero-height diagrams, and demonstrate its discriminative power through explicit computations, including irreversibility and singular-crossing cases. This invariant provides a new, computable tool for distinguishing planar knotoids and for quantifying their distance in the crossing-change metric, with potential applications in knot theory and related diagrammatic frameworks.

Abstract

As a generalization of the classical knots, knotoids are equivalence classes of immersions of the oriented unit interval in a surface. In recent years, a variety of invariants of spherical and planar knotoids have been constructed as extensions of invariants of classical and virtual knots. In this paper we introduce a three-variable transcendental invariant of planar knotoids which is defined over an index function of a Gauss diagram. We describe properties of this invariant and show that it is a Vassiliev invariant of order one. We also discuss the Gordian distance between planar knotoids and provide lower bounds on the Gordian distance of homotopic planar knotoids by using the transcendental invariant.

A three-variable transcendental invariant of planar knotoids via Gauss diagrams

TL;DR

The paper introduces , a three-variable transcendental invariant for planar knotoids defined from a Gauss-diagram index function , and proves its invariance under oriented Reidemeister moves. It shows is a Vassiliev invariant of order one and uses it to derive lower bounds on the Gordian distance between planar knotoids via a crossing-change analysis. The authors develop the behavior of under inversion and mirror, relate it to zero-height diagrams, and demonstrate its discriminative power through explicit computations, including irreversibility and singular-crossing cases. This invariant provides a new, computable tool for distinguishing planar knotoids and for quantifying their distance in the crossing-change metric, with potential applications in knot theory and related diagrammatic frameworks.

Abstract

As a generalization of the classical knots, knotoids are equivalence classes of immersions of the oriented unit interval in a surface. In recent years, a variety of invariants of spherical and planar knotoids have been constructed as extensions of invariants of classical and virtual knots. In this paper we introduce a three-variable transcendental invariant of planar knotoids which is defined over an index function of a Gauss diagram. We describe properties of this invariant and show that it is a Vassiliev invariant of order one. We also discuss the Gordian distance between planar knotoids and provide lower bounds on the Gordian distance of homotopic planar knotoids by using the transcendental invariant.
Paper Structure (13 sections, 9 theorems, 58 equations, 23 figures)

This paper contains 13 sections, 9 theorems, 58 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Knotoids and Reidemeister moves
  4. Index-type invariants of virtual knots and knotoids.
  5. Gauss diagram and $H_D (t, y, z)$ function.
  6. Singular planar knotoid diagrams.
  7. Crossing change and Gordian distance
  8. Proofs of theorems
  9. Proof of Theorem \ref{['thm-inv']}
  10. Proof of Theorem \ref{['thm-prop']}
  11. Proof of Theorem \ref{['thm-Vas']}
  12. Proof of Theorem \ref{['thm-dist']}
  13. Examples

Key Result

Theorem 1.1

Let $D$ be a planar knotoid diagram, then the function $H_D(t,y,z)$ is an invariant of $D$.

Figures (23)

  • Figure 1: The sign $\operatorname{sgn} (c)$ of a crossing $c$.
  • Figure 2: Reidemeister moves $\Omega_1$, $\Omega_2$ and $\Omega_3$.
  • Figure 3: Forbidden knotoid moves $\Phi_+$ and $\Phi_-$.
  • Figure 4: Crossings of diagram $D$ and its inverse $-D$.
  • Figure 5: Crossings of diagram $D$ and its mirror image $D^*$.
  • ...and 18 more figures

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 16 more