Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stuck Knots: Rigidity, Invariants, and Unsticking Distance

Ioannis Diamantis

Abstract

A {\it stuck knot} is a knot diagram containing designated crossings, called {\it stuck crossings}, whose incident strands are required to remain locally non-separable. These rigidity constraints restrict the allowable ambient isotopies and introduce new geometric features into the study of knot embeddings. In this paper we develop a topological framework for knots governed by such constraints. We model stuck crossings as locally rigid configurations in spatial embeddings, placing stuck knots in close relation to rigid spatial graph theory while preserving the classical over-under information and orientation of crossings. We formalize the corresponding notion of isotopy and introduce the {\it unstick move}, which releases rigidity and allows classical simplifications to occur. To detect rigid structure algebraically, we construct polynomial invariants for stuck knots, including a HOMFLYPT-type invariant and a state-sum model extending the Kauffman bracket. These invariants show that rigidity contributes independent information even when the underlying classical knot type remains fixed. We further introduce a {\it relaxed isotopy} framework and define the {\it unsticking distance}, a geometric measure quantifying the minimal number of rigidity constraints that must be released in order to relate two stuck knots. This perspective interprets stuck crossings as barriers to isotopy and highlights the role of constraint release in diagrammatic simplification.

Stuck Knots: Rigidity, Invariants, and Unsticking Distance

Abstract

A {\it stuck knot} is a knot diagram containing designated crossings, called {\it stuck crossings}, whose incident strands are required to remain locally non-separable. These rigidity constraints restrict the allowable ambient isotopies and introduce new geometric features into the study of knot embeddings. In this paper we develop a topological framework for knots governed by such constraints. We model stuck crossings as locally rigid configurations in spatial embeddings, placing stuck knots in close relation to rigid spatial graph theory while preserving the classical over-under information and orientation of crossings. We formalize the corresponding notion of isotopy and introduce the {\it unstick move}, which releases rigidity and allows classical simplifications to occur. To detect rigid structure algebraically, we construct polynomial invariants for stuck knots, including a HOMFLYPT-type invariant and a state-sum model extending the Kauffman bracket. These invariants show that rigidity contributes independent information even when the underlying classical knot type remains fixed. We further introduce a {\it relaxed isotopy} framework and define the {\it unsticking distance}, a geometric measure quantifying the minimal number of rigidity constraints that must be released in order to relate two stuck knots. This perspective interprets stuck crossings as barriers to isotopy and highlights the role of constraint release in diagrammatic simplification.
Paper Structure (18 sections, 15 theorems, 27 equations, 13 figures)

This paper contains 18 sections, 15 theorems, 27 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Stuck Knots as Rigid-Height Embeddings
  3. Spatial Realization
  4. Stuck Isotopy
  5. The Unstick Move
  6. Rigidity Filtration
  7. A Skein Theoretic Invariant for Stuck Knots
  8. A State-Sum Invariant for Stuck Knots
  9. The State Model
  10. Properties of the Stuck Bracket
  11. Detection of Rigidity.
  12. Comparison with the HOMFLYPT-Type Invariant.
  13. Stuck Knots as Rigid Spatial Graphs
  14. Rigidity Release and Unsticking Distance
  15. Relaxed Isotopy
  16. ...and 3 more sections

Key Result

Theorem 2.7

Two stuck knot diagrams represent stuck isotopic spatial embeddings if and only if they are related by a finite sequence of the local moves illustrated in Figure fig:rigid_moves. These moves consist of:

Figures (13)

  • Figure 1: Left: a classical crossing. Right: a stuck crossing represented as a rigid vertex together with a distinguished region (marked $A$) determining the height convention.
  • Figure 2: Left: The trefoil knot containing both classical and stuck crossings. Right: The unknot with one stuck crossing.
  • Figure 3: A diagrammatic stuck knot together with a spatial realization obtained via the lifting construction.
  • Figure 4: Isotopy moves for classical and stuck crossings.
  • Figure 5: The unstick move removes the rigidity constraint at a crossing while preserving the over-under structure.
  • ...and 8 more figures

Theorems & Definitions (76)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7: The analogue of the Reidemeister theorem for stuck knots
  • proof
  • Remark 2.8
  • Definition 2.9
  • ...and 66 more