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.