Tangle Insertion Invariants for Pseudoknots, Singular Knots, and Rigid Vertex Spatial Graphs

TL;DR

This paper introduces a topological invariant schema for pseudoknots, singular knots, and rigid vertex spatial graphs by replacing unknown crossings with rational tangles, aiding in their classification.

Contribution

It presents a novel invariant schema that extends to pseudoknots, singular knots, and rigid vertex spatial graphs, unifying their analysis through rational tangle replacements.

Findings

Invariant schema effectively classifies pseudoknots and related structures.

Method generalizes to various types of spatial graphs.

Provides a new tool for topological analysis of complex knots.

Abstract

The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial graphs and singular knots, that is obtained by replacing unknown crossings or vertices by rational tangles.