Crossing tribes of tangles in a thickened surface

TL;DR

This paper introduces a classification of crossings in knot diagrams based on tribes, considering their component, order, and homotopy types, and demonstrates the absence of nontrivial indices for classical knots.

Contribution

It extends the concept of tribes to tangles in fixed surfaces and analyzes their properties, linking crossings to homotopy types and component structure.

Findings

Crossings are classified by component, order, and homotopy type.

No nontrivial indices exist on classical knot diagrams.

Tribes provide a localized framework for Reidemeister move compatibility.

Abstract

Crossings of knot diagrams can be divided into classes (tribes) compatible with Reidemeister moves. Tribes can be considered as localization of the notion of weak chord index introduced by M. Xu. In the article we describe tribes of crossings for tangles in a fixed surface and show that crossings are differentiated by their component, order and homotopy types. As a consequence, we conclude that there are no nontrivial indices on diagrams of classical knots.