Crossing indices, traits and the principle of indistinguishability

TL;DR

This paper introduces a universal index for knot diagrams that remains invariant under Reidemeister moves, leading to a principle of indistinguishability where crossings of the same sign cannot be distinguished by any inherent property.

Contribution

It establishes the concept of a universal index and demonstrates that crossings of the same sign in classical knots are inherently indistinguishable.

Findings

Universal index invariance under Reidemeister moves

Crossings of the same sign are indistinguishable in classical knots

Omission of the second Reidemeister move condition in the universal index

Abstract

A (weak chord) index is a function on the crossings of knot diagrams such that: 1) the index of a crossing does not change under Reidemeister moves; 2) crossings which can be paired by a second Reidemeister move have the same index. We show that one can omit the second condition in the case of the universal index. As a consequence, we get the following principle of indistinguishability for classical knots: crossings of the same sign in a classical knot diagram can not be distinguished by any inherent property.