One-Term Parity Bracket For Braids

TL;DR

This paper introduces a simple one-term invariant formula for free braids that embodies a principle where sufficiently complex knot diagrams are self-representative within their equivalence class.

Contribution

The paper presents a novel, simplified one-term invariant for free braids that captures the self-representative property of complex knot diagrams.

Findings

The invariant formula is easy to compute.

It demonstrates the self-representative principle in free braids.

Simplifies understanding of knot diagram equivalences.

Abstract

In previous papers, the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e., is a subdiagram of any other diagram equivalent to it. This principle is realized by diagram-valued invariants [ ] of knots such that [K]=K. It turns out that in the case of free braids, the same principle can be realized an unexpectedly easy way by a one-term invariant formula.