Picture-valued parity-biquandle bracket II. Examples

TL;DR

This paper introduces new examples of knots and links with non-trivial parity-biquandle brackets, demonstrating that graph-based invariants can reveal more about link structures than traditional polynomial invariants.

Contribution

It provides the first explicit examples of non-trivial parity-biquandle brackets for knots and links, supporting the significance of graph-based invariants in knot theory.

Findings

Examples of knots and links with non-trivial parity-biquandle brackets.

Establishment of minimality theorem for these invariants.

Graphs as invariants of link diagrams instead of only polynomials.

Abstract

In [3] we constructed the parity-biquandle bracket valued in {\em pictures} (linear combinations of $4$-valent graphs). We gave no example of classical links such that the parity-biquandle bracket of which is not trivial. In the present paper we slightly change the notation of the parity-biquandle bracket and give examples of knots and links having a non-trivial parity-biquandle bracket. As a result we get the minimality theorem. This is the first evidence that graphs (link shadows) appear as invariants of link diagrams instead of just polynomials groups and other tractable objects.