# Trace Diagrams and Biquandle Brackets

## TL;DR

This paper introduces trace diagrams as a new method for computing biquandle brackets of knots and links, establishing algebraic conditions for moves and a skein relation, advancing knot invariant calculations.

## Contribution

It presents a novel approach using decorated trivalent graphs for biquandle brackets and identifies key algebraic conditions for diagram manipulations.

## Key findings

- Biquandle brackets satisfy a Homflypt-style skein relation for monochromatic crossings.
- Algebraic conditions are identified for moving strands over and under traces.
- A new stop condition for recursive expansion of trace diagrams is proposed.

## Abstract

We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving strands over and under traces and identify a new stop condition for the recursive expansion. In the case of monochromatic crossings we show that biquandle brackets satisfy a Homflypt-style skein relation and we identify algebraic conditions on the biquandle bracket coefficients to allow pass-through trace moves.