Determinant of the crossing matrix of a braid

Key Points

• This research aims to define a new braid invariant called the purified determinant and explore its properties.
• Define the purified determinant of a braid based on the crossing matrix of a pure braid.
• Analyze the relationships between purified determinants of different braids.
• Use mathematical proofs to show the validity of the invariant for any pair of braids.
• The purified determinant is established as a valid braid invariant.
• It holds true for any pair of specified braids, confirming its robustness in braid theory.