Determinant of the OU matrix of a braid diagram

TL;DR

This paper introduces the OU matrix for braid diagrams, exploring its properties and invariants, and demonstrates how its determinant relates to the layered structure of braids.

Contribution

It defines the OU matrix for braid diagrams, analyzes its determinant, and introduces new invariants for positive braids based on this matrix.

Findings

Determinant of OU matrix reflects layeredness of braid diagrams.

Determinant of layered braid's OU matrix equals product of layer determinants.

New invariants for positive braids derived from OU matrix.

Abstract

In this paper, we define the OU matrix of a braid diagram and discuss how the OU matrix reflects the warping degree or the layeredness of the braid diagram, and show that the determinant of the OU matrix of a layered braid diagram is the product of the determinants of the layers. We also introduce invariants of positive braids which are derived from the OU matrix.