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Determinant of the OU matrix of a braid diagram

Ayaka Shimizu, Yoshiro Yaguchi

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.

Determinant of the OU matrix of a braid diagram

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.

Paper Structure

This paper contains 14 sections, 26 theorems, 28 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Over-under matrix
  3. Definition of the OU matrix
  4. Permutation-independent quantities
  5. Properties of OU matrix
  6. Proof of Theorem \ref{['thm-det-layer']}
  7. Warping degree and OU matrix
  8. Linear ordering problem
  9. Weaving braids
  10. Positive braids
  11. Invariants of positive braids
  12. Positive permutation braid
  13. Fundamental braid
  14. Positive pure braid

Key Result

Theorem 1

Let $B$ be a layered braid diagram with layers $B_1$, $B_2$. The determinant of the OU matrix of $B$ is the product of the determinants of $B_1$ and $B_2$, namely,

Figures (6)

  • Figure 1: A braid diagram.
  • Figure 2: A braid diagram $B$ and its OU matrix with $\mathbf{s} = (s_1, s_2 , s_3 , s_4 , s_5)$.
  • Figure 3: A layered diagram $B= B_1 \oplus B_2$ with the layers $B_1$ and $B_2$ of the sets of strands $S_1= \{ s_1, s_3 , s_4 \}$ and $S_2 = \{ s_2 , s_5 \}$. The determinants are $\mathrm{det}(B)=-6$, $\mathrm{det}(B_1)=3$ and $\mathrm{det}(B_2 )=-2$, and we have $\mathrm{det}(B)= \mathrm{det}(B_1) \mathrm{det}(B_2)$.
  • Figure 4: A braid diagram $B$ and the OU matrices $M(B_{\mathbf{s}})$ and $M(B_{\mathbf{s}'})$ for $\mathbf{s}=( s_1, s_2 , s_3 , s_4 )$ and $\mathbf{s}' = ( s_3 , s_2 , s_1 , s_4 )$.
  • Figure 5: $B_W(7,7)$.
  • ...and 1 more figures

Theorems & Definitions (45)

  • Example 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Example 2
  • Proposition 1
  • proof
  • Corollary 1
  • Corollary 2
  • ...and 35 more