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Exploring the Varchenko Determinant of Partial Cubes

Winfried Hochstättler, Sophia Keip, Birol Yazici

TL;DR

This work generalizes the Varchenko determinant from complexes of oriented matroids (COMs) to partial cubes by defining a Varchenko matrix on partial cube vertices. It confirms a neat factorization for COMs in the form $\det(\mathfrak V)=\prod_{Y\in\mathcal L} \left(1-(\prod_{e\in z(Y)} x_e)^2\right)^{b_Y}$, but shows that this property need not hold for all partial cubes by analyzing forbidden pc-minors; notably, $Q_4^{--}(4)$ yields a non-factorizable determinant while $Q_4^{--}(1)$ remains factorizable, indicating that the factorization property is not equivalent to being the tope graph of a COM. The results motivate a broader structural characterization of partial cubes with well-behaved Varchenko determinants and raise questions about which graph-theoretic properties ensure factorization. The paper provides explicit computations and code scaffolding to facilitate further exploration of these determinants in the broader class of partial cubes.

Abstract

The Varchenko matrix is known to have a well-structured determinant for complexes of oriented matroids (COMs). COMs can be characterized as partial cubes that do not have certain forbidden pc-minors. In this work, we generalize the Varchenko matrix and its determinant to partial cubes. We identify examples of partial cubes whose Varchenko determinants lack a clean factorization, as well as those that exhibit such a structure. These findings open the door for further research into the properties and potential characterizations of partial cubes with well-behaved Varchenko determinants.

Exploring the Varchenko Determinant of Partial Cubes

TL;DR

This work generalizes the Varchenko determinant from complexes of oriented matroids (COMs) to partial cubes by defining a Varchenko matrix on partial cube vertices. It confirms a neat factorization for COMs in the form , but shows that this property need not hold for all partial cubes by analyzing forbidden pc-minors; notably, yields a non-factorizable determinant while remains factorizable, indicating that the factorization property is not equivalent to being the tope graph of a COM. The results motivate a broader structural characterization of partial cubes with well-behaved Varchenko determinants and raise questions about which graph-theoretic properties ensure factorization. The paper provides explicit computations and code scaffolding to facilitate further exploration of these determinants in the broader class of partial cubes.

Abstract

The Varchenko matrix is known to have a well-structured determinant for complexes of oriented matroids (COMs). COMs can be characterized as partial cubes that do not have certain forbidden pc-minors. In this work, we generalize the Varchenko matrix and its determinant to partial cubes. We identify examples of partial cubes whose Varchenko determinants lack a clean factorization, as well as those that exhibit such a structure. These findings open the door for further research into the properties and potential characterizations of partial cubes with well-behaved Varchenko determinants.

Paper Structure

This paper contains 5 sections, 4 theorems, 18 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Complexes of Oriented Matroids
  3. Partial Cubes
  4. The Varchenko Determinant
  5. Conclusion

Key Result

Theorem 2.1

Let $\mathcal{M} = (E,\mathcal{L})$ be a simple COM, $\mathcal{T}$ its tope set and $G = (V,E)$ its tope graph. $\mathcal{M}$ is uniquely detemined by $\mathcal{T}$ and up to reorientation by $G$.

Figures (6)

  • Figure 1: Example of a realizable COM induced by the intersection of four hyperplanes with an open convex set. The positive side of each hyperplane is indicated by an arrow. Each sign vector encodes whether the corresponding cell lies on the positive side, the negative side, or directly on the hyperplane.
  • Figure 2: Tope graph of the realizable COM from \ref{['fig:COM']}.
  • Figure 3: The color classes of the realizable COM in \ref{['fig:colorclasses']}
  • Figure 4: Pc-minors of the realizable COM from \ref{['fig:COM']}. In A) the graph is restricted to the left side of the green color class, in B) to the right side, and in C) the entire green color class is contracted.
  • Figure 5: Illustration of forbidden pc-minors: On the left, the vertex $v$ and its antipode $-v$ are highlighted within the hypercube $Q_4$. To obtain each forbidden pc-minor, $-v$ must be removed. Additionally, for $Q_4^{--}$, exactly one neighbor of $v$ is deleted. For $Q_4^{--}(1)$ and $Q_4^{--}(4)$, $v$ is removed along with 1 or 4 of its neighbors, respectively.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2: Complex of Oriented Matroids (COM)
  • Definition 2.3: Tope Graph of a COM
  • Theorem 2.1
  • Theorem 3.1
  • Definition 3.1: Djoković-Winkler-Relation djokovic1973distancewinkler1984isometric
  • Definition 3.2: Forbidden pc-minors
  • Theorem 3.2
  • Definition 4.1: Varchenko Matrix of a COM
  • Theorem 4.1
  • ...and 1 more