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An Invariant of Virtual Trivalent Spatial Graphs

Evan Carr, Nancy Scherich, Sherilyn Tamagawa

TL;DR

The paper introduces a virtual Niebrzydowski algebra to color the planar complements of virtual $Y$-oriented trivalent spatial graphs, yielding an integer-valued invariant under virtual $Y$-oriented Reidemeister moves. It synthesizes virtual tribrackets with Niebrzydowski algebras, provides a corrected, fully defined set of axioms, and proves the main invariance result (Theorem 3.6) for colorings. It also develops a practical framework for partially defined products, solves welldefinedness issues with a new axiom, and supplies extensive computational data and data sets (Latin squares/cubes, truncated algebras) to support implementation and further study. The computational results reveal the landscape of small-order structures, highlight uniqueness phenomena at low orders, and raise intriguing open questions about higher-order algebras and partial-product patterns, with implications for knot and graph invariants in low-dimensional topology.

Abstract

We create an invariant of virtual Y-oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson and Pico, and Graves, Nelson, and the second author. We provide usable data sets of Latin Cubes and virtual Niebrzydowski algebras for computational implementation.

An Invariant of Virtual Trivalent Spatial Graphs

TL;DR

The paper introduces a virtual Niebrzydowski algebra to color the planar complements of virtual -oriented trivalent spatial graphs, yielding an integer-valued invariant under virtual -oriented Reidemeister moves. It synthesizes virtual tribrackets with Niebrzydowski algebras, provides a corrected, fully defined set of axioms, and proves the main invariance result (Theorem 3.6) for colorings. It also develops a practical framework for partially defined products, solves welldefinedness issues with a new axiom, and supplies extensive computational data and data sets (Latin squares/cubes, truncated algebras) to support implementation and further study. The computational results reveal the landscape of small-order structures, highlight uniqueness phenomena at low orders, and raise intriguing open questions about higher-order algebras and partial-product patterns, with implications for knot and graph invariants in low-dimensional topology.

Abstract

We create an invariant of virtual Y-oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson and Pico, and Graves, Nelson, and the second author. We provide usable data sets of Latin Cubes and virtual Niebrzydowski algebras for computational implementation.

Paper Structure

This paper contains 10 sections, 3 theorems, 10 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Virtual Y-oriented Reidemeister Moves
  3. From Tribrackets to Virtual Niebrzydowski Algebras
  4. Partial Products and Welldefinedness
  5. Computational Results
  6. Latin Squares and Cubes.
  7. Tribrackets
  8. Niebrzydowski Algebras
  9. Virtual Niebrzydowski Algebras
  10. Small mysteries for the interested reader.

Key Result

Theorem 3.5

Let $X$ be a virtual Niebrzydowski algebra and $\Gamma$ a virtual $Y$-oriented trivalent spatial graph. The number $\Phi_X(\Gamma)$ of $X$-colorings of the planar complement of $\Gamma$ is invariant under the virtual $Y$-oriented Reidemeister moves, and is therefore an integer-valued invariant of vi

Figures (7)

  • Figure 1: Two virtual Y-oriented trivalent spatial graphs.
  • Figure 2: A subset of the virtual Y-oriented Reidemeister moves
  • Figure 3: A graphical interpretation for coloring instructions of planar complements of diagrams with tribrackets, virtual tribrackets, and Niebrzydowski algebras.
  • Figure 4: Two virtual $Y$-oriented Reidemeister moves which describe the virtual Niebrzydowski algebra relations $vR5.1$, $vR5.2$, $vR5.3$, and $vR5.4$.
  • Figure 5: Two virtual Y-oriented trivalent spatial graphs $K_1$ and $K_2$. Three planar regions in the complement of $K_1$ have been labeled with $a,b$, and $c$. Five planar regions in the complement of $K_2$ have been labeled by $a,b,c,d$, and $e$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5
  • Definition 4.1
  • Corollary 4.2
  • Definition 4.3
  • Corollary 4.4
  • Example 4.5