Theta-Graph Tricolorability Invariants

ISEF Category: Mathematics

Subcategory: Geometry and Topology  ·  Difficulty: Advanced  ·  Setup: University Lab  ·  Time: Full Year

The Hook

Some knots look different to your eye, but math says they can still be the same. Spatial theta-graphs are even trickier, because three edges meet at two vertices, so regular knot tricks do not always separate them. A good invariant acts like a fingerprint. Your job is to build one that can tell apart graphs that standard tools miss.

What Is It?

A spatial theta-graph is a three-edge network drawn in three-dimensional space. Picture a bicycle frame made from three paths between the same two junctions. Two graphs can look similar in a drawing, but still be different when you study how they sit in space.

A tricolorability-style invariant checks whether the graph can be colored in a rule-based way that stays unchanged under allowed moves. Think of it like a puzzle score. If two graphs give different scores, then they cannot be the same up to the moves you care about. The "plus" version adds extra structure from knot Floer-style ideas, which gives you more ways to separate hard cases.

For a student project, the main goal is not to invent the whole theory from scratch. You can build a small Python tabulator, test known examples, and see whether your invariant separates pairs of small theta-graphs that other quick checks treat as equal. That makes the project part math, part computation, and part careful classification.

Why This Is a Good Topic

This is a strong science fair topic because you can ask exact yes-or-no questions, then test them with code and known graph data. You are not collecting noisy lab measurements, so your results can be clean and repeatable. The project connects to topology, graph theory, and computer science, and it gives you room to make a real contribution by testing examples other students may not have checked. You can also learn how mathematicians build invariants, compare algorithms, and verify claims with outside databases.

Research Questions

• How does the tricolorability+ invariant change across small spatial theta-graphs with different crossing structures?
• What is the effect of using different diagram simplification rules on the invariant value?
• Does the invariant distinguish pairs of theta-graphs that share the same basic tricolorability count?
• To what extent does your Python tabulator agree with KnotInfo entries for known examples?
• Which families of small theta-graphs are separated by the invariant, and which remain indistinguishable?
• What is the effect of adding or removing a single local twist on the invariant for a fixed theta-graph?

Basic Materials

• Laptop or desktop computer with Python installed.
• Python package manager, such as pip.
• Text editor or notebook environment, such as VS Code or Jupyter.
• Access to KnotInfo or a saved copy of relevant theta-graph data.
• Spreadsheet software for recording outputs and comparisons.
• Graph paper or digital drawing tool for sketching diagram conventions.
• Version control tool, such as Git, for tracking code changes.

Advanced Materials

• University computer access for larger tabulation runs.
• Symbolic algebra software, such as SageMath, for checking algebraic steps.
• Access to topology papers or preprints that define the invariant in detail.
• A database of spatial graph examples with crossing data.
• High-performance Python environment for batch testing many diagrams.
• LaTeX for writing clean proof notes and result tables.
• Shared research storage for code, logs, and exported tables.

Software & Tools

• Python: Runs the tabulator, stores graph encodings, and compares invariant values across examples.
• Jupyter Notebook: Lets you test code blocks, inspect outputs, and keep notes in one place.
• ImageJ: Helps if you need to measure or inspect diagram images before turning them into data.
• Git: Tracks code revisions, so you can see when a change altered your results.
• NumPy: Speeds up array and table operations for larger batches of graph computations.

Experiment Steps

1. Define the exact graph class you will study, including how you will encode theta-graphs from diagrams.
2. Choose one invariant formula or rule set, then write down the inputs it needs and the outputs it produces.
3. Build a small parser or data-entry method that turns a drawn graph into a machine-readable form.
4. Set up a comparison table with known examples so you can check whether your code matches published results.
5. Plan a test set of closely related theta-graphs that differ by one local move or one twist.
6. Decide how you will judge success, such as separation power, agreement with KnotInfo, or consistency across diagram choices.

Common Pitfalls

• Encoding the same theta-graph two different ways, which makes the tabulator seem inconsistent when the input is the real problem.
• Mixing up graph equivalence moves with ordinary drawing changes, which can make two equivalent diagrams look different in your code.
• Trusting a single output from one implementation run, which hides bugs in the invariant calculation.
• Comparing your results to KnotInfo without matching the same labeling convention, which leads to false mismatches.
• Testing only easy examples, which can make the invariant look stronger than it really is on small theta-graphs.

What Makes This Competitive

A strong version of this project goes beyond making code that works on a few examples. You would explain exactly which diagrams your invariant separates, which ones it does not, and why. Better still, you would test a carefully chosen family of graphs and compare your results with published tables using clear statistics or classification counts. If you can find a new pair that the invariant separates, or a pattern in when it succeeds, the project becomes much stronger.

Project Variations

• Test the same invariant on spatial handcuff graphs instead of theta-graphs to compare separation power.
• Compare the tricolorability+ invariant with a simpler coloring invariant to see how much extra information the plus version adds.
• Focus on one family of small theta-graphs and study how the invariant changes under a single crossing reversal.

Learn More

• KnotInfo: Search this database for knot and spatial graph tables to compare against your own computed examples.
• MIT OpenCourseWare: Search for topology or graph theory lecture notes that explain invariants and equivalence moves.
• arXiv: Search for preprints on spatial graphs, knot Floer homology, and graph invariants.
• Journal of Knot Theory and Its Ramifications: Search recent papers on theta-graphs, coloring invariants, and computational topology.
• PubMed: Not a main source for this topic, but useful only if you need general guidance on scientific writing or data analysis methods.