What question did this study set out to answer?

This research aims to define and calculate the determinant of theta curves, particularly when they are simple and include an unknot.

January 18, 2026

Determinants of Simple Theta Curves and Symmetric Graphs

Key Points

This research aims to define and calculate the determinant of theta curves, particularly when they are simple and include an unknot.
Defined the determinant of theta curves based on their first homology.
Used combinatorial proofs grounded in Kirchhoff’s Matrix Tree Theorem.
Analyzed spanning tree enumeration for symmetric, signed planar graphs.
Proved that the determinant of a simple theta curve is the product of the determinants of its constituent knots.
Established a mathematical framework for understanding theta curves in relation to symmetric graphs.

Abstract

A theta curve is a spatial embedding of the θ-graph in the threesphere, taken up to ambient isotopy. We define the determinant of a theta curve as an integer-valued invariant arising from the first homology of its Klein cover. When a theta curve is simple, containing a constituent unknot, we prove that the determinant of the theta curve is the product of the determinants of the constituent knots. Our proofs are combinatorial, relying on Kirchhoff’s Matrix Tree Theorem and spanning tree enumeration results for symmetric, signed, planar graphs.

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