Determinants of Simple Theta Curves and Symmetric Graphs

TL;DR

This paper introduces a new invariant called the determinant for theta curves, relating it to constituent knots in simple theta curves, and uses combinatorial methods to establish its properties.

Contribution

It defines the determinant of a theta curve via Klein covers and proves its multiplicative property for simple theta curves containing an unknot.

Findings

Determinant is an integer-valued invariant from the first homology of the Klein cover.

For simple theta curves with an unknot, the determinant equals the product of the determinants of the constituent knots.

Uses combinatorial proofs based on Kirchhoff's Matrix Tree Theorem and spanning tree enumeration.

Abstract

A theta curve is a spatial embedding of the $\theta$-graph in the three-sphere, 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.