Double branched covers of theta-curves

TL;DR

This paper establishes a criterion for the primality of certain theta-curves in the 3-sphere based on their double branched covers, and applies it to a specific example, Kinoshita's theta-curve.

Contribution

It proves a folklore theorem relating the primality of theta-curves with their double branched covers, providing a new tool for analyzing theta-curve primality.

Findings

Primality of theta-curves can be characterized via their double branched covers.

The criterion is applied successfully to Kinoshita's theta-curve.

The result links knot theory and 3-manifold topology through branched covers.

Abstract

We prove a folklore theorem of W. Thurston which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot U is prime if and only if lifting the third arc of the theta-curve to the double branched cover over U produces a prime knot. We apply this result to Kinoshita's theta-curve.