New examples of Brunnian theta graphs

TL;DR

This paper introduces a new family of Brunnian theta graphs, expanding the known examples by linking them to a subgroup of the pure braid group, and proves their non-isotopy.

Contribution

It presents a novel family of Brunnian theta graphs parameterized by a subgroup of the pure braid group, demonstrating their infinite non-isotopic instances.

Findings

New family of Brunnian theta graphs constructed

Infinitely many non-isotopic examples proven

Connection established with pure braid group subgroup

Abstract

The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by a certain subgroup of the pure braid group on four strands. We prove that infinitely many of these diagrams give rise to non-isotopic Brunnian theta graphs.