Search arXivSearch

arXiv · 1902.01719

Abstractly Planar Spatial Graphs

Abstract

This is a survey article for the forthcoming `A Concise Encyclopedia of Knot Theory.' We focus on the topology of spatial graphs with few vertices and edges, paying particular attention to Brunnian $θ$-graphs.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Scott A. Taylor. 2019-02-05. Abstractly Planar Spatial Graphs. https://arxiv.org/abs/1902.01719

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

The Burau representation of the braid group is faithful for n = 4

In this paper we use ideas introduced earlier by Moody, Long, Long-Paton, and Bigelow to prove the theorem of the title, that the Burau representation of the classical braid group is faithful for n = 4. An immediate corollary is that the Jones representation of the braid group is also faithful for n = 4.

math.GT

The Lorenz braid index and hyperbolic volume

A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

math.GT