The Hexatangle

TL;DR

This paper classifies when Dehn surgery on certain closed pure 3-braids yields the 3-sphere, by analyzing a complex link configuration called the Hexatangle, generalizing previous work on the Pentangle.

Contribution

It explicitly determines the integral surgeries on a specific six-component link that produce the 3-sphere, extending the understanding of Dehn surgeries on pure 3-braids.

Findings

Identifies all integral surgeries on the link L that yield S^3.

Establishes the Hexatangle as a generalization of the Pentangle.

Provides explicit solutions for trivial knot fillings in the Hexatangle.

Abstract

We are interested in knowing what type of manifolds are obtained by doing Dehn surgery on closed pure 3-braids in the 3-sphere. In particular, we want to determine when we get the 3-sphere by surgery on such a link. We consider links which are small closed pure 3-braids; these are the closure of 3-braids of the form $({\sigma_1}^{2e_1})({\sigma_2}^{2f_1})(\sigma_2\sigma_1\sigma_2)^{2e}$, where $\sigma_1$, $\sigma_2$ are the generators of the 3-braid group and $e_1$, $f_1$, $e$ are integers. We study Dehn surgeries on these links, and determine exactly which ones admit an integral surgery producing the 3-sphere. This is equivalent to determining the surgeries of some type on a certain six component link $L$ that produce $S^3$. The link $L$ is strongly invertible and its exterior double branch covers a certain configuration of arcs and spheres, which we call the Hexatangle. Our problem is equivalent to determine which fillings of the spheres by integral tangles produce the trivial knot, which is what we explicitly solve. This hexatangle is a generalization of the Pentangle, which is studied by Gordon and Luecke.