## The Hexatangle

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Armas-Sanabria, Lorena

Eudave-Munoz, Mario

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##### Description

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.

Comment: 31 pages, 9 figures

Comment: 31 pages, 9 figures

##### Keywords

Mathematics - Geometric Topology, 57M25