Closures of 1-tangles and annulus twists

TL;DR

This paper investigates the conditions under which closures of 1-tangles in unknotted solid tori result in the unknot, providing a classification of such closures and applications to annulus twists.

Contribution

It introduces a method to identify all nontrivial 1-tangles with two unknot closures and applies sutured manifold theory to analyze annulus twists and their effects on knots.

Findings

At most two closures of a nontrivial 1-tangle are the unknot.

Twisting an unknot around an incompressible annulus yields at most one unknotting twist.

The Krebes 1-tangle does not admit an unknot closure.

Abstract

A 1-tangle is a properly embedded arc $\psi$ in an unknotted solid torus $V$ in $S^3$. Attaching an arc $\phi$ in the complementary solid torus $W$ to its endpoints creates a knot $K(\phi)$ called the closure of $\psi$. We show that for a given nontrivial 1-tangle $\psi$ there exist at most two closures that are the unknot. We give a general method for producing nontrivial 1-tangles admitting two distinct closures and show that our construction accounts for all such examples. As an application, we show that if we twist an unknot $q \neq 0$ times around an unknotted sufficiently incompressible annulus intersecting it exactly once, then there is at most one $q$ such that the resulting knot is unknotted and, if there is such, then $q = \pm 1$. With additional work, we also show that the Krebes 1-tangle does not admit an unknot closure. Our key tools are the ``wrapping index'' which compares how two complementary 1-tangles $\phi_1$ and $\phi_2$ wrap around $W$, a theorem of the author's from sutured manifold theory, and theorems of Gabai and Scharlemann concerning band sums.