On symmetry and exterior problems of knotted handlebodies

TL;DR

This paper investigates the symmetries and exterior properties of knotted handlebodies in 3-spheres, providing classifications and solutions for specific hyperbolic knot cases and generalizing known families.

Contribution

It classifies symmetries of genus two handlebodies from hyperbolic knots with non-integral toroidal surgeries and addresses the exterior problem for these cases.

Findings

Classified symmetries of certain knotted handlebodies.

Solved the exterior problem for hyperbolic knots with specific surgeries.

Generalized Lee-Lee family of knotted handlebodies.

Abstract

The paper concerns two classical problems in knot theory pertaining to knot symmetry and knot exteriors. In the context of a knotted handlebody $V$ in a $3$-sphere $S^3$, the symmetry problem seeks to classify the mapping class group of the pair $(S^3,V)$, whereas the exterior problem examines to what extent the exterior $E(V)$ determines or fails to determine the isotopy type of $V$. The paper determines the symmetries of knotted genus two handlebodies arising from hyperbolic knots with non-integral toroidal Dehn surgeries, and solve the knot exterior problem for them. A new interpretation and generalization of a Lee-Lee family of knotted handlebodies is provided.