On handlebody-knot pairs which realize exteriors of knotted surfaces in $S^3$

TL;DR

This paper explores the relationship between handlebody-knot pairs and embedded surfaces in $S^3$, demonstrating properties of irreducibility and reducibility, and constructing specific bi-knotted surfaces with given handlebody-knot pairs.

Contribution

It establishes a connection between handlebody-knot pairs and embedded surfaces, proving irreducibility/reducibility properties and constructing bi-knotted surfaces with prescribed handlebody-knot pairs.

Findings

One handlebody-knot in the pair is irreducible, the other is reducible.

Construction of prime bi-knotted surfaces with specified handlebody-knot pairs.

The relation between surface embeddings and handlebody-knot theory is clarified.

Abstract

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in $S^3$ in the sense that their exterior components are pairwise homeomorphic. We show that for every handlebody-knot pair associated to a genus two "prime bi-knotted" surface, one is irreducible, and the other is reducible. Furthermore, for given two genus two handlebody-knots $H_1$ and $H_2$ satisfying certain conditions, we will construct a "prime bi-knotted" surface whose associated handlebody-knot pair coincides with $H_1$ and $H_2$.