Knotting trivial knots and resulting knot types

Kimihiko Motegi

Vol. 161 (1993), No. 2, 371–383

DOI: 10.2140/pjm.1993.161.371

Abstract

Let (V,K) be a pattern (i.e. V is a standardly embedded solid torus in oriented S³ and K is a knot in V ) and f an orientation preserving emdedding from V into S³ such that f(V ) is knotted.

In this paper answers to the following questions will be given depending upon whether the winding number of K₂ in V is zero or not.

1. Suppose that K₁ is unknotted and K₂ is knotted in S³. Can f(K₁) be ambient isotopic to f(K₂) in S³ for some embedding f : V ↪S³?
2. Suppose that K₁ and K₂ are both unknotted in S³. How are (V,K₁) and (V,K₂) related if f(K₁) is ambient isotopic to f(K₂) in S³ for some embedding f : V ↪S³?

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 24 July 1991

Revised: 7 September 1992

Published: 1 December 1993