Abstract

Let K be a knot in the 3-sphere S 3 , and Δ a disk in S 3 meeting K transversely in the interior. For non-triviality we assume that |Δ∩K|≥2 over all isotopies of K in S 3 -∂Δ. Let K Δ,n (⊂S 3 ) be a knot obtained from K by n twistings along the disk Δ. If the original knot is unknotted in S 3 , we call K Δ,n a twisted knot. We describe for which pair (K,Δ) and an integer n, the twisted knot K Δ,n is a torus knot, a satellite knot or a hyperbolic knot.

Highlights

  • Let K be a knot in the 3-sphere S 3 and ∆ a disk in S 3 meeting K transversely in the interior

  • Let K∆,n (⊂ S 3 ) be a knot obtained from K by n twistings along the disk ∆, in other words, an image of K

  • If K is a trivial knot in S 3, we call (K, ∆) a twisting pair and call K∆,n a twisted knot, see Figure 1

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Summary

Introduction

Let K be a knot in the 3-sphere S 3 and ∆ a disk in S 3 meeting K transversely in the interior. The result is a consequence of Proposition 1.3 below, [19] and [2, Theorem 1.1]. If K∆,n is a satellite knot, by Proposition 1.3, it would be a cable of a torus knot This contradicts [2, Theorem 1.1] which asserts that K∆,n (|n| > 1) cannot be a graph knot. Let us give some examples of hyperbolic twisting pairs (K, ∆) such that K∆,1 is not a hyperbolic knot. Example 1 (Producing torus knots from hyperbolic pairs). In [5, Theorem 1.3], [29, p.2293], we find other examples of hyperbolic pairs (K, ∆) such that K∆,1 is a torus knot. The results of this paper has been announced in [1]

Twistings on non-hyperbolic twisting pairs
Twistings on hyperbolic twisting pairs
Topology of black disk faces supported by annuli
Topology of white primitive disk faces
Existence of white primitive disk faces
Existence of black disk faces supported by annuli
Oriented dual graphs
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