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
Summary
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]
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