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Let K be a nontrivial knot in S 3 and t ( K ) its tunnel number. For any ( p ≥ 2 , q ) -slope in the torus boundary of a closed regular neighborhood of K in S 3 , denoted by K ⋆ , it is a nontrivial cable knot in S 3 . Though t ( K ⋆ ) ≤ t ( K ) + 1 , Example 1.1 in Section 1 shows that in some case, t ( K ⋆ ) ≤ t ( K ) . So it is interesting to know when t ( K ⋆ ) = t ( K ) + 1 . After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot K ⋆ and its companion K, t ( K ⋆ ) ≥ t ( K ) ; (2) if either K admits a high distance Heegaard splitting or p / q is far away from a fixed subset in the Farey graph, then t ( K ⋆ ) = t ( K ) + 1 . Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large.

We study the knot invariant called trunk, as defined by Ozawa, and the relation of the trunk of a satellite knot with the trunk of its companion knot. Our first result is ${\rm trunk}(K) \geq n \cdot {\rm trunk}(J)$ where ${\rm trunk}(\cdot)$ denotes the trunk of a knot, $K$ is a satellite knot with companion $J$, and $n$ is the winding number of $K$. To upgrade winding number to wrapping number, which we denote by $m$, we must include an extra factor of $\frac{1}{2}$ in our second result ${\rm trunk}(K) > \frac{1}{2} m\cdot {\rm trunk}(J)$ since $m \geq n$. We also discuss generalizations of the second result., 21 pages, 5 figures

Recently Stoimenow showed that for every knot K and any n ∈ N and u 0 ⩾ u ( K ) there is a prime knot K n , u o which is n-equivalent to the knot K and has unknotting number u ( K n , u o ) equal to u 0 . The similar result has been obtained for the 4-ball genus g s of a knot. Stoimenow also proved that any admissible value of the Tristram–Levine signature σ ξ can be realized by a knot with the given Vassiliev invariants of bounded order. In this paper, we show that for every knot K with genus g ( K ) and any n ∈ N and m ⩾ g ( K ) there exists a prime knot L which is n-equivalent to K and has genus g ( L ) equal to m.

Let K be a knot in the 3-sphere S3, and D a disk in S3 meeting K transversely in the interior. For non-triviality we assume that |D∩K|⩾2 over all isotopies of K in S3−∂D. Let KD,n(⊂S3) be the knot obtained from K by n twisting along the disk D. If the original knot is unknotted in S3, we call KD,n a twisted unknot. We describe for which pairs (K,D) and integers n, the twisted unknot KD,n is a torus knot, a satellite knot or a hyperbolic knot. To cite this article: M. Ait Nouh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

For certain knots J in S 1 × D 2 , the dual knot J ∗ in S 1 × D 2 is defined. Let J ( O ) be the satellite knot of the unknot O with pattern J , and K be the satellite of J ( O ) with pattern J ∗ . The knot K then bounds a smooth disk in a 4-ball, but is not obviously a ribbon knot. We show that K is, in fact, ribbon. We also show that the connected sum J(O) # J ∗ (O) is a nonribbon knot for which all known algebraic obstructions to sliceness vanish.

The authors show that for every k⩾/0 there are knots intrinsically of depth k, i.e., whose complements admit smooth, taut foliations of depth k but do not admit C2, taut foliations of any depth less than k. They also give examples of knots whose complements do not admit taut, C2 foliations, with leaves meeting the boundary of the knot complement in circles, of any finite depth.