I apologize in advance for my rudimentary knowledge of knot theory, but I've been trying to find out about the significance (if any) of taking a knot (particularly a torus knot), cutting it, and forming another knot using it. Some examples:

trefoil knot of a trefoil knot

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and another "double trefoil" knot

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Is there a generally accepted name for this? Is the resulting knot prime? Are there any other generalizations one can make about these?

Thank you for any information you have on this.

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    $\begingroup$ Knot knot is kyes. $\endgroup$ – Asaf Karagila Jan 18 '16 at 12:27
  • $\begingroup$ I should have seen that coming. $\endgroup$ – Rick Rothenberg Jan 18 '16 at 12:31
  • $\begingroup$ Well, only if you assume the law of excluded kmiddle. In intuitionistic knot theory, knot knot can be slightly weaker than a kyes (so I wouldn't use it to secure a repel line, for example). $\endgroup$ – Asaf Karagila Jan 18 '16 at 12:32
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    $\begingroup$ I think you're looking for 'satellite knots'. $\endgroup$ – HJRW Jan 18 '16 at 12:59
  • $\begingroup$ Besides satellite knots suggested by HJRW you probably want to look up framed knots as well. You can consider framing as the knots being made of orientable ribbon rather than a line. Framed knots are central to some topology topics, such as Kirby Calculus. $\endgroup$ – Michael Jan 18 '16 at 18:21

A satellite knot (or link) is more general that your construction. To make a satellite knot, you take a knot in a solid torus (not in a ball within the solid torus), and embed the solid torus so that it is knotted.

A doubled knot (or Whitehead double of a knot) is special type of satellite knot that is different from yours.

Your knots are cablings of a knot, where the knot in the solid torus is a torus knot following a curve on the boundary. The slope is usually parametrized $(p,q)$ and yours are $(2,n)$ cablings for some $n$. Nontrivial cablings are prime (see Burde and Zieschang, Knots, p. 93) so your knots are prime.

Thanks to Ryan Budney for pointing out that my initial definition of satellite know was more restricted than a more common one.

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    $\begingroup$ All satellite knots are prime.... except for the ones that are not! :) Connect-sums are satellite knots, and they are not prime. $\endgroup$ – Ryan Budney Jan 18 '16 at 22:36
  • $\begingroup$ @Ryan Budney: Yes, that's one definition, but I don't like that one. I said that the meridianal disk has to intersect the knot at least twice, which I think rules out connected sums and gives a $\pi_1$-injective torus. $\endgroup$ – Douglas Zare Jan 18 '16 at 23:05
  • $\begingroup$ You need to say it a bit more strongly than how you did it, Douglas, since you could take a messy embedding of the knot into the torus which is still isotopic to one that meets some disk in one or zero points. $\endgroup$ – Jim Conant Jan 18 '16 at 23:18
  • $\begingroup$ Oh, okay. But readers should be aware that your defnition is a special case of the ones you see in the original articles (Schubert) as well as most textbooks and research articles that use the notion. $\endgroup$ – Ryan Budney Jan 18 '16 at 23:23
  • $\begingroup$ @JimConant: By meridianal disk, I meant topological meridianal disk. $\endgroup$ – Douglas Zare Jan 18 '16 at 23:23

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